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mw-ui-icon-wikimedia-expand"></span> <span>Toggle History and formulation of the laws subsection</span> </button> <ul id="toc-History_and_formulation_of_the_laws-sublist" class="vector-toc-list"> <li id="toc-Weber's_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weber's_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Weber's law</span> </div> </a> <ul id="toc-Weber's_law-sublist" class="vector-toc-list"> <li id="toc-Weber_contrast" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Weber_contrast"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Weber contrast</span> </div> </a> <ul id="toc-Weber_contrast-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fechner's_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fechner's_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Fechner's law</span> </div> </a> <ul id="toc-Fechner's_law-sublist" class="vector-toc-list"> <li id="toc-Deriving_Fechner's_law" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Deriving_Fechner's_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Deriving Fechner's law</span> </div> </a> <ul id="toc-Deriving_Fechner's_law-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Types_of_perception" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_perception"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Types of perception</span> </div> </a> <button aria-controls="toc-Types_of_perception-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Types of perception subsection</span> </button> <ul id="toc-Types_of_perception-sublist" class="vector-toc-list"> <li id="toc-Weight_perception" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weight_perception"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Weight perception</span> </div> </a> <ul id="toc-Weight_perception-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sound" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sound"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Sound</span> </div> </a> <ul id="toc-Sound-sublist" class="vector-toc-list"> <li id="toc-Limitation_of_Weber's_law_in_the_auditory_system" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Limitation_of_Weber's_law_in_the_auditory_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Limitation of Weber's law in the auditory system</span> </div> </a> <ul id="toc-Limitation_of_Weber's_law_in_the_auditory_system-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vision" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vision"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Vision</span> </div> </a> <ul id="toc-Vision-sublist" class="vector-toc-list"> <li id="toc-Limitations_of_Weber's_law_in_visual_regularity_perception" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Limitations_of_Weber's_law_in_visual_regularity_perception"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Limitations of Weber's law in visual regularity perception</span> </div> </a> <ul id="toc-Limitations_of_Weber's_law_in_visual_regularity_perception-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitation_of_Weber's_law_at_low_light_levels" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Limitation_of_Weber's_law_at_low_light_levels"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Limitation of Weber's law at low light levels</span> </div> </a> <ul id="toc-Limitation_of_Weber's_law_at_low_light_levels-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Logarithmic_coding_schemes_for_neurons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Logarithmic_coding_schemes_for_neurons"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Logarithmic coding schemes for neurons</span> </div> </a> <button aria-controls="toc-Logarithmic_coding_schemes_for_neurons-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Logarithmic coding schemes for neurons subsection</span> </button> <ul id="toc-Logarithmic_coding_schemes_for_neurons-sublist" class="vector-toc-list"> <li id="toc-Lognormal_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lognormal_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Lognormal distributions</span> </div> </a> <ul id="toc-Lognormal_distributions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other applications</span> </div> </a> <button aria-controls="toc-Other_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Other applications subsection</span> </button> <ul id="toc-Other_applications-sublist" class="vector-toc-list"> <li id="toc-Numerical_cognition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numerical_cognition"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Numerical cognition</span> </div> </a> <ul id="toc-Numerical_cognition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pharmacology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pharmacology"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Pharmacology</span> </div> </a> <ul id="toc-Pharmacology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Public_finance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Public_finance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Public finance</span> </div> </a> <ul id="toc-Public_finance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Emotion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Emotion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Emotion</span> </div> </a> <ul id="toc-Emotion-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"> <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 id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Weber–Fechner law</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 30 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-30" 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">30 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D9%88%D9%8A%D8%A8%D8%B1-%D9%81%D9%8A%D8%B4%D9%86%D8%B1" title="قانون ويبر-فيشنر – Arabic" lang="ar" hreflang="ar" data-title="قانون ويبر-فيشنر" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%BD%D0%B0_%D0%92%D0%B5%D0%B1%D0%B5%D1%80-%D0%A4%D0%B5%D1%85%D0%BD%D0%B5%D1%80" title="Закон на Вебер-Фехнер – Bulgarian" lang="bg" hreflang="bg" data-title="Закон на Вебер-Фехнер" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llei_de_Weber-Fechner" title="Llei de Weber-Fechner – Catalan" lang="ca" hreflang="ca" data-title="Llei de Weber-Fechner" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Weber%C5%AFv%E2%80%93Fechner%C5%AFv_z%C3%A1kon" title="Weberův–Fechnerův zákon – Czech" lang="cs" hreflang="cs" data-title="Weberův–Fechnerův zákon" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Weber%E2%80%93Fechners_lov" title="Weber–Fechners lov – Danish" lang="da" hreflang="da" data-title="Weber–Fechners lov" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Weber-Fechner-Gesetz" title="Weber-Fechner-Gesetz – German" lang="de" hreflang="de" data-title="Weber-Fechner-Gesetz" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ley_de_Weber-Fechner" title="Ley de Weber-Fechner – Spanish" lang="es" hreflang="es" data-title="Ley de Weber-Fechner" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_de_Weber-Fechner" title="Loi de Weber-Fechner – French" lang="fr" hreflang="fr" data-title="Loi de Weber-Fechner" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A0%EB%B2%84-%ED%8E%98%ED%9E%88%EB%84%88%EC%9D%98_%EB%B2%95%EC%B9%99" title="베버-페히너의 법칙 – Korean" lang="ko" hreflang="ko" data-title="베버-페히너의 법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D5%A5%D5%AD%D5%B6%D5%A5%D6%80-%D5%8E%D5%A5%D5%A2%D5%A5%D6%80%D5%AB_%D6%85%D6%80%D5%A5%D5%B6%D6%84" title="Ֆեխներ-Վեբերի օրենք – Armenian" lang="hy" hreflang="hy" data-title="Ֆեխներ-Վեբերի օրենք" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hukum_Weber%E2%80%93Fechner" title="Hukum Weber–Fechner – Indonesian" lang="id" hreflang="id" data-title="Hukum Weber–Fechner" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Legge_di_Weber-Fechner" title="Legge di Weber-Fechner – Italian" lang="it" hreflang="it" data-title="Legge di Weber-Fechner" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%A7_%D7%95%D7%91%D7%A8-%D7%A4%D7%9B%D7%A0%D7%A8" title="חוק ובר-פכנר – Hebrew" lang="he" hreflang="he" data-title="חוק ובר-פכנר" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/V%C4%93bera-Fehnera_likums" title="Vēbera-Fehnera likums – Latvian" lang="lv" hreflang="lv" data-title="Vēbera-Fehnera likums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Weber-t%C3%B6rv%C3%A9ny" title="Weber-törvény – Hungarian" lang="hu" hreflang="hu" data-title="Weber-törvény" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B5%86%E0%B4%AC%E0%B5%BC-%E0%B4%AB%E0%B5%86%E0%B4%95%E0%B5%8D%E0%B4%95%E0%B5%8D%E0%B4%A8%E0%B5%BC_%E0%B4%A8%E0%B4%BF%E0%B4%AF%E0%B4%AE%E0%B4%82" title="വെബർ-ഫെക്ക്നർ നിയമം – Malayalam" lang="ml" hreflang="ml" data-title="വെബർ-ഫെക്ക്നർ നിയമം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wet_van_Weber" title="Wet van Weber – Dutch" lang="nl" hreflang="nl" data-title="Wet van Weber" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%B4%E3%82%A7%E3%83%BC%E3%83%90%E3%83%BC%E2%80%90%E3%83%95%E3%82%A7%E3%83%92%E3%83%8A%E3%83%BC%E3%81%AE%E6%B3%95%E5%89%87" title="ヴェーバー‐フェヒナーの法則 – Japanese" lang="ja" hreflang="ja" data-title="ヴェーバー‐フェヒナーの法則" 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/Weber-Fechners_lov" title="Weber-Fechners lov – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Weber-Fechners lov" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Veber-fexner_qonuni" title="Veber-fexner qonuni – Uzbek" lang="uz" hreflang="uz" data-title="Veber-fexner qonuni" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Laj_%C3%ABd_Weber-Fechner" title="Laj ëd Weber-Fechner – Piedmontese" lang="pms" hreflang="pms" data-title="Laj ëd Weber-Fechner" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Prawo_Webera-Fechnera" title="Prawo Webera-Fechnera – Polish" lang="pl" hreflang="pl" data-title="Prawo Webera-Fechnera" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Lei_de_Weber-Fechner" title="Lei de Weber-Fechner – Portuguese" lang="pt" hreflang="pt" data-title="Lei de Weber-Fechner" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%92%D0%B5%D0%B1%D0%B5%D1%80%D0%B0_%E2%80%94_%D0%A4%D0%B5%D1%85%D0%BD%D0%B5%D1%80%D0%B0" title="Закон Вебера — Фехнера – Russian" lang="ru" hreflang="ru" data-title="Закон Вебера — Фехнера" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Weber-Fechner_law" title="Weber-Fechner law – Simple English" lang="en-simple" hreflang="en-simple" data-title="Weber-Fechner law" data-language-autonym="Simple English" data-language-local-name="Simple English" 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Related laws in the field of psychophysics</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Weber-Fechner_law_demo_-_dots.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Weber-Fechner_law_demo_-_dots.svg/220px-Weber-Fechner_law_demo_-_dots.svg.png" decoding="async" width="220" height="237" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Weber-Fechner_law_demo_-_dots.svg/330px-Weber-Fechner_law_demo_-_dots.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Weber-Fechner_law_demo_-_dots.svg/440px-Weber-Fechner_law_demo_-_dots.svg.png 2x" data-file-width="843" data-file-height="907" /></a><figcaption>An illustration of the Weber–Fechner law. On each side, the lower square contains 10 more dots than the upper one. However the perception is different: On the left side, the difference between upper and lower square is clearly visible. On the right side, the two squares look almost the same.</figcaption></figure> <p>The <b>Weber–Fechner laws</b> are two related <a href="/wiki/Scientific_law" title="Scientific law">scientific laws</a> in the field of <a href="/wiki/Psychophysics" title="Psychophysics">psychophysics</a>, known as <b>Weber's law</b> and <b>Fechner's law</b>. Both relate to human perception, more specifically the relation between the actual change in a physical <a href="/wiki/Stimulus_(physiology)" title="Stimulus (physiology)">stimulus</a> and the perceived change. This includes stimuli to all senses: vision, hearing, taste, touch, and smell. </p><p><a href="/wiki/Ernst_Heinrich_Weber" title="Ernst Heinrich Weber">Ernst Heinrich Weber</a> states that "the minimum increase of stimulus which will produce a perceptible increase of sensation is <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> to the pre-existent stimulus," while <a href="/wiki/Gustav_Fechner" title="Gustav Fechner">Gustav Fechner</a>'s law is an inference from Weber's law (with additional assumptions) which states that the intensity of our sensation increases as the <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> of an increase in energy rather than as rapidly as the increase.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History_and_formulation_of_the_laws">History and formulation of the laws</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=1" title="Edit section: History and formulation of the laws"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Both Weber's law and Fechner's law were formulated by <a href="/wiki/Gustav_Fechner" title="Gustav Fechner">Gustav Theodor Fechner</a> (1801–1887). They were first published in 1860 in the work <i>Elemente der Psychophysik</i> (<i>Elements of Psychophysics</i>). This publication was the first work ever in this field, and where Fechner coined the term <a href="/wiki/Psychophysics" title="Psychophysics">psychophysics</a> to describe the interdisciplinary study of how humans perceive physical magnitudes.<sup id="cite_ref-Fechner1_2-0" class="reference"><a href="#cite_note-Fechner1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> He made the claim that "...psycho-physics is an exact doctrine of the relation of function or dependence between body and soul."