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id="toc-Sampling_a_dichotomous_population-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algorithms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Algorithms</span> </div> </a> <ul id="toc-Algorithms-sublist" class="vector-toc-list"> </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"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul 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Available in 16 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-16" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">16 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lihtne_juhuslik_valik" title="Lihtne juhuslik valik – Estonian" lang="et" hreflang="et" data-title="Lihtne juhuslik valik" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zorizko_laginketa_sinple" title="Zorizko laginketa sinple – Basque" lang="eu" hreflang="eu" data-title="Zorizko laginketa sinple" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A8%EC%88%9C_%EB%AC%B4%EC%9E%91%EC%9C%84_%EC%B6%94%EC%B6%9C%EB%B2%95" title="단순 무작위 추출법 – Korean" lang="ko" hreflang="ko" data-title="단순 무작위 추출법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%93%D7%92%D7%9D_%D7%9E%D7%A7%D7%A8%D7%99_%D7%A4%D7%A9%D7%95%D7%98" title="מדגם מקרי פשוט – Hebrew" lang="he" hreflang="he" data-title="מדגם מקרי פשוט" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Atsitiktin%C4%97_imtis" title="Atsitiktinė imtis – Lithuanian" lang="lt" hreflang="lt" data-title="Atsitiktinė imtis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B5%D0%BD_%D0%BF%D1%80%D0%B8%D0%BC%D0%B5%D1%80%D0%BE%D0%BA" title="Прост случаен примерок – Macedonian" lang="mk" hreflang="mk" data-title="Прост случаен примерок" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%84%A1%E4%BD%9C%E7%82%BA%E6%8A%BD%E5%87%BA" title="無作為抽出 – Japanese" lang="ja" hreflang="ja" data-title="無作為抽出" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Enkelt_tilfeldig_utvalg" title="Enkelt tilfeldig utvalg – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Enkelt tilfeldig utvalg" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Enkelt_tilfeldig_utval" title="Enkelt tilfeldig utval – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Enkelt tilfeldig utval" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Amostra_aleat%C3%B3ria_simples" title="Amostra aleatória simples – Portuguese" lang="pt" hreflang="pt" data-title="Amostra aleatória simples" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Sampel_acak_basajan" title="Sampel acak basajan – Sundanese" lang="su" hreflang="su" data-title="Sampel acak basajan" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Obundet_slumpm%C3%A4ssigt_urval" title="Obundet slumpmässigt urval – Swedish" lang="sv" hreflang="sv" 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searchaux" style="display:none">Sampling technique</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Simple_random_sample" title="Special:EditPage/Simple random sample">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Simple+random+sample%22">"Simple random sample"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Simple+random+sample%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Simple+random+sample%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Simple+random+sample%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Simple+random+sample%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Simple+random+sample%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">November 2011</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, a <b>simple random sample</b> (or <b>SRS</b>) is a <a href="/wiki/Subset" title="Subset">subset</a> of <a href="/wiki/Individuals" class="mw-redirect" title="Individuals">individuals</a> (a <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">sample</a>) chosen from a larger <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> (a <a href="/wiki/Statistical_population" title="Statistical population">population</a>) in which a subset of individuals are chosen <a href="/wiki/Randomization" title="Randomization">randomly</a>, all with the same probability. It is a process of selecting a sample in a random way. In SRS, each subset of <i>k</i> individuals has the same probability of being chosen for the sample as any other subset of <i>k</i> individuals.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Simple random sampling is a basic type of sampling and can be a component of other more complex sampling methods.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simple_random_sample&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The principle of simple random sampling is that every set with the same number of items has the same probability of being chosen. For example, suppose <i>N</i> college students want to get a ticket for a basketball game, but there are only <i>X</i> < <i>N</i> tickets for them, so they decide to have a fair way to see who gets to go. Then, everybody is given a number in the range from 0 to <i>N</i>-1, and random numbers are generated, either electronically or from a table of random numbers. Numbers outside the range from 0 to <i>N</i>-1 are ignored, as are any numbers previously selected. The first <i>X</i> numbers would identify the lucky ticket winners. </p><p>In small populations and often in large ones, such sampling is typically done "<b>without replacement</b>", i.e., one