Algorithms for calculating variance

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vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Naïve_algorithm" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Naïve_algorithm"> <div class="vector-toc-text"> 1 Naïve algorithm </div> </a> <button aria-controls="toc-Naïve_algorithm-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Naïve algorithm subsection </button> <ul id="toc-Naïve_algorithm-sublist" class="vector-toc-list"> <li id="toc-Computing_shifted_data" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computing_shifted_data"> <div class="vector-toc-text"> 1.1 Computing shifted data </div> </a> <ul id="toc-Computing_shifted_data-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Two-pass_algorithm" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Two-pass_algorithm"> <div class="vector-toc-text"> 2 Two-pass algorithm </div> </a> <ul id="toc-Two-pass_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Welford's_online_algorithm" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Welford's_online_algorithm"> <div class="vector-toc-text"> 3 Welford's online algorithm </div> </a> <ul id="toc-Welford's_online_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weighted_incremental_algorithm" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Weighted_incremental_algorithm"> <div class="vector-toc-text"> 4 Weighted incremental algorithm </div> </a> <ul id="toc-Weighted_incremental_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parallel_algorithm" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Parallel_algorithm"> <div class="vector-toc-text"> 5 Parallel algorithm </div> </a> <ul id="toc-Parallel_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> 6 Example </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher-order_statistics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher-order_statistics"> <div class="vector-toc-text"> 7 Higher-order statistics </div> </a> <ul id="toc-Higher-order_statistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Covariance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Covariance"> <div class="vector-toc-text"> 8 Covariance </div> </a> <button aria-controls="toc-Covariance-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Covariance subsection </button> <ul id="toc-Covariance-sublist" class="vector-toc-list"> <li id="toc-Naïve_algorithm_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Naïve_algorithm_2"> <div class="vector-toc-text"> 8.1 Naïve algorithm </div> </a> <ul id="toc-Naïve_algorithm_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-With_estimate_of_the_mean" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#With_estimate_of_the_mean"> <div class="vector-toc-text"> 8.2 With estimate of the mean </div> </a> <ul id="toc-With_estimate_of_the_mean-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Two-pass" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two-pass"> <div class="vector-toc-text"> 8.3 Two-pass </div> </a> <ul id="toc-Two-pass-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Online" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Online"> <div class="vector-toc-text"> 8.4 Online </div> </a> <ul id="toc-Online-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weighted_batched_version" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weighted_batched_version"> <div class="vector-toc-text"> 8.5 Weighted batched version </div> </a> <ul id="toc-Weighted_batched_version-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 9 See 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A key difficulty in the design of good <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> for this problem is that formulas for the <a href="/wiki/Variance" title="Variance">variance</a> may involve sums of squares, which can lead to <a href="/wiki/Numerical_instability" class="mw-redirect" title="Numerical instability">numerical instability</a> as well as to <a href="/wiki/Arithmetic_overflow" class="mw-redirect" title="Arithmetic overflow">arithmetic overflow</a> when dealing with large values. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Naïve_algorithm">Naïve algorithm</h2>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=1" title="Edit section: Naïve algorithm">edit</a>]</div> A formula for calculating the variance of an entire <a href="/wiki/Statistical_population" title="Statistical population">population</a> of size N is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}={\overline {(x^{2})}}-{\bar {x}}^{2}={\frac {\sum _{i=1}^{N}x_{i}^{2}-(\sum _{i=1}^{N}x_{i})^{2}/N}{N}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mo stretchy="false">(</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}={\overline {(x^{2})}}-{\bar {x}}^{2}={\frac {\sum _{i=1}^{N}x_{i}^{2}-(\sum _{i=1}^{N}x_{i})^{2}/N}{N}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a52e70e4e67e5e2d8fe32a71f7c21ebec27ac3f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.523ex; height:6.176ex;" alt="{\displaystyle \sigma ^{2}={\overline {(x^{2})}}-{\bar {x}}^{2}={\frac {\sum _{i=1}^{N}x_{i}^{2}-(\sum _{i=1}^{N}x_{i})^{2}/N}{N}}.}"></dd></dl> Using <a href="/wiki/Bessel%27s_correction" title="Bessel's correction">Bessel's correction</a> to calculate an <a href="/wiki/Estimator_bias" class="mw-redirect" title="Estimator bias">unbiased</a> estimate of the population variance from a finite <a href="/wiki/Statistical_sample" class="mw-redirect" title="Statistical sample">sample</a> of n observations, the formula is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}=\left({\frac {\sum _{i=1}^{n}x_{i}^{2}}{n}}-\left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)^{2}\right)\cdot {\frac {n}{n-1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}=\left({\frac {\sum _{i=1}^{n}x_{i}^{2}}{n}}-\left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)^{2}\right)\cdot {\frac {n}{n-1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1e93f1e1e6aebfaa9301c0bb6995e96a8e482d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:43.113ex; height:8.509ex;" alt="{\displaystyle s^{2}=\left({\frac {\sum _{i=1}^{n}x_{i}^{2}}{n}}-\left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)^{2}\right)\cdot {\frac {n}{n-1}}.}"></dd></dl> Therefore, a naïve algorithm to calculate the estimated variance is given by the following: <div style="margin-left: 35px; width: 600px"> <style data-mw-deduplicate="TemplateStyles:r1239945218">.mw-parser-output .framebox-container{margin-bottom:1.25em}.mw-parser-output .framebox-header{height:8px;margin:0;border:0;padding:0;font-size:1px}.mw-parser-output .framebox-inner{padding:5px;font-size:small}</style> <div class="framebox-container" style="width: auto; margin-left: auto; border:1px solid #8898BF; background: transparent; color: var(--color-base, #202122)"> <div class="framebox-header" style="border-bottom:1px solid #8898BF; background: #C8D8FF"></div> <div class="framebox-inner"> <ul><li>Let n ← 0, Sum ← 0, SumSq ← 0</li> <li>For each datum x: <ul><li>n ← n + 1</li> <li>Sum ← Sum + x</li> <li>SumSq ← SumSq + x × x</li></ul></li> <li>Var = (SumSq − (Sum × Sum) / n) / (n − 1)</li></ul> </div></div> </div> This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line. Because SumSq and (Sum×Sum)/n can be very similar numbers, <a href="/wiki/Catastrophic_cancellation" title="Catastrophic cancellation">cancellation</a> can lead to the <a href="/wiki/Precision_(arithmetic)" class="mw-redirect" title="Precision (arithmetic)">precision</a> of the result to be much less than the inherent precision of the <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">floating-point arithmetic</a> used to perform the computation. Thus this algorithm should not be used in practice,<a href="#cite_note-Einarsson2005-1">[1]</a><a href="#cite_note-Chan1983-2">[2]</a> and several alternate, numerically stable, algorithms have been proposed.<a href="#cite_note-:1-3">[3]</a> This is particularly bad if the standard deviation is small relative to the mean. <div class="mw-heading mw-heading3"><h3 id="Computing_shifted_data">Computing shifted data</h3>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=2" title="Edit section: Computing shifted data">edit</a>]</div> The variance is <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariant</a> with respect to changes in a <a href="/wiki/Location_parameter" title="Location parameter">location parameter</a>, a property which can be used to avoid the catastrophic cancellation in this formula. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} (X-K)=\operatorname {Var} (X).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (X-K)=\operatorname {Var} (X).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d982570b4d0dbc752123570fdb4e87bf620cc8fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.865ex; height:2.843ex;" alt="{\displaystyle \operatorname {Var} (X-K)=\operatorname {Var} (X).