<sup id="cite_ref-FOOTNOTEPringle-Pattison1911458_3-0" class="reference"><a href="#cite_note-FOOTNOTEPringle-Pattison1911458-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Weber's_law"><span id="Weber.27s_law"></span>Weber's law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=2" title="Edit section: Weber's law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Ernst_Heinrich_Weber" title="Ernst Heinrich Weber">Ernst Heinrich Weber</a> (1795–1878) was one of the first persons to approach the study of the human response to a <a href="/wiki/Stimulus_(physiology)" title="Stimulus (physiology)">physical stimulus</a> in a <a href="/wiki/Quantitative_research" title="Quantitative research">quantitative</a> fashion. Fechner was a student of Weber and named his first law in honor of his mentor, since it was Weber who had conducted the experiments needed to formulate the law.<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> </p><p>Fechner formulated several versions of the law, all communicating the same idea. One formulation states: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p> Simple differential sensitivity is inversely proportional to the size of the components of the difference; relative differential sensitivity remains the same regardless of size.<sup id="cite_ref-Fechner1_2-1" class="reference"><a href="#cite_note-Fechner1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>What this means is that the perceived change in stimuli is inversely proportional to the initial stimuli. </p><p>Weber's law also incorporates the <a href="/wiki/Just-noticeable_difference" title="Just-noticeable difference">just-noticeable difference</a> (JND). This is the smallest change in stimuli that can be perceived. As stated above, the JND <span class="texhtml mvar" style="font-style:italic;">dS</span> is proportional to the initial stimuli intensity <span class="texhtml mvar" style="font-style:italic;">S</span>. Mathematically, it can be described as <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 dS=K\cdot S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mi>K</mi> <mo>⋅</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS=K\cdot S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4649ecabc1de1f105ac220f11139f60e2671901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.058ex; height:2.176ex;" alt="{\displaystyle dS=K\cdot S}"></span> where <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is the reference stimulus and <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/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}"></span> is a constant.<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> It may be written as <span class="texhtml">Ψ = <i>k</i> log<i>S</i></span>, with <span class="texhtml">Ψ</span> being the sensation, <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> being a constant, and <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> being the physical intensity of the stimulus. </p><p>Weber's law always fails at low intensities, near and below the absolute detection threshold, and often also at high intensities, but may be approximately true across a wide middle range of intensities.<sup id="cite_ref-norris_6-0" class="reference"><a href="#cite_note-norris-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Weber_contrast">Weber contrast</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=3" title="Edit section: Weber contrast"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although Weber's law includes a statement of the proportionality of a perceived change to initial stimuli, Weber only refers to this as a rule of thumb regarding human perception. It was Fechner who formulated this statement as a mathematical expression referred to as <b>Weber contrast</b>.<sup id="cite_ref-Fechner1_2-2" class="reference"><a href="#cite_note-Fechner1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Fechner2_7-0" class="reference"><a href="#cite_note-Fechner2-7"><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><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> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dp={\frac {dS}{S}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mi>S</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dp={\frac {dS}{S}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6771613eaefa7361a6b8c397ea6bdc487063a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:9.422ex; height:5.509ex;" alt="{\displaystyle dp={\frac {dS}{S}}\,\!}"></span> Weber contrast is not part of Weber's law.<sup id="cite_ref-Fechner1_2-3" class="reference"><a href="#cite_note-Fechner1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Fechner2_7-1" class="reference"><a href="#cite_note-Fechner2-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Fechner's_law"><span id="Fechner.27s_law"></span>Fechner's law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=4" title="Edit section: Fechner's law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fechner noticed in his own studies that different individuals have different sensitivity to certain stimuli. For example, the ability to perceive differences in light intensity could be related to how good that individual's vision is.<sup id="cite_ref-Fechner1_2-4" class="reference"><a href="#cite_note-Fechner1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> He also noted that how the human sensitivity to stimuli changes depends on which sense is affected. He used this to formulate another version of Weber's law that he named <i>die Maßformel</i>, the "measurement formula". Fechner's law states that the subjective sensation is proportional to the logarithm of the stimulus intensity. According to this law, human perceptions of sight and sound work as follows: Perceived loudness/brightness is proportional to logarithm of the actual intensity measured with an accurate nonhuman instrument.<sup id="cite_ref-Fechner2_7-2" class="reference"><a href="#cite_note-Fechner2-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=k\ln {\frac {S}{S_{0}}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=k\ln {\frac {S}{S_{0}}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f147c0889bee33aebccdcd86e0417eae0cb36357" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; margin-right: -0.387ex; width:11.984ex; height:5.843ex;" alt="{\displaystyle p=k\ln {\frac {S}{S_{0}}}\,\!