deliberately avoids choosing any member of the population more than once. Although simple random sampling can be conducted with replacement instead, this is less common and would normally be described more fully as simple random sampling <b>with replacement</b>. Sampling done without replacement is no longer independent, but still satisfies <a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">exchangeability</a>, hence most results of mathematical statistics still hold. Further, for a small sample from a large population, sampling without replacement is approximately the same as sampling with replacement, since the probability of choosing the same individual twice is low. Survey methodology textbooks generally consider simple random sampling without replacement as the benchmark to compute the relative efficiency of other sampling approaches.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>An unbiased random selection of individuals is important so that if many samples were drawn, the average sample would accurately represent the population. However, this does not guarantee that a particular sample is a perfect representation of the population. Simple random sampling merely allows one to draw externally valid conclusions about the entire population based on the sample. The concept can be extended when the population is a geographic area.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In this case, <a href="/wiki/Area_sampling_frame" title="Area sampling frame">area sampling frames</a> are relevant. </p><p>Conceptually, simple random sampling is the simplest of the probability sampling techniques. It requires a complete <a href="/wiki/Sampling_frame" title="Sampling frame">sampling frame</a>, which may not be available or feasible to construct for large populations. Even if a complete frame is available, more efficient approaches may be possible if other useful information is available about the units in the population. </p><p>Advantages are that it is free of classification error, and it requires minimum previous knowledge of the population other than the frame. Its simplicity also makes it relatively easy to interpret data collected in this manner. For these reasons, simple random sampling best suits situations where not much information is available about the population and data collection can be efficiently conducted on randomly distributed items, or where the cost of sampling is small enough to make efficiency less important than simplicity. If these conditions do not hold, <a href="/wiki/Stratified_sampling" title="Stratified sampling">stratified sampling</a> or <a href="/wiki/Cluster_sampling" title="Cluster sampling">cluster sampling</a> may be a better choice. </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_between_simple_random_sample_and_other_methods">Relationship between simple random sample and other methods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simple_random_sample&action=edit&section=2" title="Edit section: Relationship between simple random sample and other methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Equal_probability_sampling_(epsem)"><span id="Equal_probability_sampling_.28epsem.29"></span>Equal probability sampling (epsem)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simple_random_sample&action=edit&section=3" title="Edit section: Equal probability sampling (epsem)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A sampling method for which each individual unit has the same chance of being selected is called <b>equal probability sampling</b> (epsem for short). </p><p>Using a simple random sample will always lead to an epsem, but not all epsem samples are SRS. For example, if a teacher has a class arranged in 5 rows of 6 columns and she wants to take a random sample of 5 students she might pick one of the 6 columns at random. This would be an epsem sample but not all subsets of 5 pupils are equally likely here, as only the subsets that are arranged as a single column are eligible for selection. There are also ways of constructing <a href="/wiki/Multistage_sampling" title="Multistage sampling">multistage sampling</a>, that are not srs, while the final sample will be epsem.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> For example, <a href="/wiki/Systematic_sampling" title="Systematic sampling">systematic random sampling</a> produces a sample for which each individual unit has the same probability of inclusion, but different sets of units have different probabilities of being selected. </p><p>Samples that are epsem are <b>self weighting</b>, meaning that the inverse of selection probability for each sample is equal. </p> <div class="mw-heading mw-heading3"><h3 id="Distinction_between_a_systematic_random_sample_and_a_simple_random_sample">Distinction between a <a href="/wiki/Systematic_sampling" title="Systematic sampling">systematic random sample</a> and a simple random sample</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simple_random_sample&action=edit&section=4" title="Edit section: Distinction between a systematic random sample and a simple random sample"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a school with 1000 students, and suppose that a researcher wants to select 100 of them for further study. All their names might be put in a bucket and then 100 names might be pulled out. Not only does each person have an equal chance