}"></dd></dl> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> any constant, which leads to the new formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}={\frac {\sum _{i=1}^{n}(x_{i}-K)^{2}-(\sum _{i=1}^{n}(x_{i}-K))^{2}/n}{n-1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>K</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>K</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}={\frac {\sum _{i=1}^{n}(x_{i}-K)^{2}-(\sum _{i=1}^{n}(x_{i}-K))^{2}/n}{n-1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f7d88759e45941fae0243041da5fd043a49308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.68ex; height:6.009ex;" alt="{\displaystyle \sigma ^{2}={\frac {\sum _{i=1}^{n}(x_{i}-K)^{2}-(\sum _{i=1}^{n}(x_{i}-K))^{2}/n}{n-1}}.}"></dd></dl> the closer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is to the mean value the more accurate the result will be, but just choosing a value inside the samples range will guarantee the desired stability. If the values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i}-K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i}-K)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f216c0fd06db99d732c12d0721a71e9fe6cc47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.845ex; height:2.843ex;" alt="{\displaystyle (x_{i}-K)}"> are small then there are no problems with the sum of its squares, on the contrary, if they are large it necessarily means that the variance is large as well. In any case the second term in the formula is always smaller than the first one therefore no cancellation may occur.<a href="#cite_note-Chan1983-2">[2]</a> If just the first sample is taken as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> the algorithm can be written in <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python programming language</a> as <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def shifted_data_variance(data): if len(data) < 2: return 0.0 K = data[0] n = Ex = Ex2 = 0.0 for x in data: n += 1 Ex += x - K Ex2 += (x - K) ** 2 variance = (Ex2 - Ex**2 / n) / (n - 1) # use n instead of (n-1) if want to compute the exact variance of the given data # use (n-1) if data are samples of a larger population return variance </pre></div> This formula also facilitates the incremental computation that can be expressed as <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>K = Ex = Ex2 = 0.0 n = 0 def add_variable(x): global K, n, Ex, Ex2 if n == 0: K = x n += 1 Ex += x - K Ex2 += (x - K) ** 2 def remove_variable(x): global K, n, Ex, Ex2 n -= 1 Ex -= x - K Ex2 -= (x - K) ** 2 def get_mean(): global K, n, Ex return K + Ex / n def get_variance(): global n, Ex, Ex2 return (Ex2 - Ex**2 / n) / (n - 1) </pre></div> <div class="mw-heading mw-heading2"><h2 id="Two-pass_algorithm">Two-pass algorithm</h2>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=3" title="Edit section: Two-pass algorithm">edit</a>]</div> An alternative approach, using a different formula for the variance, first computes the sample mean, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}={\frac {\sum _{j=1}^{n}x_{j}}{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}={\frac {\sum _{j=1}^{n}x_{j}}{n}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12587e2bf25e9d8bb9525f9e877a0ba5e1212a92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.002ex; height:6.176ex;" alt="{\displaystyle {\bar {x}}={\frac {\sum _{j=1}^{n}x_{j}}{n}},}"></dd></dl> and then computes the sum of the squares of the differences from the mean, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{sample variance}}=s^{2}={\dfrac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}{n-1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>sample variance</mtext> </mrow> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{sample variance}}=s^{2}={\dfrac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}{n-1}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30561183657c1c3967c0f2b107d951e210482c19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.377ex; height:6.009ex;" alt="{\displaystyle {\text{sample variance}}=s^{2}={\dfrac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}{n-1}},}"></dd></dl> where s is the standard deviation. This is given by the following code: <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def two_pass_variance(data): n = len(data) mean = sum(data) / n variance = sum((x - mean) ** 2 for x in data) / (n - 1) return variance </pre></div> This algorithm is numerically stable if n is small.<a href="#cite_note-Einarsson2005-1">[1]</a><a href="#cite_note-4">[4]</a> However, the results of both of these simple algorithms ("naïve" and "two-pass") can depend inordinately on the ordering of the data and can give poor results for very large data sets due to repeated roundoff error in the accumulation of the sums. Techniques such as <a href="/wiki/Compensated_summation" class="mw-redirect" title="Compensated summation">compensated summation</a> can be used to combat this error to a degree. <div class="mw-heading mw-heading2"><h2 id="Welford's_online_algorithm">Welford's online algorithm</h2>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=4" title="Edit section: Welford's online algorithm">edit</a>]</div> It is often useful to be able to compute the variance in a <a href="/wiki/One-pass_algorithm" title="One-pass algorithm">single pass</a>, inspecting each value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"> only once; for example, when the data is being collected without enough storage to keep all the values, or when costs of memory access dominate those of computation. For such an <a href="/wiki/Online_algorithm" title="Online algorithm">online algorithm</a>, a <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a> is required between quantities from which the required statistics can be calculated in a numerically stable fashion. The following formulas can be used to update the <a href="/wiki/Mean" title="Mean">mean</a> and (estimated) variance of the sequence, for an additional element xn. Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\overline {x}}_{n}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\overline {x}}_{n}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e20b5b37b79c22901150b0fc1c8d3860b051af1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.842ex; height:3.343ex;" alt="{\textstyle {\overline {x}}_{n}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}"> denotes the sample mean of the first n samples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\dots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\dots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56204ada1ecf9dd5c6beddfdfd0f341cd69ff632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{1},\dots ,x_{n})}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma _{n}^{2}={\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}_{n}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sigma _{n}^{2}={\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}_{n}\right)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7b8fd464261fd905bd2fa6a736153ec94d8560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.091ex; height:3.509ex;" alt="{\textstyle \sigma _{n}^{2}={\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}_{n}\right)^{2}}"> their <a href="/wiki/Variance#Biased_sample_variance" title="Variance">biased sample variance</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s_{n}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}_{n}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle s_{n}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}_{n}\right)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7830d411286cc2e250a564f445ddcc0ddb51bb1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:25.955ex; height:3.843ex;" alt="{\textstyle s_{n}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}\left(x_{i}-{\overline {x}}_{n}\right)^{2}}"> their <a href="/wiki/Variance#Unbiased_sample_variance" title="Variance">unbiased sample variance</a>. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}_{n}={\frac {(n-1)\,{\bar {x}}_{n-1}+x_{n}}{n}}={\bar {x}}_{n-1}+{\frac {x_{n}-{\bar {x}}_{n-1}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}_{n}={\frac {(n-1)\,{\bar {x}}_{n-1}+x_{n}}{n}}={\bar {x}}_{n-1}+{\frac {x_{n}-{\bar {x}}_{n-1}}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01872e3782e05f2f00af847071b044d419d557ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.575ex; height:5.676ex;" alt="{\displaystyle {\bar {x}}_{n}={\frac {(n-1)\,{\bar {x}}_{n-1}+x_{n}}{n}}={\bar {x}}_{n-1}+{\frac {x_{n}-{\bar {x}}_{n-1}}{n}}}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{n}^{2}={\frac {(n-1)\,\sigma _{n-1}^{2}+(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})}{n}}=\sigma _{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})-\sigma _{n-1}^{2}}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{n}^{2}={\frac {(n-1)\,\sigma _{n-1}^{2}+(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})}{n}}=\sigma _{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})-\sigma _{n-1}^{2}}{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a07187029f774ee0da058f3b6ab6e5d3e7131df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:84.301ex; height:6.009ex;" alt="{\displaystyle \sigma _{n}^{2}={\frac {(n-1)\,\sigma _{n-1}^{2}+(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})}{n}}=\sigma _{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})-\sigma _{n-1}^{2}}{n}}.