}"></span> </p><p>The relationship between stimulus and perception is <a href="/wiki/Logarithmic_scale" title="Logarithmic scale">logarithmic</a>. This logarithmic relationship means that if a stimulus varies as a <a href="/wiki/Geometric_progression" title="Geometric progression">geometric progression</a> (i.e., multiplied by a fixed factor), the corresponding perception is altered in an <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a> (i.e., in additive constant amounts). For example, if a stimulus is tripled in strength (i.e., <span class="nowrap">3 × 1</span>), the corresponding perception may be two times as strong as its original value (i.e., <span class="nowrap">1 + 1</span>). If the stimulus is again tripled in strength (i.e., <span class="nowrap">3 × 3 × 1</span>), the corresponding perception will be three times as strong as its original value (i.e., <span class="nowrap">1 + 1 + 1</span>). Hence, for multiplications in stimulus strength, the strength of perception only adds. The mathematical derivations of the torques on a simple beam balance produce a description that is strictly compatible with Weber's law.<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><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> </p><p>Since Weber's law fails at low intensity, so does Fechner's law.<sup id="cite_ref-norris_6-1" class="reference"><a href="#cite_note-norris-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>An early reference to "Fechner's ... law" was in 1875 by <a href="/wiki/Ludimar_Hermann" title="Ludimar Hermann">Ludimar Hermann</a> in <i>Elements of Human Physiology</i>.<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> </p> <div class="mw-heading mw-heading4"><h4 id="Deriving_Fechner's_law"><span id="Deriving_Fechner.27s_law"></span>Deriving Fechner's law</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=5" title="Edit section: Deriving Fechner's law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fechner's law is a mathematical derivation of Weber contrast. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dp=k{\frac {dS}{S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>p</mi> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>S</mi> </mrow> <mi>S</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dp=k{\frac {dS}{S}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/228de662aeb2c5c290c38acffffab3ee784efd65" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.246ex; height:5.509ex;" alt="{\displaystyle dp=k{\frac {dS}{S}}}"></span> </p><p>Integrating the mathematical expression for Weber contrast gives: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=k\ln {S}+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=k\ln {S}+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82f903319fb33fb2548812b90dc1dcfbf50d63c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.388ex; height:2.509ex;" alt="{\displaystyle p=k\ln {S}+C}"></span> </p><p>where <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> is a <a href="/wiki/Constant_of_integration" title="Constant of integration">constant of integration</a> and <i>ln</i> is the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>. </p><p>To solve for <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>, assume that the perceived stimulus becomes zero at some threshold stimulus <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 S_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe0ac45a38c4437bd2689a14ec434cd499e7e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{0}}"></span>. Using this as a constraint, set <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e6ac10fa45fb984d886065f959a6bdd467b5e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=0}"></span> and <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 S=S_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=S_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa07dbad9716a470adce6814bffc39ffe00eb4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.077ex; height:2.509ex;" alt="{\displaystyle S=S_{0}}"></span>. This gives: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=-k\ln {S_{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=-k\ln {S_{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b68e54240c0a07472523cc17b79cfb4b41df7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.077ex; height:2.509ex;" alt="{\displaystyle C=-k\ln {S_{0}}}"></span> </p><p>Substituting <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> in the integrated expression for Weber's law, the expression can be written as: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=k\ln {\frac {S}{S_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=k\ln {\frac {S}{S_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577d1610c2a6f0f98585d921a8751b93aca678de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:11.597ex; height:5.843ex;" alt="{\displaystyle p=k\ln {\frac {S}{S_{0}}}}"></span> </p><p>The constant k is sense-specific and must be determined depending on the sense and type of stimulus.<sup id="cite_ref-Fechner2_7-3" class="reference"><a href="#cite_note-Fechner2-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_perception">Types of perception</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=6" title="Edit section: Types of perception"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Weber and Fechner conducted research on differences in light intensity and the perceived difference in weight.<sup id="cite_ref-Fechner1_2-5" class="reference"><a href="#cite_note-Fechner1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Other sense modalities provide only mixed support for either Weber's law or Fechner's law. </p> <div class="mw-heading mw-heading3"><h3 id="Weight_perception">Weight perception</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=7" title="Edit section: Weight perception"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Weber found that the <a href="/wiki/Just_noticeable_difference" class="mw-redirect" title="Just noticeable difference">just noticeable difference</a> (JND) between two weights was approximately proportional to the weights. Thus, if the weight of 105 g can (only just) be distinguished from that of 100 g, the JND (or differential threshold) is 5 g. If the mass is doubled, the differential threshold also doubles to 10 g, so that 210 g can be distinguished from 200 g. In this example, a weight (any weight) seems to have to increase by 5% for someone to be able to reliably detect the increase, and this minimum required fractional increase (of 5/100 of the original weight) is referred to as the "Weber fraction" for detecting changes in weight. Other discrimination tasks, such as detecting changes in brightness, or in tone height (pure tone frequency), or in the length of a line shown on a screen, may have different Weber fractions, but they all obey Weber's law in that observed values need to change by at least some small but constant proportion of the current value to ensure human observers will reliably be able to detect that change. </p><p>Fechner did not conduct any experiments on how perceived heaviness increased with the mass of the stimulus. Instead, he assumed that all JNDs are subjectively equal, and argued mathematically that this would produce a logarithmic relation between the stimulus intensity and the sensation. These assumptions have both been questioned.