of being selected, we can also easily calculate the probability (<i>P</i>) of a given person being chosen, since we know the sample size (<i>n</i>) and the population (<i>N</i>): </p><p>1. In the case that any given person can only be selected once (i.e., after selection a person is removed from the selection pool): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P&=1-{\frac {N-1}{N}}\cdot {\frac {N-2}{N-1}}\cdot \cdots \cdot {\frac {N-n}{N-(n-1)}}\\[8pt]&{\stackrel {\text{Canceling:}}{=}}1-{\frac {N-n}{N}}\\[8pt]&={\frac {n}{N}}\\[8pt]&={\frac {100}{1000}}\\[8pt]&=10\%\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="1.1em 1.1em 1.1em 1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>P</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>N</mi> </mfrac> </mrow> <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> <mo>⋅</mo> <mo>⋯</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Canceling:</mtext> </mrow> </mover> </mrow> </mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>−</mo> <mi>n</mi> </mrow> <mi>N</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>N</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>1000</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>10</mn> <mi mathvariant="normal">%</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P&=1-{\frac {N-1}{N}}\cdot {\frac {N-2}{N-1}}\cdot \cdots \cdot {\frac {N-n}{N-(n-1)}}\\[8pt]&{\stackrel {\text{Canceling:}}{=}}1-{\frac {N-n}{N}}\\[8pt]&={\frac {n}{N}}\\[8pt]&={\frac {100}{1000}}\\[8pt]&=10\%\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999bd18a28af58f74ae2b187752f49d64b367762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.671ex; width:44.111ex; height:32.509ex;" alt="{\displaystyle {\begin{aligned}P&=1-{\frac {N-1}{N}}\cdot {\frac {N-2}{N-1}}\cdot \cdots \cdot {\frac {N-n}{N-(n-1)}}\\[8pt]&{\stackrel {\text{Canceling:}}{=}}1-{\frac {N-n}{N}}\\[8pt]&={\frac {n}{N}}\\[8pt]&={\frac {100}{1000}}\\[8pt]&=10\%\end{aligned}}}"></span></dd></dl> <p>2. In the case that any selected person is returned to the selection pool (i.e., can be picked more than once): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=1-\left(1-{\frac {1}{N}}\right)^{n}=1-\left({\frac {999}{1000}}\right)^{100}=0.0952\dots \approx 9.5\%}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>999</mn> <mn>1000</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>100</mn> </mrow> </msup> <mo>=</mo> <mn>0.0952</mn> <mo>⋯</mo> <mo>≈</mo> <mn>9.5</mn> <mi mathvariant="normal">%</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=1-\left(1-{\frac {1}{N}}\right)^{n}=1-\left({\frac {999}{1000}}\right)^{100}=0.0952\dots \approx 9.5\%}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c967b3e8593c38a0a218c6a02a611911eecb0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.77ex; height:6.509ex;" alt="{\displaystyle P=1-\left(1-{\frac {1}{N}}\right)^{n}=1-\left({\frac {999}{1000}}\right)^{100}=0.0952\dots \approx 9.5\%}"></span></dd></dl> <p>This means that every student in the school has in any case approximately a 1 in 10 chance of being selected using this method. Further, any combination of 100 students has the same probability of selection. </p><p>If a systematic pattern is introduced into random sampling, it is referred to as "systematic (random) sampling". An example would be if the students in the school had numbers attached to their names ranging from 0001 to 1000, and we chose a random starting point, e.g. 0533, and then picked every 10th name thereafter to give us our sample of 100 (starting over with 0003 after reaching 0993). In this sense, this technique is similar to cluster sampling, since the choice of the first unit will determine the remainder. This is no longer simple random sampling, because some combinations of 100 students have a larger selection probability than others – for instance, {3, 13, 23, ..., 993} has a 1/10 chance of selection, while {1, 2, 3, ..., 100} cannot be selected under this method. </p> <div class="mw-heading mw-heading2"><h2 id="Sampling_a_dichotomous_population">Sampling a dichotomous population</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simple_random_sample&action=edit&section=5" title="Edit section: Sampling a dichotomous population"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the members of the population come in three kinds, say "blue" "red" and "black", the number of red elements in a sample of given size will vary by sample and hence is a random variable whose distribution can be studied. That distribution depends on the numbers of red and black elements in the full population. For a simple random sample <i>with</i> replacement, the distribution is a <i><a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a></i>. For a simple random sample <i>without</i> replacement, one obtains a <i><a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">hypergeometric distribution</a></i>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Algorithms">Algorithms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simple_random_sample&action=edit&section=6" title="Edit section: Algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several efficient algorithms for simple random sampling have been developed.