}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}^{2}={\frac {n-2}{n-1}}\,s_{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})^{2}}{n}}=s_{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})^{2}}{n}}-{\frac {s_{n-1}^{2}}{n-1}},\quad n>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}^{2}={\frac {n-2}{n-1}}\,s_{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})^{2}}{n}}=s_{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})^{2}}{n}}-{\frac {s_{n-1}^{2}}{n-1}},\quad n>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f407c573792f57f6a8b9799e70d9b13193bdc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:75.186ex; height:6.176ex;" alt="{\displaystyle s_{n}^{2}={\frac {n-2}{n-1}}\,s_{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})^{2}}{n}}=s_{n-1}^{2}+{\frac {(x_{n}-{\bar {x}}_{n-1})^{2}}{n}}-{\frac {s_{n-1}^{2}}{n-1}},\quad n>1}"></dd></dl> These formulas suffer from numerical instability [<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>], as they repeatedly subtract a small number from a big number which scales with n. A better quantity for updating is the sum of squares of differences from the current mean, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=1}^{n}(x_{i}-{\bar {x}}_{n})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{n}(x_{i}-{\bar {x}}_{n})^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ec4abc32a30ce412c810feb80ce851571cd1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.735ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{n}(x_{i}-{\bar {x}}_{n})^{2}}">, here denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{2,n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2,n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8df02a452d69ff8d1a763700158247c25550e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.752ex; height:2.843ex;" alt="{\displaystyle M_{2,n}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}M_{2,n}&=M_{2,n-1}+(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})\\[4pt]\sigma _{n}^{2}&={\frac {M_{2,n}}{n}}\\[4pt]s_{n}^{2}&={\frac {M_{2,n}}{n-1}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}M_{2,n}&=M_{2,n-1}+(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})\\[4pt]\sigma _{n}^{2}&={\frac {M_{2,n}}{n}}\\[4pt]s_{n}^{2}&={\frac {M_{2,n}}{n-1}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f2ba2132cbd007a99e52e50211c2afa3f29ea65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.481ex; margin-bottom: -0.19ex; width:39.887ex; height:16.509ex;" alt="{\displaystyle {\begin{aligned}M_{2,n}&=M_{2,n-1}+(x_{n}-{\bar {x}}_{n-1})(x_{n}-{\bar {x}}_{n})\\[4pt]\sigma _{n}^{2}&={\frac {M_{2,n}}{n}}\\[4pt]s_{n}^{2}&={\frac {M_{2,n}}{n-1}}\end{aligned}}}"></dd></dl> This algorithm was found by Welford,<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> and it has been thoroughly analyzed.<a href="#cite_note-Chan1983-2">[2]</a><a href="#cite_note-7">[7]</a> It is also common to denote <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k}={\bar {x}}_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{k}={\bar {x}}_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b104ec723606827f53d9897e8e264df36307beca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.86ex; height:2.509ex;" alt="{\displaystyle M_{k}={\bar {x}}_{k}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{k}=M_{2,k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{k}=M_{2,k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b580d502ee7c4c5cf3e496a47333be779c46ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.234ex; height:2.843ex;" alt="{\displaystyle S_{k}=M_{2,k}}">.<a href="#cite_note-8">[8]</a> An example Python implementation for Welford's algorithm is given below. <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre># For a new value new_value, compute the new count, new mean, the new M2. # mean accumulates the mean of the entire dataset # M2 aggregates the squared distance from the mean # count aggregates the number of samples seen so far def update(existing_aggregate, new_value): (count, mean, M2) = existing_aggregate count += 1 delta = new_value - mean mean += delta / count delta2 = new_value - mean M2 += delta * delta2 return (count, mean, M2) # Retrieve the mean, variance and sample variance from an aggregate def finalize(existing_aggregate): (count, mean, M2) = existing_aggregate if count < 2: return float("nan") else: (mean, variance, sample_variance) = (mean, M2 / count, M2 / (count - 1)) return (mean, variance, sample_variance) </pre></div> This algorithm is much less prone to loss of precision due to <a href="/wiki/Catastrophic_cancellation" title="Catastrophic cancellation">catastrophic cancellation</a>, but might not be as efficient because of the division operation inside the loop. For a particularly robust two-pass algorithm for computing the variance, one can first compute and subtract an estimate of the mean, and then use this algorithm on the residuals. The <a href="#Parallel_algorithm">parallel algorithm</a> below illustrates how to merge multiple sets of statistics calculated online. <div class="mw-heading mw-heading2"><h2 id="Weighted_incremental_algorithm">Weighted incremental algorithm</h2>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=5" title="Edit section: Weighted incremental algorithm">edit</a>]</div> The algorithm can be extended to handle unequal sample weights, replacing the simple counter n with the sum of weights seen so far. West (1979)<a href="#cite_note-9">[9]</a> suggests this <a href="/wiki/Incremental_computing" title="Incremental computing">incremental algorithm</a>: <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def weighted_incremental_variance(data_weight_pairs): w_sum = w_sum2 = mean = S = 0 for x, w in data_weight_pairs: w_sum = w_sum + w w_sum2 = w_sum2 + w**2 mean_old = mean mean = mean_old + (w / w_sum) * (x - mean_old) S = S + w * (x - mean_old) * (x - mean) population_variance = S / w_sum # Bessel's correction for weighted samples # for integer frequency weights sample_frequency_variance = S / (w_sum - 1) </pre></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Weighted_arithmetic_mean#Weighted_sample_variance" title="Weighted arithmetic mean">Weighted arithmetic mean § Weighted sample variance</a></div> <div class="mw-heading mw-heading2"><h2 id="Parallel_algorithm">Parallel algorithm</h2>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=6" title="Edit section: Parallel algorithm">edit</a>]</div> Chan et al.<a href="#cite_note-:0-10">[10]</a> note that Welford's online algorithm detailed above is a special case of an algorithm that works for combining arbitrary sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}n_{AB}&=n_{A}+n_{B}\\\delta &={\bar {x}}_{B}-{\bar {x}}_{A}\\{\bar {x}}_{AB}&={\bar {x}}_{A}+\delta \cdot {\frac {n_{B}}{n_{AB}}}\\M_{2,AB}&=M_{2,A}+M_{2,B}+\delta ^{2}\cdot {\frac {n_{A}n_{B}}{n_{AB}}}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>δ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}n_{AB}&=n_{A}+n_{B}\\\delta &={\bar {x}}_{B}-{\bar {x}}_{A}\\{\bar {x}}_{AB}&={\bar {x}}_{A}+\delta \cdot {\frac {n_{B}}{n_{AB}}}\\M_{2,AB}&=M_{2,A}+M_{2,B}+\delta ^{2}\cdot {\frac {n_{A}n_{B}}{n_{AB}}}\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd88631832bace8b86e5b41ffcfa78f50f1b6602" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:36.145ex; height:16.509ex;" alt="{\displaystyle {\begin{aligned}n_{AB}&=n_{A}+n_{B}\\\delta &={\bar {x}}_{B}-{\bar {x}}_{A}\\{\bar {x}}_{AB}&={\bar {x}}_{A}+\delta \cdot {\frac {n_{B}}{n_{AB}}}\\M_{2,AB}&=M_{2,A}+M_{2,B}+\delta ^{2}\cdot {\frac {n_{A}n_{B}}{n_{AB}}}\\\end{aligned}}}">.