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Following the work of S. S. Stevens, many researchers came to believe in the 1960s that the <a href="/wiki/Stevens%27s_power_law" title="Stevens's power law">Stevens's power law</a> was a more general psychophysical principle than Fechner's logarithmic law. </p> <div class="mw-heading mw-heading3"><h3 id="Sound">Sound</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=8" title="Edit section: Sound"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Weber's law does not quite hold for <a href="/wiki/Loudness" title="Loudness">loudness</a>. It is a fair approximation for higher intensities, but not for lower amplitudes.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Limitation_of_Weber's_law_in_the_auditory_system"><span id="Limitation_of_Weber.27s_law_in_the_auditory_system"></span>Limitation of Weber's law in the auditory system</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=9" title="Edit section: Limitation of Weber's law in the auditory system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Weber's law does not hold at perception of higher intensities. Intensity discrimination improves at higher intensities. The first demonstration of the phenomena was presented by Riesz in 1928, in Physical Review. This deviation of the Weber's law is known as the "near miss" of the Weber's law. This term was coined by McGill and Goldberg in their paper of 1968 in Perception & Psychophysics. Their study consisted of intensity discrimination in pure tones. Further studies have shown that the near miss is observed in noise stimuli as well. Jesteadt et al. (1977)<sup id="cite_ref-Jesteadt_Walt,_Wier_Craig_C.,_Green_David_M._1977_169–77_16-0" class="reference"><a href="#cite_note-Jesteadt_Walt,_Wier_Craig_C.,_Green_David_M._1977_169–77-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> demonstrated that the near miss holds across all the frequencies, and that the intensity discrimination is not a function of frequency, and that the change in discrimination with level can be represented by a single function across all frequencies: <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 \Delta I/I=0.463{(I/I_{0})}^{-0.072}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> <mo>=</mo> <mn>0.463</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>0.072</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta I/I=0.463{(I/I_{0})}^{-0.072}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8edb87e8d7c7f205fa8c90557474e6c1a7b9bfd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.314ex; height:3.343ex;" alt="{\displaystyle \Delta I/I=0.463{(I/I_{0})}^{-0.072}}"></span>.<sup id="cite_ref-Jesteadt_Walt,_Wier_Craig_C.,_Green_David_M._1977_169–77_16-1" class="reference"><a href="#cite_note-Jesteadt_Walt,_Wier_Craig_C.,_Green_David_M._1977_169–77-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Vision">Vision</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=10" title="Edit section: Vision"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:408px;max-width:408px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Before_prism_lighting.jpg" class="mw-file-description"><img alt="Woodcut? of a deep, shadowy room with light only near the window at the far end" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Before_prism_lighting.jpg/200px-Before_prism_lighting.jpg" decoding="async" width="200" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/d/d3/Before_prism_lighting.jpg 1.5x" data-file-width="256" data-file-height="233" /></a></span></div><div class="thumbcaption">Uneven light from a window.</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:After_prism_lighting.jpg" class="mw-file-description"><img alt="Woodcut? of the same room, but much more evenly lit, with diffuse light in the former shadows" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/After_prism_lighting.jpg/200px-After_prism_lighting.jpg" decoding="async" width="200" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/a2/After_prism_lighting.jpg 1.5x" data-file-width="251" data-file-height="233" /></a></span></div><div class="thumbcaption">The same light, redistributed by <a href="/wiki/Prism_tile" class="mw-redirect" title="Prism tile">prism tiles</a> in the window.</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Because of the ~logarithmic perception of light levels, if light is redistributed from the brightest parts of a room to the dimmest, the room appears brighter overall, and more space can be given a useful and comfortable level of illumination.</div></div></div></div> <p>The eye senses <a href="/wiki/Brightness" title="Brightness">brightness</a> approximately logarithmically over a moderate range and <a href="/wiki/Stellar_magnitude" class="mw-redirect" title="Stellar magnitude">stellar magnitude</a> is measured on a logarithmic scale.<sup id="cite_ref-bhatia_17-0" class="reference"><a href="#cite_note-bhatia-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> This magnitude scale was invented by the ancient Greek astronomer <a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a> in about 150 B.C. He ranked the stars he could see in terms of their brightness, with 1 representing the brightest down to 6 representing the faintest, though now the scale has been extended beyond these limits; an increase in 5 magnitudes corresponds to a decrease in brightness by a factor of 100.<sup id="cite_ref-bhatia_17-1" class="reference"><a href="#cite_note-bhatia-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Modern researchers have attempted to incorporate such perceptual effects into mathematical models of vision.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Limitations_of_Weber's_law_in_visual_regularity_perception"><span id="Limitations_of_Weber.27s_law_in_visual_regularity_perception"></span>Limitations of Weber's law in visual regularity perception</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=11" title="Edit section: Limitations of Weber's law in visual regularity perception"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Perception of <a href="/w/index.php?title=Glass_patterns&action=edit&redlink=1" class="new" title="Glass patterns (page does not exist)">Glass patterns</a><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> and mirror symmetries in the presence of noise follows Weber's law in the middle range of regularity-to-noise ratios (<i>S</i>), but in both outer ranges, sensitivity to variations is disproportionally lower. As Maloney, Mitchison, & Barlow (1987)<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> showed for Glass patterns, and as van der Helm (2010)<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> showed for mirror symmetries, perception of these visual regularities in the whole range of regularity-to-noise ratios follows the law <i>p</i> = <i>g</i>/(2+1/<i>S</i>) with parameter <i>g</i> to be estimated using experimental data. </p> <div class="mw-heading mw-heading4"><h4 id="Limitation_of_Weber's_law_at_low_light_levels"><span id="Limitation_of_Weber.27s_law_at_low_light_levels"></span>Limitation of Weber's law at low light levels</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=12" title="Edit section: Limitation of Weber's law at low light levels"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:CrumeyFig3.pdf" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/CrumeyFig3.pdf/page1-220px-CrumeyFig3.pdf.jpg" decoding="async" width="220" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/CrumeyFig3.pdf/page1-330px-CrumeyFig3.pdf.