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> A naive algorithm is the draw-by-draw algorithm where at each step we remove the item at that step from the set with equal probability and put the item in the sample. We continue until we have a sample of desired size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>. The drawback of this method is that it requires random access in the set. </p><p>The selection-rejection algorithm developed by Fan et al. in 1962<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> requires a single pass over data; however, it is a sequential algorithm and requires knowledge of total count of items <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, which is not available in streaming scenarios. </p><p>A very simple random sort algorithm was proved by Sunter in 1977.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The algorithm simply assigns a random number drawn from uniform distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span> as a key to each item, then sorts all items using the key and selects the smallest <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> items. </p><p>J. Vitter in 1985<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> proposed <a href="/wiki/Reservoir_sampling" title="Reservoir sampling">reservoir sampling</a> algorithms, which are widely used. This algorithm does not require knowledge of the size of the population <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in advance, and uses constant space. </p><p>Random sampling can also be accelerated by sampling from the distribution of gaps between samples<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and skipping over the gaps. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simple_random_sample&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Multistage_sampling" title="Multistage sampling">Multistage sampling</a></li> <li><a href="/wiki/Nonprobability_sampling" title="Nonprobability sampling">Nonprobability sampling</a></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Quantitative_marketing_research" title="Quantitative marketing research">Quantitative marketing research</a></li> <li><a href="/wiki/Sampling_design" title="Sampling design">Sampling design</a></li> <li><a href="/wiki/Bernoulli_sampling" title="Bernoulli sampling">Bernoulli sampling</a></li> <li><a href="/wiki/Poisson_sampling" title="Poisson sampling">Poisson sampling</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Simple_random_sample&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFYatesDavid_S._MooreDaren_S._Starnes2008" class="citation book cs1">Yates, Daniel S.; David S. 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Paediatric and perinatal epidemiology 9.2 (1995): 219-224.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAsh2008" class="citation book cs1">Ash, Robert B. (2008). <i>Basic probability theory</i>. Mineola, N.Y: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-46628-6" title="Special:BookSources/978-0-486-46628-6"><bdi>978-0-486-46628-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/190785258">190785258</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+probability+theory&rft.place=Mineola%2C+N.Y&rft.pub=Dover+Publications&rft.date=2008&rft_id=info%3Aoclcnum%2F190785258&rft.isbn=978-0-486-46628-6&rft.aulast=Ash&rft.aufirst=Robert+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+random+sample" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTilleTillé2006" class="citation book cs1">Tille, Yves; Tillé, Yves (2006-01-01). <i>Sampling Algorithms - Springer</i>. Springer Series in Statistics. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F0-387-34240-0">10.1007/0-387-34240-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-30814-2" title="Special:BookSources/978-0-387-30814-2"><bdi>978-0-387-30814-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sampling+Algorithms+-+Springer&rft.series=Springer+Series+in+Statistics&rft.date=2006-01-01&rft_id=info%3Adoi%2F10.1007%2F0-387-34240-0&rft.isbn=978-0-387-30814-2&rft.aulast=Tille&rft.aufirst=Yves&rft.au=Till%C3%A9%2C+Yves&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+random+sample" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeng2013" class="citation journal cs1">Meng, Xiangrui (2013). <a rel="nofollow" class="external text" href="http://jmlr.org/proceedings/papers/v28/meng13a.pdf">"Scalable Simple Random Sampling and Stratified Sampling"</a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of the 30th International Conference on Machine Learning (ICML-13)</i>: 531–539.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+30th+International+Conference+on+Machine+Learning+%28ICML-13%29&rft.atitle=Scalable+Simple+Random+Sampling+and+Stratified+Sampling&rft.pages=531-539&rft.date=2013&rft.aulast=Meng&rft.aufirst=Xiangrui&rft_id=http%3A%2F%2Fjmlr.org%2Fproceedings%2Fpapers%2Fv28%2Fmeng13a.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+random+sample" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFanMullerRezucha1962" class="citation journal cs1">Fan, C. 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(1985-03-01). "Random Sampling with a Reservoir". <i>ACM Trans. Math. Softw</i>. <b>11</b> (1): 37–57. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.138.784">10.1.1.138.784</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F3147.3165">10.1145/3147.3165</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0098-3500">0098-3500</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+Trans.+Math.+Softw.