</dd></dl> This may be useful when, for example, multiple processing units may be assigned to discrete parts of the input. Chan's method for estimating the mean is numerically unstable when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{A}\approx n_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>≈</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{A}\approx n_{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07640e8cffbc13372a410aa20e8055a1d51a7051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.832ex; height:2.009ex;" alt="{\displaystyle n_{A}\approx n_{B}}"> and both are large, because the numerical error in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ={\bar {x}}_{B}-{\bar {x}}_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ={\bar {x}}_{B}-{\bar {x}}_{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/131dd14741b4b49e517032681385a6265ba05552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.591ex; height:2.676ex;" alt="{\displaystyle \delta ={\bar {x}}_{B}-{\bar {x}}_{A}}"> is not scaled down in the way that it is in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{B}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{B}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0c6e28d168c8b42599074a2aa8147c03983306" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.135ex; height:2.509ex;" alt="{\displaystyle n_{B}=1}"> case. In such cases, prefer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\bar {x}}_{AB}={\frac {n_{A}{\bar {x}}_{A}+n_{B}{\bar {x}}_{B}}{n_{AB}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\bar {x}}_{AB}={\frac {n_{A}{\bar {x}}_{A}+n_{B}{\bar {x}}_{B}}{n_{AB}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0e21c304ed2a315b6957149af1b975249446ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:17.802ex; height:4.509ex;" alt="{\textstyle {\bar {x}}_{AB}={\frac {n_{A}{\bar {x}}_{A}+n_{B}{\bar {x}}_{B}}{n_{AB}}}}">. <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def parallel_variance(n_a, avg_a, M2_a, n_b, avg_b, M2_b): n = n_a + n_b delta = avg_b - avg_a M2 = M2_a + M2_b + delta**2 * n_a * n_b / n var_ab = M2 / (n - 1) return var_ab </pre></div> This can be generalized to allow parallelization with <a href="/wiki/Advanced_Vector_Extensions" title="Advanced Vector Extensions">AVX</a>, with <a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">GPUs</a>, and <a href="/wiki/Computer_cluster" title="Computer cluster">computer clusters</a>, and to covariance.<a href="#cite_note-:1-3">[3]</a> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=7" title="Edit section: Example">edit</a>]</div> Assume that all floating point operations use standard <a href="/wiki/IEEE_754#Double-precision_64_bit" title="IEEE 754">IEEE 754 double-precision</a> arithmetic. Consider the sample (4, 7, 13, 16) from an infinite population. Based on this sample, the estimated population mean is 10, and the unbiased estimate of population variance is 30. Both the naïve algorithm and two-pass algorithm compute these values correctly. Next consider the sample (108 + 4, 108 + 7, 108 + 13, 108 + 16), which gives rise to the same estimated variance as the first sample. The two-pass algorithm computes this variance estimate correctly, but the naïve algorithm returns 29.333333333333332 instead of 30. While this loss of precision may be tolerable and viewed as a minor flaw of the naïve algorithm, further increasing the offset makes the error catastrophic. Consider the sample (109 + 4, 109 + 7, 109 + 13, 109 + 16). Again the estimated population variance of 30 is computed correctly by the two-pass algorithm, but the naïve algorithm now computes it as −170.66666666666666. This is a serious problem with naïve algorithm and is due to <a href="/wiki/Catastrophic_cancellation" title="Catastrophic cancellation">catastrophic cancellation</a> in the subtraction of two similar numbers at the final stage of the algorithm. <div class="mw-heading mw-heading2"><h2 id="Higher-order_statistics">Higher-order statistics</h2>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=8" title="Edit section: Higher-order statistics">edit</a>]</div> Terriberry<a href="#cite_note-11">[11]</a> extends Chan's formulae to calculating the third and fourth <a href="/wiki/Central_moment" title="Central moment">central moments</a>, needed for example when estimating <a href="/wiki/Skewness" title="Skewness">skewness</a> and <a href="/wiki/Kurtosis" title="Kurtosis">kurtosis</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}M_{3,X}=M_{3,A}+M_{3,B}&{}+\delta ^{3}{\frac {n_{A}n_{B}(n_{A}-n_{B})}{n_{X}^{2}}}+3\delta {\frac {n_{A}M_{2,B}-n_{B}M_{2,A}}{n_{X}}}\\[6pt]M_{4,X}=M_{4,A}+M_{4,B}&{}+\delta ^{4}{\frac {n_{A}n_{B}\left(n_{A}^{2}-n_{A}n_{B}+n_{B}^{2}\right)}{n_{X}^{3}}}\\[6pt]&{}+6\delta ^{2}{\frac {n_{A}^{2}M_{2,B}+n_{B}^{2}M_{2,A}}{n_{X}^{2}}}+4\delta {\frac {n_{A}M_{3,B}-n_{B}M_{3,A}}{n_{X}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>X</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>+</mo> <mn>3</mn> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>A</mi> </mrow> </msub> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mi>X</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mi>B</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mrow> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>6</mn> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>A</mi> </mrow> </msub> </mrow> <msubsup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>+</mo> <mn>4</mn> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>A</mi> </mrow> </msub> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}M_{3,X}=M_{3,A}+M_{3,B}&{}+\delta ^{3}{\frac {n_{A}n_{B}(n_{A}-n_{B})}{n_{X}^{2}}}+3\delta {\frac {n_{A}M_{2,B}-n_{B}M_{2,A}}{n_{X}}}\\[6pt]M_{4,X}=M_{4,A}+M_{4,B}&{}+\delta ^{4}{\frac {n_{A}n_{B}\left(n_{A}^{2}-n_{A}n_{B}+n_{B}^{2}\right)}{n_{X}^{3}}}\\[6pt]&{}+6\delta ^{2}{\frac {n_{A}^{2}M_{2,B}+n_{B}^{2}M_{2,A}}{n_{X}^{2}}}+4\delta {\frac {n_{A}M_{3,B}-n_{B}M_{3,A}}{n_{X}}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d15ceb6b5c7f6a8a1b50f89b1f66ae6dc6b055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.27ex; margin-bottom: -0.235ex; width:71.873ex; height:24.176ex;" alt="{\displaystyle {\begin{aligned}M_{3,X}=M_{3,A}+M_{3,B}&{}+\delta ^{3}{\frac {n_{A}n_{B}(n_{A}-n_{B})}{n_{X}^{2}}}+3\delta {\frac {n_{A}M_{2,B}-n_{B}M_{2,A}}{n_{X}}}\\[6pt]M_{4,X}=M_{4,A}+M_{4,B}&{}+\delta ^{4}{\frac {n_{A}n_{B}\left(n_{A}^{2}-n_{A}n_{B}+n_{B}^{2}\right)}{n_{X}^{3}}}\\[6pt]&{}+6\delta ^{2}{\frac {n_{A}^{2}M_{2,B}+n_{B}^{2}M_{2,A}}{n_{X}^{2}}}+4\delta {\frac {n_{A}M_{3,B}-n_{B}M_{3,A}}{n_{X}}}\end{aligned}}}"></dd></dl> Here the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43175c381ee6ba2694520af1f1a26c676a2726ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.343ex; height:2.509ex;" alt="{\displaystyle M_{k}}"> are again the sums of powers of differences from the mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum (x-{\overline {x}})^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum (x-{\overline {x}})^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/388d8967b3837a7cf01900ab82325a1a487632eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.967ex; height:3.009ex;" alt="{\textstyle \sum (x-{\overline {x}})^{k}}">, giving <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\text{skewness}}=g_{1}={\frac {{\sqrt {n}}M_{3}}{M_{2}^{3/2}}},\\[4pt]&{\text{kurtosis}}=g_{2}={\frac {nM_{4}}{M_{2}^{2}}}-3.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>skewness</mtext> </mrow> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>kurtosis</mtext> </mrow> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mn>3.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\text{skewness}}=g_{1}={\frac {{\sqrt {n}}M_{3}}{M_{2}^{3/2}}},\\[4pt]&{\text{kurtosis}}=g_{2}={\frac {nM_{4}}{M_{2}^{2}}}-3.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c838ea34d2f10fe9f8ab6f78607e0b84c99b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:27.279ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}&{\text{skewness}}=g_{1}={\frac {{\sqrt {n}}M_{3}}{M_{2}^{3/2}}},\\[4pt]&{\text{kurtosis}}=g_{2}={\frac {nM_{4}}{M_{2}^{2}}}-3.\end{aligned}}}"></dd></dl> For the incremental case (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{x\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a8bd2b763f0b6509ea8303801d61105aecaf0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.517ex; height:2.843ex;" alt="{\displaystyle B=\{x\}}">), this simplifies to: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\delta &=x-m\\[5pt]m'&=m+{\frac {\delta }{n}}\\[5pt]M_{2}'&=M_{2}+\delta ^{2}{\frac {n-1}{n}}\\[5pt]M_{3}'&=M_{3}+\delta ^{3}{\frac {(n-1)(n-2)}{n^{2}}}-{\frac {3\delta M_{2}}{n}}\\[5pt]M_{4}'&=M_{4}+{\frac {\delta ^{4}(n-1)(n^{2}-3n+3)}{n^{3}}}+{\frac {6\delta ^{2}M_{2}}{n^{2}}}-{\frac {4\delta M_{3}}{n}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.8em 0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>δ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>m</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>m</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>δ</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>δ</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\delta &=x-m\\[5pt]m'&=m+{\frac {\delta }{n}}\\[5pt]M_{2}'&=M_{2}+\delta ^{2}{\frac {n-1}{n}}\\[5pt]M_{3}'&=M_{3}+\delta ^{3}{\frac {(n-1)(n-2)}{n^{2}}}-{\frac {3\delta M_{2}}{n}}\\[5pt]M_{4}'&=M_{4}+{\frac {\delta ^{4}(n-1)(n^{2}-3n+3)}{n^{3}}}+{\frac {6\delta ^{2}M_{2}}{n^{2}}}-{\frac {4\delta