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/CrumeyFig3.pdf/page1-440px-CrumeyFig3.pdf.jpg 2x" data-file-width="1350" data-file-height="1066" /></a><figcaption>Threshold increment versus background luminance for various target diameters (in arcmin). Data from tables 4 and 8 of Blackwell (1946), plotted in Crumey (2014).</figcaption></figure> <p>For vision, Weber's law implies constancy of <a href="/wiki/Luminance" title="Luminance">luminance</a> <a href="/wiki/Contrast_(vision)" title="Contrast (vision)">contrast</a>. Suppose a target object is set against a background luminance <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 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></span><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}"></span>. In order to be just visible, the target must be brighter or fainter than the background by some small amount <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 \Delta B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dcbfb0d19f9ccb6d536a630cdb0b478dad636f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.7ex; height:2.176ex;" alt="{\displaystyle \Delta B}"></span>. The Weber contrast is defined as <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 C=\Delta B/B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\Delta B/B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c50beb12fcd9eb5aaad7a0c9521d52180ee478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.491ex; height:2.843ex;" alt="{\displaystyle C=\Delta B/B}"></span>, and Weber's law says that <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> should be constant for all <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 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></span><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}"></span>. </p><p>Human vision follows Weber's law closely at normal daylight levels (i.e. in the <a href="/wiki/Photopic_vision" title="Photopic vision">photopic range</a>) but begins to break down at twilight levels (the <a href="/wiki/Mesopic_vision" title="Mesopic vision">mesopic</a> range) and is completely inapplicable at low light levels (<a href="/wiki/Scotopic_vision" title="Scotopic vision">scotopic vision</a>). This can be seen in data collected by Blackwell<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> and plotted by <a href="/wiki/Andrew_Crumey" title="Andrew Crumey">Crumey</a>,<sup id="cite_ref-crumey_24-0" class="reference"><a href="#cite_note-crumey-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> showing threshold increment <span class="texhtml">log</span><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 \Delta B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dcbfb0d19f9ccb6d536a630cdb0b478dad636f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.7ex; height:2.176ex;" alt="{\displaystyle \Delta B}"></span> versus background luminance <span class="texhtml">log</span><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 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></span><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}"></span> for various targets sizes. At daylight levels, the curves are approximately straight with slope 1, i.e. <span class="texhtml">log</span><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 \Delta B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dcbfb0d19f9ccb6d536a630cdb0b478dad636f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.7ex; height:2.176ex;" alt="{\displaystyle \Delta B}"></span> = <span class="texhtml">log</span><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 B+const.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>+</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B+const.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1acae552a7e3198d0d6beb93f8a2a9e86dd0c2f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.71ex; height:2.343ex;" alt="{\displaystyle B+const.}"></span>, implying <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 C=\Delta B/B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\Delta B/B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c50beb12fcd9eb5aaad7a0c9521d52180ee478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.491ex; height:2.843ex;" alt="{\displaystyle C=\Delta B/B}"></span> is constant. At the very darkest background levels (<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 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></span><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}"></span> ≲ 10<sup>− 5</sup> cd m<sup>−2</sup>, approximately 25 mag arcsec<sup>−2</sup>)<sup id="cite_ref-crumey_24-1" class="reference"><a href="#cite_note-crumey-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> the curves are flat - this is where the only visual perception is the observer's own neural noise (<a href="/wiki/Eigengrau" title="Eigengrau">'dark light'</a>). In the intermediate range, a portion can be approximated by the <a href="/wiki/De_Vries_-_Rose_Law" class="mw-redirect" title="De Vries - Rose Law">De Vries - Rose law</a>, related to <a href="/wiki/Ricco%27s_law" title="Ricco's law">Ricco's law</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Logarithmic_coding_schemes_for_neurons">Logarithmic coding schemes for neurons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=13" title="Edit section: Logarithmic coding schemes for neurons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Lognormal_distributions">Lognormal distributions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=14" title="Edit section: Lognormal distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Activation of neurons by sensory stimuli in many parts of the brain is by a proportional law: neurons change their spike rate by about 10–30%, when a stimulus (e.g. a natural scene for <a href="/wiki/Visual_perception" title="Visual perception">vision</a>) has been applied. However, as Scheler (2017)<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> showed, the population distribution of the <a href="/w/index.php?title=Intrinsic_excitability&action=edit&redlink=1" class="new" title="Intrinsic