&rft.atitle=Random+Sampling+with+a+Reservoir&rft.volume=11&rft.issue=1&rft.pages=37-57&rft.date=1985-03-01&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.138.784%23id-name%3DCiteSeerX&rft.issn=0098-3500&rft_id=info%3Adoi%2F10.1145%2F3147.3165&rft.aulast=Vitter&rft.aufirst=Jeffrey+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+random+sample" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVitter1984" class="citation journal cs1">Vitter, Jeffrey S. (1984-07-01). "Faster methods for random sampling". <i>Communications of the ACM</i>. <b>27</b> (7): 703–718. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.329.6400">10.1.1.329.6400</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F358105.893">10.1145/358105.893</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-0782">0001-0782</a>.</cite><span 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href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">Statistical sample</a></li> <li><a href="/wiki/Survey_sampling" title="Survey sampling">Sampling for surveys</a> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Random sampling</a></li> <li><a class="mw-selflink selflink">Simple random sampling</a></li> <li><a href="/wiki/Quota_sampling" title="Quota sampling">Quota sampling</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified sampling</a></li> <li><a href="/wiki/Nonprobability_sampling" title="Nonprobability sampling">Nonprobability sampling</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li></ul></li> <li><a href="/wiki/Research_design" title="Research design">Research design</a> <ul><li><a href="/wiki/Longitudinal_study" title="Longitudinal study">Panel study</a></li> <li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Cross-sequential_study" title="Cross-sequential study">Cross-sequential study</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Total_survey_error" title="Total survey error">Survey errors</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_error" title="Sampling error">Sampling error</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li> <li><a href="/wiki/Sampling_bias" title="Sampling bias">Sampling bias</a></li> <li><a href="/wiki/Systematic_errors" class="mw-redirect" title="Systematic errors">Systematic errors</a></li> <li><a href="/wiki/Non-sampling_error" title="Non-sampling error">Non-sampling error</a> <ul><li><a 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class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical data</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Level_of_measurement" title="Level of measurement">Level of measurement</a></li> <li><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></li> <li><a href="/wiki/Exploratory_data_analysis" title="Exploratory data analysis">Exploratory data analysis</a></li> <li><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate statistics</a></li> <li><a href="/wiki/Psychometrics" title="Psychometrics">Psychometrics</a></li> <li><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical models</a> <ul><li><a href="/wiki/Graphical_model" title="Graphical model">Graphical</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Audience_measurement" title="Audience measurement">Audience measurement</a></li> <li><a href="/wiki/Demography" title="Demography">Demography</a></li> <li><a href="/wiki/Market_research" title="Market research">Market research</a></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Public_opinion" title="Public opinion">Public opinion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Major surveys</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_comparative_social_surveys" title="List of comparative social surveys">List of comparative social surveys</a></li> <li><a href="/wiki/Afrobarometer" title="Afrobarometer">Afrobarometer</a></li> <li><a href="/wiki/American_National_Election_Studies" title="American National Election Studies">American National Election Studies</a></li> <li><a href="/wiki/Asian_Barometer_Survey" title="Asian Barometer Survey">Asian Barometer Survey</a></li> <li><a href="/wiki/Comparative_Study_of_Electoral_Systems" title="Comparative Study of Electoral Systems">Comparative Study of Electoral Systems</a></li> <li>Emerson College Polling</li> <li><a href="/wiki/Eurobarometer" title="Eurobarometer">Eurobarometer</a></li> <li><a href="/wiki/European_Social_Survey" title="European Social Survey">European Social Survey</a></li> <li><a href="/wiki/Gallup_(company)" class="mw-redirect" title="Gallup (company)">Gallup Poll</a></li> <li><a href="/wiki/General_Social_Survey" title="General Social Survey">General Social Survey</a></li> <li><a href="/wiki/Household,_Income_and_Labour_Dynamics_in_Australia_Survey" title="Household, Income and Labour Dynamics in Australia Survey">Household, Income and Labour Dynamics in Australia Survey</a></li> <li><a href="/wiki/International_Social_Survey_Programme" title="International Social Survey Programme">International Social Survey</a></li> <li><a href="/wiki/Latinobar%C3%B3metro" title="Latinobarómetro">Latinobarómetro</a></li> <li><a href="/wiki/List_of_household_surveys_in_the_United_States" title="List of household surveys in the United States">List of household surveys in the United States</a></li> <li><a href="/wiki/National_Health_and_Nutrition_Examination_Survey" title="National Health and Nutrition Examination Survey">National Health and Nutrition 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