M_{3}}{n}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acfbbe12768419f004d32d11750e940fa0af584e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.838ex; width:56.568ex; height:30.843ex;" alt="{\displaystyle {\begin{aligned}\delta &=x-m\\[5pt]m'&=m+{\frac {\delta }{n}}\\[5pt]M_{2}'&=M_{2}+\delta ^{2}{\frac {n-1}{n}}\\[5pt]M_{3}'&=M_{3}+\delta ^{3}{\frac {(n-1)(n-2)}{n^{2}}}-{\frac {3\delta M_{2}}{n}}\\[5pt]M_{4}'&=M_{4}+{\frac {\delta ^{4}(n-1)(n^{2}-3n+3)}{n^{3}}}+{\frac {6\delta ^{2}M_{2}}{n^{2}}}-{\frac {4\delta M_{3}}{n}}\end{aligned}}}"></dd></dl> By preserving the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta /n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta /n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e6c8cc7925797c3b5c5f5ef6d34f0772dd692d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.606ex; height:2.843ex;" alt="{\displaystyle \delta /n}">, only one division operation is needed and the higher-order statistics can thus be calculated for little incremental cost. An example of the online algorithm for kurtosis implemented as described is: <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def online_kurtosis(data): n = mean = M2 = M3 = M4 = 0 for x in data: n1 = n n = n + 1 delta = x - mean delta_n = delta / n delta_n2 = delta_n**2 term1 = delta * delta_n * n1 mean = mean + delta_n M4 = M4 + term1 * delta_n2 * (n**2 - 3*n + 3) + 6 * delta_n2 * M2 - 4 * delta_n * M3 M3 = M3 + term1 * delta_n * (n - 2) - 3 * delta_n * M2 M2 = M2 + term1 # Note, you may also calculate variance using M2, and skewness using M3 # Caution: If all the inputs are the same, M2 will be 0, resulting in a division by 0. kurtosis = (n * M4) / (M2**2) - 3 return kurtosis </pre></div> Pébaÿ<a href="#cite_note-12">[12]</a> further extends these results to arbitrary-order <a href="/wiki/Central_moment" title="Central moment">central moments</a>, for the incremental and the pairwise cases, and subsequently Pébaÿ et al.<a href="#cite_note-13">[13]</a> for weighted and compound moments. One can also find there similar formulas for <a href="/wiki/Covariance" title="Covariance">covariance</a>. Choi and Sweetman<a href="#cite_note-Choi2010-14">[14]</a> offer two alternative methods to compute the skewness and kurtosis, each of which can save substantial computer memory requirements and CPU time in certain applications. The first approach is to compute the statistical moments by separating the data into bins and then computing the moments from the geometry of the resulting histogram, which effectively becomes a <a href="/wiki/One-pass_algorithm" title="One-pass algorithm">one-pass algorithm</a> for higher moments. One benefit is that the statistical moment calculations can be carried out to arbitrary accuracy such that the computations can be tuned to the precision of, e.g., the data storage format or the original measurement hardware. A relative histogram of a random variable can be constructed in the conventional way: the range of potential values is divided into bins and the number of occurrences within each bin are counted and plotted such that the area of each rectangle equals the portion of the sample values within that bin: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x_{k})={\frac {h(x_{k})}{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>A</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x_{k})={\frac {h(x_{k})}{A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0823ffbd15a8a29d670291ee96ad80b27532e4db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.793ex; height:5.843ex;" alt="{\displaystyle H(x_{k})={\frac {h(x_{k})}{A}}}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x_{k})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7cafe3a7018ccbd18072363647649a125b4a142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.567ex; height:2.843ex;" alt="{\displaystyle h(x_{k})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(x_{k})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62078d290756950d0275e7071dfd09a4b1390fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.291ex; height:2.843ex;" alt="{\displaystyle H(x_{k})}"> represent the frequency and the relative frequency at bin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2b88c64c76a03611549fb9b4cf4ed060b56002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A=\sum _{k=1}^{K}h(x_{k})\,\Delta x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle A=\sum _{k=1}^{K}h(x_{k})\,\Delta x_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfdd56dbc14317429ae0bfb00fe1ef509c771d93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.18ex; height:3.509ex;" alt="{\textstyle A=\sum _{k=1}^{K}h(x_{k})\,\Delta x_{k}}"> is the total area of the histogram. After this normalization, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> raw moments and central moments of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}"> can be calculated from the relative histogram: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{n}^{(h)}=\sum _{k=1}^{K}x_{k}^{n}H(x_{k})\,\Delta x_{k}={\frac {1}{A}}\sum _{k=1}^{K}x_{k}^{n}h(x_{k})\,\Delta x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>A</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{n}^{(h)}=\sum _{k=1}^{K}x_{k}^{n}H(x_{k})\,\Delta x_{k}={\frac {1}{A}}\sum _{k=1}^{K}x_{k}^{n}h(x_{k})\,\Delta x_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8be2914ad152e7d545420ea2beeada2eb284cbfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.583ex; height:7.343ex;" alt="{\displaystyle m_{n}^{(h)}=\sum _{k=1}^{K}x_{k}^{n}H(x_{k})\,\Delta x_{k}={\frac {1}{A}}\sum _{k=1}^{K}x_{k}^{n}h(x_{k})\,\Delta x_{k}}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{n}^{(h)}=\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}\,H(x_{k})\,\Delta x_{k}={\frac {1}{A}}\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}h(x_{k})\,\Delta x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>A</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{n}^{(h)}=\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}\,H(x_{k})\,\Delta x_{k}={\frac {1}{A}}\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}h(x_{k})\,\Delta x_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4544936b1ebb6e1e373030ed8834d912c04705ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:69.427ex; height:7.343ex;" alt="{\displaystyle \theta _{n}^{(h)}=\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}\,H(x_{k})\,\Delta x_{k}={\frac {1}{A}}\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}h(x_{k})\,\Delta x_{k}}"></dd></dl> where the superscript <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{(h)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{(h)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aafde7bf67b35ed1d7e04342d5ee4337ea891bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.458ex; height:2.676ex;" alt="{\displaystyle ^{(h)}}"> indicates the moments are calculated from the histogram. For constant bin width <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x_{k}=\Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x_{k}=\Delta x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/996e11040922984df3ad46a027dcc354339dc725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.718ex; height:2.509ex;" alt="{\displaystyle \Delta x_{k}=\Delta x}"> these two expressions can be simplified using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=A/\Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=A/\Delta x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e557049f9f0077352eaa203d96d8258bbfe3fb66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.441ex; height:2.843ex;" alt="{\displaystyle I=A/\Delta x}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{n}^{(h)}={\frac {1}{I}}\sum _{k=1}^{K}x_{k}^{n}\,h(x_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>I</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{n}^{(h)}={\frac {1}{I}}\sum _{k=1}^{K}x_{k}^{n}\,h(x_{k})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa350e390ed68b7c81872067921d57ddbe4bf9f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.236ex; height:7.343ex;" alt="{\displaystyle m_{n}^{(h)}={\frac {1}{I}}\sum _{k=1}^{K}x_{k}^{n}\,h(x_{k})}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{n}^{(h)}={\frac {1}{I}}\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}h(x_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>I</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{n}^{(h)}={\frac {1}{I}}\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}h(x_{k})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b446a58253762907ec85555174c2aba3b60a1d94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.103ex; height:7.343ex;" alt="{\displaystyle \theta _{n}^{(h)}={\frac {1}{I}}\sum _{k=1}^{K}{\Big (}x_{k}-m_{1}^{(h)}{\Big )}^{n}h(x_{k})}"></dd></dl> The second approach from Choi and Sweetman<a href="#cite_note-Choi2010-14">[14]</a> is an analytical methodology to combine statistical moments from individual segments of a time-history such that the resulting overall moments are those of the complete time-history. This methodology could be used for parallel computation of statistical moments with subsequent combination of those moments, or for combination of statistical moments computed at sequential times. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> sets of statistical moments are known: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\gamma _{0,q},\mu _{q},\sigma _{q}^{2},\alpha _{3,q},\alpha _{4,q})\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\gamma _{0,q},\mu _{q},\sigma _{q}^{2},\alpha _{3,q},\alpha _{4,q})\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c2af50d87c5001f01adf601568f9198f167f019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.026ex; height:3.176ex;" alt="{\displaystyle (\gamma _{0,q},\mu _{q},\sigma _{q}^{2},\alpha _{3,q},\alpha _{4,q})\quad }"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=1,2,\ldots ,Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=1,2,\ldots ,Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0504071b4afecbf12daf70f1c275813c0b3ed8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.543ex; height:2.509ex;" alt="{\displaystyle q=1,2,\ldots ,Q}">, then each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84a6d83f89515bf8d1a053639dc484e620f42036" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.423ex; height:2.176ex;" alt="{\displaystyle \gamma _{n}}"> can be expressed in terms of the equivalent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> raw moments: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{n,q}=m_{n,q}\gamma _{0,q}\qquad \quad {\textrm {for}}\quad n=1,2,3,4\quad {\text{ and }}\quad q=1,2,\dots ,Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mspace width="2em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> </mrow> <mspace width="1em" /> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{n,q}=m_{n,q}\gamma _{0,q}\qquad \quad {\textrm {for}}\quad n=1,2,3,4\quad {\text{ and }}\quad q=1,2,\dots ,Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b984dcf1f94bc80eb489e32213b711f1cf2b2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.098ex; height:2.843ex;" alt="{\displaystyle \gamma _{n,q}=m_{n,q}\gamma _{0,q}\qquad \quad {\textrm {for}}\quad n=1,2,3,4\quad {\text{ and }}\quad q=1,2,\dots ,Q}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{0,q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{0,q}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13223d94d4f3bd4b58bc9c68151b888795db5875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.472ex; height:2.343ex;" alt="{\displaystyle \gamma _{0,q}}"> is generally taken to be the duration of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{th}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>h</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{th}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/215a844f887a954d9eb533647a71f02da11355db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.852ex; height:3.009ex;" alt="{\displaystyle q^{th}}"> time-history, or the number of points if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}"> is constant. The benefit of expressing the statistical moments in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"> is that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> sets can be combined by addition, and there is no upper limit on the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}">. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{n,c}=\sum _{q=1}^{Q}\gamma _{n,q}\quad \quad {\text{for }}n=0,1,2,3,4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </munderover> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{n,c}=\sum _{q=1}^{Q}\gamma _{n,q}\quad \quad {\text{for }}n=0,1,2,3,4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40bff45d3a920f17239a410fc171bbfd760f5445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.522ex; height:7.843ex;" alt="{\displaystyle \gamma _{n,c}=\sum _{q=1}^{Q}\gamma _{n,q}\quad \quad {\text{for }}n=0,1,2,3,4}"></dd></dl> where the subscript <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle _{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ac95da3d27b8f3d4540ba99845ecc3990bfb0b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:0.944ex; height:1.343ex;" alt="{\displaystyle _{c}}"> represents the concatenated time-history or combined <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }">. These combined values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"> can then be inversely transformed into raw moments representing the complete concatenated time-history <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{n,c}={\frac {\gamma _{n,c}}{\gamma _{0,c}}}\quad {\text{for }}n=1,2,3,4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>c</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mi>c</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{n,c}={\frac {\gamma _{n,c}}{\gamma _{0,c}}}\quad {\text{for }}n=1,2,3,4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669f8cce2b9d6af600bc364397a1006bb8a8c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.889ex; height:5.676ex;" alt="{\displaystyle m_{n,c}={\frac {\gamma _{n,c}}{\gamma _{0,c}}}\quad {\text{for }}n=1,2,3,4}"></dd></dl> Known relationships between the raw moments (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8181c5cf15da902bbaa5c2291aeb8167fe78ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.259ex; height:2.009ex;" alt="{\displaystyle m_{n}}">) and the central moments (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{n}=\operatorname {E} [(x-\mu )^{n}])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{n}=\operatorname {E} [(x-\mu )^{n}])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6c61d2b10f7f76d3bd29dbbc2d52b1decf3053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.788ex; height:2.843ex;" alt="{\displaystyle \theta _{n}=\operatorname {E} [(x-\mu )^{n}])}">) are then used to compute the central moments of the concatenated time-history. Finally, the statistical moments of the concatenated history are computed from the central moments: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{c}=m_{1,c}\qquad \sigma _{c}^{2}=\theta _{2,c}\qquad \alpha _{3,c}={\frac {\theta _{3,c}}{\sigma _{c}^{3}}}\qquad \alpha _{4,c}={\frac {\theta _{4,c}}{\sigma _{c}^{4}}}-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mspace width="2em" /> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mspace width="2em" /> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>c</mi> </mrow> </msub> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mfrac> </mrow> <mspace width="2em" /> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mi>c</mi> </mrow> </msub> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{c}=m_{1,c}\qquad \sigma _{c}^{2}=\theta _{2,c}\qquad \alpha _{3,c}={\frac {\theta _{3,c}}{\sigma _{c}^{3}}}\qquad \alpha _{4,c}={\frac {\theta _{4,c}}{\sigma _{c}^{4}}}-3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1de2de6cf52fb1852f3da5aa4e607a894f9ab3f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:58.364ex; height:6.509ex;" alt="{\displaystyle \mu _{c}=m_{1,c}\qquad \sigma _{c}^{2}=\theta _{2,c}\qquad \alpha _{3,c}={\frac {\theta _{3,c}}{\sigma _{c}^{3}}}\qquad \alpha _{4,c}={\frac {\theta _{4,c}}{\sigma _{c}^{4}}}-3}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Covariance">Covariance</h2>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=9" title="Edit section: Covariance">edit</a>]</div> Very similar algorithms can be used to compute the <a href="/wiki/Covariance" title="Covariance">covariance</a>. <div class="mw-heading mw-heading3"><h3 id="Naïve_algorithm_2">Naïve algorithm</h3>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=10" title="Edit section: Naïve algorithm">edit</a>]</div> The naïve algorithm is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cov} (X,Y)={\frac {\sum _{i=1}^{n}x_{i}y_{i}-(\sum _{i=1}^{n}x_{i})(\sum _{i=1}^{n}y_{i})/n}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} (X,Y)={\frac {\sum _{i=1}^{n}x_{i}y_{i}-(\sum _{i=1}^{n}x_{i})(\sum _{i=1}^{n}y_{i})/n}{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa56e6b8a568e85a091819226af741ab18c273e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.622ex; height:5.843ex;" alt="{\displaystyle \operatorname {Cov} (X,Y)={\frac {\sum _{i=1}^{n}x_{i}y_{i}-(\sum _{i=1}^{n}x_{i})(\sum _{i=1}^{n}y_{i})/n}{n}}.