excitability (page does not exist)">intrinsic excitability</a> or gain of a neuron is a <a href="/wiki/Heavy_tail_distribution" class="mw-redirect" title="Heavy tail distribution">heavy tail distribution</a>, more precisely a <a href="/wiki/Lognormal" class="mw-redirect" title="Lognormal">lognormal</a> shape, which is equivalent to a logarithmic coding scheme. Neurons may therefore spike with 5–10 fold different mean rates. Obviously, this increases the dynamic range of a neuronal population, while stimulus-derived changes remain small and linear proportional. </p><p>An analysis<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> of the length of comments in internet discussion boards across several languages shows that comment lengths obey the lognormal distribution with great precision. The authors explain the distribution as a manifestation of the Weber–Fechner law. </p> <div class="mw-heading mw-heading2"><h2 id="Other_applications">Other applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=15" title="Edit section: Other applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Weber–Fechner law has been applied in other fields of research than just the human senses. </p> <div class="mw-heading mw-heading3"><h3 id="Numerical_cognition">Numerical cognition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=16" title="Edit section: Numerical cognition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Psychological studies show that it becomes increasingly difficult to discriminate between two numbers as the difference between them decreases. This is called the <i>distance effect</i>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> This is important in areas of magnitude estimation, such as dealing with large scales and estimating distances. It may also play a role in explaining why consumers neglect to shop around to save a small percentage on a large purchase, but will shop around to save a large percentage on a small purchase which represents a much smaller absolute dollar amount.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Pharmacology">Pharmacology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=17" title="Edit section: Pharmacology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It has been hypothesized that <a href="/wiki/Dose-response_relationship" class="mw-redirect" title="Dose-response relationship">dose-response</a> relationships can follow Weber's Law<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> which suggests this law – which is often applied at the sensory level – originates from underlying <a href="/wiki/Chemoreceptor" title="Chemoreceptor">chemoreceptor</a> responses to <a href="/wiki/Cell_signaling" title="Cell signaling">cellular signaling</a> dose relationships within the body. Dose response can be related to the <a href="/wiki/Hill_equation_(biochemistry)" title="Hill equation (biochemistry)">Hill equation</a>, which is closer to a power law. </p> <div class="mw-heading mw-heading3"><h3 id="Public_finance">Public finance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=18" title="Edit section: Public finance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a new branch of the literature on public finance hypothesizing that the Weber–Fechner law can explain the increasing levels of public expenditures in mature democracies. Election after election, voters demand more public goods to be effectively impressed; therefore, politicians try to increase the magnitude of this "signal" of competence – the size and composition of public expenditures – in order to collect more votes.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Emotion">Emotion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=19" title="Edit section: Emotion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Preliminary research has found that pleasant emotions adhere to Weber’s Law, with accuracy in judging their intensity decreasing as pleasantness increases. However, this pattern wasn't observed for unpleasant emotions, suggesting a survival-related need for accurately discerning high-intensity negative emotions.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </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=Weber%E2%80%93Fechner_law&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Human_nature" title="Human nature">Human nature</a></li> <li><a href="/wiki/Level_(logarithmic_quantity)" title="Level (logarithmic quantity)">Level (logarithmic quantity)</a></li> <li><a href="/wiki/Nervous_system" title="Nervous system">Nervous system</a></li> <li><a href="/wiki/Ricco%27s_law" title="Ricco's law">Ricco's law</a></li> <li><a href="/wiki/Stevens%27s_power_law" title="Stevens's power law">Stevens's power law</a></li> <li><a href="/wiki/Sone" title="Sone">Sone</a></li> <li><a href="/wiki/Neural_coding" title="Neural coding">Neural coding</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=Weber%E2%80%93Fechner_law&action=edit&section=21" 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">Jeans, James (1968/1937). <i>Science & Music</i>, pp. 222, 224. Dover Publications.</span> </li> <li id="cite_note-Fechner1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fechner1_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fechner1_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Fechner1_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Fechner1_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Fechner1_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Fechner1_2-5"><sup><i><b>f</b></i></sup></a></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="CITEREFFechner1966" class="citation book cs1">Fechner, Gustav Theodor (1966) [First published .1860]. Howes, D H; Boring, E G (eds.). <i>Elements of psychophysics</i> [<i>Elemente der Psychophysik</i>]. Vol. 1. Translated by Adler, H E. United States of America: Holt, Rinehart and Winston.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+psychophysics&rft.place=United+States+of+America&rft.pub=Holt%2C+Rinehart+and+Winston&rft.date=1966&rft.aulast=Fechner&rft.aufirst=Gustav+Theodor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeber%E2%80%93Fechner+law" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEPringle-Pattison1911458-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPringle-Pattison1911458_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPringle-Pattison1911">Pringle-Pattison 1911</a>, p. 458.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Ross, H.E. and Murray, D. J.(Ed. and Transl.) (1996)<i>E.H.Weber on the tactile senses</i>. 2nd ed. Hove: Erlbaum (UK) Taylor & Francis;</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKandelJessellSchwartzSiegelbaum2013" class="citation book cs1">Kandel, Eric R.; Jessell, Thomas M.; Schwartz, James H.; Siegelbaum, Steven A.; Hudspeth, A. J. (2013). <i>Principles of neural science</i>. Kandel, Eric R. (5th ed.). 