}"></dd></dl> For the algorithm above, one could use the following Python code: <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def naive_covariance(data1, data2): n = len(data1) sum1 = sum(data1) sum2 = sum(data2) sum12 = sum([i1 * i2 for i1, i2 in zip(data1, data2)]) covariance = (sum12 - sum1 * sum2 / n) / n return covariance </pre></div> <div class="mw-heading mw-heading3"><h3 id="With_estimate_of_the_mean">With estimate of the mean</h3>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=11" title="Edit section: With estimate of the mean">edit</a>]</div> As for the variance, the covariance of two random variables is also shift-invariant, so given any two constant values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adcbcc5a960cbbd063fb012b087a8fdef03066d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.509ex;" alt="{\displaystyle k_{x}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{y},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{y},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff5bb5ce28c07dffba22fc642174f9eba5485d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.907ex; height:2.843ex;" alt="{\displaystyle k_{y},}"> it can be written: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cov} (X,Y)=\operatorname {Cov} (X-k_{x},Y-k_{y})={\dfrac {\sum _{i=1}^{n}(x_{i}-k_{x})(y_{i}-k_{y})-(\sum _{i=1}^{n}(x_{i}-k_{x}))(\sum _{i=1}^{n}(y_{i}-k_{y}))/n}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mi>Y</mi> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} (X,Y)=\operatorname {Cov} (X-k_{x},Y-k_{y})={\dfrac {\sum _{i=1}^{n}(x_{i}-k_{x})(y_{i}-k_{y})-(\sum _{i=1}^{n}(x_{i}-k_{x}))(\sum _{i=1}^{n}(y_{i}-k_{y}))/n}{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94d06c7f798b0612a368b11cd49c307d36fa2cfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:100.436ex; height:5.843ex;" alt="{\displaystyle \operatorname {Cov} (X,Y)=\operatorname {Cov} (X-k_{x},Y-k_{y})={\dfrac {\sum _{i=1}^{n}(x_{i}-k_{x})(y_{i}-k_{y})-(\sum _{i=1}^{n}(x_{i}-k_{x}))(\sum _{i=1}^{n}(y_{i}-k_{y}))/n}{n}}.}"></dd></dl> and again choosing a value inside the range of values will stabilize the formula against catastrophic cancellation as well as make it more robust against big sums. Taking the first value of each data set, the algorithm can be written as: <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def shifted_data_covariance(data_x, data_y): n = len(data_x) if n < 2: return 0 kx = data_x[0] ky = data_y[0] Ex = Ey = Exy = 0 for ix, iy in zip(data_x, data_y): Ex += ix - kx Ey += iy - ky Exy += (ix - kx) * (iy - ky) return (Exy - Ex * Ey / n) / n </pre></div> <div class="mw-heading mw-heading3"><h3 id="Two-pass">Two-pass</h3>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=12" title="Edit section: Two-pass">edit</a>]</div> The two-pass algorithm first computes the sample means, and then the covariance: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}=\sum _{i=1}^{n}x_{i}/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}=\sum _{i=1}^{n}x_{i}/n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb385f363a8b4a96a189de33d5b4a62c1854bd58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.857ex; height:6.843ex;" alt="{\displaystyle {\bar {x}}=\sum _{i=1}^{n}x_{i}/n}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {y}}=\sum _{i=1}^{n}y_{i}/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {y}}=\sum _{i=1}^{n}y_{i}/n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2646ae762bb8b57823ffe1d3b969c7e32f55fa22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.639ex; height:6.843ex;" alt="{\displaystyle {\bar {y}}=\sum _{i=1}^{n}y_{i}/n}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cov} (X,Y)={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} (X,Y)={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a45cc6687d7f659ddebdee3df2b9dde2e87c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.6ex; height:5.843ex;" alt="{\displaystyle \operatorname {Cov} (X,Y)={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{n}}.}"></dd></dl> The two-pass algorithm may be written as: <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def two_pass_covariance(data1, data2): n = len(data1) mean1 = sum(data1) / n mean2 = sum(data2) / n covariance = 0 for i1, i2 in zip(data1, data2): a = i1 - mean1 b = i2 - mean2 covariance += a * b / n return covariance </pre></div> A slightly more accurate compensated version performs the full naive algorithm on the residuals. The final sums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/842460409f03764d17d9e1c0cee98837469cb1d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.77ex; height:3.009ex;" alt="{\textstyle \sum _{i}x_{i}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}y_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dac3620693928c4eb931881989b65b5d57bd7e50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.58ex; height:3.009ex;" alt="{\textstyle \sum _{i}y_{i}}"> should be zero, but the second pass compensates for any small error. <div class="mw-heading mw-heading3"><h3 id="Online">Online</h3>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=13" title="Edit section: Online">edit</a>]</div> A stable one-pass algorithm exists, similar to the online algorithm for computing the variance, that computes co-moment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{n}=\sum _{i=1}^{n}(x_{i}-{\bar {x}}_{n})(y_{i}-{\bar {y}}_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle C_{n}=\sum _{i=1}^{n}(x_{i}-{\bar {x}}_{n})(y_{i}-{\bar {y}}_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/850927eedc9fafa60339507410ea8a691bf0ce2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.769ex; height:3.176ex;" alt="{\textstyle C_{n}=\sum _{i=1}^{n}(x_{i}-{\bar {x}}_{n})(y_{i}-{\bar {y}}_{n})}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{2}{\bar {x}}_{n}&={\bar {x}}_{n-1}&\,+\,&{\frac {x_{n}-{\bar {x}}_{n-1}}{n}}\\[5pt]{\bar {y}}_{n}&={\bar {y}}_{n-1}&\,+\,&{\frac {y_{n}-{\bar {y}}_{n-1}}{n}}\\[5pt]C_{n}&=C_{n-1}&\,+\,&(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n-1})\\[5pt]&=C_{n-1}&\,+\,&(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n})\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="0.8em 0.8em 0.8em 0.3em" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}{\bar {x}}_{n}&={\bar {x}}_{n-1}&\,+\,&{\frac {x_{n}-{\bar {x}}_{n-1}}{n}}\\[5pt]{\bar {y}}_{n}&={\bar {y}}_{n-1}&\,+\,&{\frac {y_{n}-{\bar {y}}_{n-1}}{n}}\\[5pt]C_{n}&=C_{n-1}&\,+\,&(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n-1})\\[5pt]&=C_{n-1}&\,+\,&(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n})\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f440f49beea73d0aeec36d0101caa20f89e864aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.838ex; width:35.668ex; height:20.843ex;" alt="{\displaystyle {\begin{alignedat}{2}{\bar {x}}_{n}&={\bar {x}}_{n-1}&\,+\,&{\frac {x_{n}-{\bar {x}}_{n-1}}{n}}\\[5pt]{\bar {y}}_{n}&={\bar {y}}_{n-1}&\,+\,&{\frac {y_{n}-{\bar {y}}_{n-1}}{n}}\\[5pt]C_{n}&=C_{n-1}&\,+\,&(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n-1})\\[5pt]&=C_{n-1}&\,+\,&(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n})\end{alignedat}}}"></dd></dl> The apparent asymmetry in that last equation is due to the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (x_{n}-{\bar {x}}_{n})={\frac {n-1}{n}}(x_{n}-{\bar {x}}_{n-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (x_{n}-{\bar {x}}_{n})={\frac {n-1}{n}}(x_{n}-{\bar {x}}_{n-1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62c6438fed70eadeed78060c49d1693b6f44d7a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.614ex; height:3.509ex;" alt="{\textstyle (x_{n}-{\bar {x}}_{n})={\frac {n-1}{n}}(x_{n}-{\bar {x}}_{n-1})}">, so both update terms are equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {n-1}{n}}(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {n-1}{n}}(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n-1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a791a6b260b85a31f1f9c883704ff77680accccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.398ex; height:3.509ex;" alt="{\textstyle {\frac {n-1}{n}}(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n-1})}">. Even greater accuracy can be achieved by first computing the means, then using the stable one-pass algorithm on the residuals. Thus the covariance can be computed as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {Cov} _{N}(X,Y)={\frac {C_{N}}{N}}&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n-1})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+{\frac {N-1}{N}}(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n-1})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+{\frac {N}{N-1}}(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n})}{N}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mi>N</mi> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>N</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>N</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>N</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>N</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Cov} _{N}(X,Y)={\frac {C_{N}}{N}}&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n-1})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+{\frac {N-1}{N}}(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n-1})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+{\frac {N}{N-1}}(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n})}{N}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/241549051c7dfd419243943194d0912785b9fbf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.005ex; width:79.053ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Cov} _{N}(X,Y)={\frac {C_{N}}{N}}&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n-1})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+{\frac {N-1}{N}}(x_{n}-{\bar {x}}_{n-1})(y_{n}-{\bar {y}}_{n-1})}{N}}\\&={\frac {\operatorname {Cov} _{N-1}(X,Y)\cdot (N-1)+{\frac {N}{N-1}}(x_{n}-{\bar {x}}_{n})(y_{n}-{\bar {y}}_{n})}{N}}.\end{aligned}}}"></dd></dl> <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def online_covariance(data1, data2): meanx = meany = C = n = 0 for x, y in zip(data1, data2): n += 1 dx = x - meanx meanx += dx / n meany += (y - meany) / n C += dx * (y - meany) population_covar = C / n # Bessel's correction for sample variance sample_covar = C / (n - 1) </pre></div> A small modification can also be made to compute the weighted covariance: <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def online_weighted_covariance(data1, data2, data3): meanx = meany = 0 wsum = wsum2 = 0 C = 0 for x, y, w in zip(data1, data2, data3): wsum += w wsum2 += w * w dx = x - meanx meanx += (w / wsum) * dx meany += (w / wsum) * (y - meany) C += w * dx * (y - meany) population_covar = C / wsum # Bessel's correction for sample variance # Frequency weights sample_frequency_covar = C / (wsum - 1) # Reliability weights sample_reliability_covar = C / (wsum - wsum2 / wsum) </pre></div> Likewise, there is a formula for combining the covariances of two sets that can be used to parallelize the computation:<a href="#cite_note-:1-3">[3]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{X}=C_{A}+C_{B}+({\bar {x}}_{A}-{\bar {x}}_{B})({\bar {y}}_{A}-{\bar {y}}_{B})\cdot {\frac {n_{A}n_{B}}{n_{X}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{X}=C_{A}+C_{B}+({\bar {x}}_{A}-{\bar {x}}_{B})({\bar {y}}_{A}-{\bar {y}}_{B})\cdot {\frac {n_{A}n_{B}}{n_{X}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f170d41f6631ce258a60d3025e9ad73064a3332" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:47.689ex; height:5.009ex;" alt="{\displaystyle C_{X}=C_{A}+C_{B}+({\bar {x}}_{A}-{\bar {x}}_{B})({\bar {y}}_{A}-{\bar {y}}_{B})\cdot {\frac {n_{A}n_{B}}{n_{X}}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Weighted_batched_version">Weighted batched version</h3>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=14" title="Edit section: Weighted batched version">edit</a>]</div> A version of the weighted online algorithm that does batched updated also exists: let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{1},\dots w_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{1},\dots w_{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20e0597fff43d471b464b7ffcf3cdaa35ff68543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.218ex; height:2.009ex;" alt="{\displaystyle w_{1},\dots w_{N}}"> denote the weights, and write <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{2}{\bar {x}}_{n+k}&={\bar {x}}_{n}&\,+\,&{\frac {\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n})}{\sum _{i=1}^{n+k}w_{i}}}\\{\bar {y}}_{n+k}&={\bar {y}}_{n}&\,+\,&{\frac {\sum _{i=n+1}^{n+k}w_{i}(y_{i}-{\bar {y}}_{n})}{\sum _{i=1}^{n+k}w_{i}}}\\C_{n+k}&=C_{n}&\,+\,&\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n+k})(y_{i}-{\bar {y}}_{n})\\&=C_{n}&\,+\,&\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n})(y_{i}-{\bar {y}}_{n+k})\\\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mi></mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mi></mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}{\bar {x}}_{n+k}&={\bar {x}}_{n}&\,+\,&{\frac {\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n})}{\sum _{i=1}^{n+k}w_{i}}}\\{\bar {y}}_{n+k}&={\bar {y}}_{n}&\,+\,&{\frac {\sum _{i=n+1}^{n+k}w_{i}(y_{i}-{\bar {y}}_{n})}{\sum _{i=1}^{n+k}w_{i}}}\\C_{n+k}&=C_{n}&\,+\,&\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n+k})(y_{i}-{\bar {y}}_{n})\\&=C_{n}&\,+\,&\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n})(y_{i}-{\bar {y}}_{n+k})\\\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ce541f496b3df80b7d5f6bf6ecacd86bcd0a64a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.838ex; width:43.07ex; height:30.843ex;" alt="{\displaystyle {\begin{alignedat}{2}{\bar {x}}_{n+k}&={\bar {x}}_{n}&\,+\,&{\frac {\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n})}{\sum _{i=1}^{n+k}w_{i}}}\\{\bar {y}}_{n+k}&={\bar {y}}_{n}&\,+\,&{\frac {\sum _{i=n+1}^{n+k}w_{i}(y_{i}-{\bar {y}}_{n})}{\sum _{i=1}^{n+k}w_{i}}}\\C_{n+k}&=C_{n}&\,+\,&\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n+k})(y_{i}-{\bar {y}}_{n})\\&=C_{n}&\,+\,&\sum _{i=n+1}^{n+k}w_{i}(x_{i}-{\bar {x}}_{n})(y_{i}-{\bar {y}}_{n+k})\\\end{alignedat}}}"></dd></dl> The covariance can then be computed as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cov} _{N}(X,Y)={\frac {C_{N}}{\sum _{i=1}^{N}w_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Cov</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} _{N}(X,Y)={\frac {C_{N}}{\sum _{i=1}^{N}w_{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c62280dd9d52ecccf6d9911e41880fbb42e64fe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.496ex; height:6.676ex;" alt="{\displaystyle \operatorname {Cov} _{N}(X,Y)={\frac {C_{N}}{\sum _{i=1}^{N}w_{i}}}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=15" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Kahan_summation_algorithm" title="Kahan summation algorithm">Kahan summation algorithm</a></li> <li><a href="/wiki/Squared_deviations_from_the_mean" title="Squared deviations from the mean">Squared deviations from the mean</a></li> <li><a href="/wiki/Yamartino_method" title="Yamartino method">Yamartino method</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Algorithms_for_calculating_variance&action=edit&section=16" title="Edit section: References">edit</a>]</div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Einarsson2005-1">^ <a href="#cite_ref-Einarsson2005_1-0">a</a> <a href="#cite_ref-Einarsson2005_1-1">b</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFEinarsson2005" 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SIAM. p. 47. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-584-2" title="Special:BookSources/978-0-89871-584-2"><bdi>978-0-89871-584-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Accuracy+and+Reliability+in+Scientific+Computing&rft.pages=47&rft.pub=SIAM&rft.date=2005&rft.isbn=978-0-89871-584-2&rft.aulast=Einarsson&rft.aufirst=Bo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8hrDV5EbrEsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithms+for+calculating+variance" class="Z3988"> </li> <li id="cite_note-Chan1983-2">^ <a href="#cite_ref-Chan1983_2-0">a</a> <a href="#cite_ref-Chan1983_2-1">b</a> <a href="#cite_ref-Chan1983_2-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChanGolubLeVeque1983" class="citation journal cs1"><a href="/wiki/Tony_F._Chan" title="Tony F. Chan">Chan, Tony F.</a>; <a href="/wiki/Gene_H._Golub" title="Gene H. Golub">Golub, Gene H.</a>; LeVeque, Randall J. (1983). <a rel="nofollow" class="external text" href="http://cpsc.yale.edu/sites/default/files/files/tr222.pdf">"Algorithms for computing the sample variance: Analysis and recommendations"</a> (PDF). The American Statistician. 37 (3): 242–247. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00031305.1983.10483115">10.1080/00031305.1983.10483115</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2683386">2683386</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://cpsc.yale.edu/sites/default/files/files/tr222.pdf">Archived</a> (PDF) from the original on 9 October 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=Algorithms+for+computing+the+sample+variance%3A+Analysis+and+recommendations&rft.volume=37&rft.issue=3&rft.pages=242-247&rft.date=1983&rft_id=info%3Adoi%2F10.1080%2F00031305.1983.10483115&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2683386%23id-name%3DJSTOR&rft.aulast=Chan&rft.aufirst=Tony+F.&rft.au=Golub%2C+Gene+H.&rft.au=LeVeque%2C+Randall+J.&rft_id=http%3A%2F%2Fcpsc.yale.edu%2Fsites%2Fdefault%2Ffiles%2Ffiles%2Ftr222.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgorithms+for+calculating+variance" class="Z3988"> </li> <li id="cite_note-:1-3">^ <a href="#cite_ref-:1_3-0">a</a> <a href="#cite_ref-:1_3-1">b</a> <a href="#cite_ref-:1_3-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchubertGertz2018" class="citation book cs1">Schubert, Erich; Gertz, Michael (9 July 2018). <a rel="nofollow" class="external text" href="http://dl.acm.org/citation.cfm?id=3221269.3223036">Numerically stable parallel computation of (co-)variance</a>. 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