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Murray Lyon (1923). <a rel="nofollow" class="external text" href="http://jpet.aspetjournals.org/content/21/4/229.abstract">"Does the reaction to adrenalin obey Weber's law?"</a>. <i>Journal of Pharmacology and Experimental Therapeutics</i>. <b>21</b> (4): 229–235.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Pharmacology+and+Experimental+Therapeutics&rft.atitle=Does+the+reaction+to+adrenalin+obey+Weber%27s+law%3F&rft.volume=21&rft.issue=4&rft.pages=229-235&rft.date=1923&rft.au=D.+Murray+Lyon&rft_id=http%3A%2F%2Fjpet.aspetjournals.org%2Fcontent%2F21%2F4%2F229.abstract&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeber%E2%80%93Fechner+law" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMourao,_P.2012" class="citation journal cs1">Mourao, P. (2012). <a rel="nofollow" class="external text" href="https://doi.org/10.18267%2Fj.pep.425">"The Weber-Fechner Law and Public Expenditures Impact to the Win-Margins at Parliamentary Elections"</a>. <i>Prague Economic Papers</i>. <b>21</b> (3): 290–308. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.18267%2Fj.pep.425">10.18267/j.pep.425</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Prague+Economic+Papers&rft.atitle=The+Weber-Fechner+Law+and+Public+Expenditures+Impact+to+the+Win-Margins+at+Parliamentary+Elections&rft.volume=21&rft.issue=3&rft.pages=290-308&rft.date=2012&rft_id=info%3Adoi%2F10.18267%2Fj.pep.425&rft.au=Mourao%2C+P.&rft_id=https%3A%2F%2Fdoi.org%2F10.18267%252Fj.pep.425&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeber%E2%80%93Fechner+law" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHedrih2023" class="citation web cs1">Hedrih, Vladimir (2023-12-25). <a rel="nofollow" class="external text" href="https://www.psypost.org/2023/12/intriguing-new-study-reveals-positive-emotions-follow-a-classical-psychophysical-law-215155">"Intriguing new study reveals positive emotions follow a classical psychophysical law"</a>. <i>PsyPost</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-12-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PsyPost&rft.atitle=Intriguing+new+study+reveals+positive+emotions+follow+a+classical+psychophysical+law&rft.date=2023-12-25&rft.aulast=Hedrih&rft.aufirst=Vladimir&rft_id=https%3A%2F%2Fwww.psypost.org%2F2023%2F12%2Fintriguing-new-study-reveals-positive-emotions-follow-a-classical-psychophysical-law-215155&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeber%E2%80%93Fechner+law" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=22" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRies1962" class="citation book cs1 cs1-prop-foreign-lang-source">Ries, Clemens (1962). <i>Normung nach Normzahlen</i> [<i>Standardization by preferred numbers</i>] (in German) (1 ed.). Berlin l: <a href="/w/index.php?title=Duncker_%26_Humblot_Verlag&action=edit&redlink=1" class="new" title="Duncker & Humblot Verlag (page does not exist)">Duncker & Humblot Verlag</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://de.wikipedia.org/wiki/Duncker_%26_Humblot_Verlag" class="extiw" title="de:Duncker & Humblot Verlag">de</a>]</span>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-42801242-8" title="Special:BookSources/978-3-42801242-8"><bdi>978-3-42801242-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Normung+nach+Normzahlen&rft.place=Berlin+l&rft.edition=1&rft.pub=Duncker+%26+Humblot+Verlag%3Cspan+class%3D%22noprint%22+style%3D%22font-size%3A85%25%3B+font-style%3A+normal%3B+%22%3E+%26%2391%3Bde%26%2393%3B%3C%2Fspan%3E&rft.date=1962&rft.isbn=978-3-42801242-8&rft.aulast=Ries&rft.aufirst=Clemens&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeber%E2%80%93Fechner+law" class="Z3988"></span> (135 pages)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaulin2007" class="citation book cs1 cs1-prop-foreign-lang-source">Paulin, Eugen (2007-09-01). <a rel="nofollow" class="external text" href="http://www.rechenschieber.org/Normzahlen.pdf"><i>Logarithmen, Normzahlen, Dezibel, Neper, Phon – natürlich verwandt!</i></a> [<i>Logarithms, preferred numbers, decibel, neper, phon – naturally related!</i>] <span class="cs1-format">(PDF)</span> (in German). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161218223050/http://www.rechenschieber.org/Normzahlen.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2016-12-18<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-12-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logarithmen%2C+Normzahlen%2C+Dezibel%2C+Neper%2C+Phon+%E2%80%93+nat%C3%BCrlich+verwandt%21&rft.date=2007-09-01&rft.aulast=Paulin&rft.aufirst=Eugen&rft_id=http%3A%2F%2Fwww.rechenschieber.org%2FNormzahlen.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeber%E2%80%93Fechner+law" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Weber%E2%80%93Fechner_law&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikisource-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/15px-Wikisource-logo.svg.png" decoding="async" width="15" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/23px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/30px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></a></span> Texts on Wikisource: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1">"<a href="https://en.wikisource.org/wiki/The_Encyclopedia_Americana_(1920)/Weber%27s_Law" class="extiw" title="s:The Encyclopedia Americana (1920)/Weber's Law">Weber's Law</a>". <i><a href="/wiki/Encyclopedia_Americana" title="Encyclopedia Americana">Encyclopedia Americana</a></i>. 1920.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Weber%27s+Law&rft.btitle=Encyclopedia+Americana&rft.date=1920&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeber%E2%80%93Fechner+law" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPringle-Pattison1911" class="citation encyclopaedia cs1"><a href="/wiki/Andrew_Seth_Pringle-Pattison" title="Andrew Seth Pringle-Pattison">Pringle-Pattison, Andrew Seth</a> (1911). "<a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Weber%27s_Law" class="extiw" title="s:1911 Encyclopædia Britannica/Weber's Law">Weber's Law</a>". In <a href="/wiki/Hugh_Chisholm" title="Hugh Chisholm">Chisholm, Hugh</a> (ed.). <i><a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a></i>. Vol. 28 (11th ed.). Cambridge University Press. pp. 458–459.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Weber%27s+Law&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.pages=458-459&rft.edition=11th&rft.pub=Cambridge+University+Press&rft.date=1911&rft.aulast=Pringle-Pattison&rft.aufirst=Andrew+Seth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeber%E2%80%93Fechner+law" class="Z3988"></span> This contains a detailed contemporaneous description of the law's applicability, and several additional references.</li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFry2018" class="citation web cs1"><a href="/wiki/Hannah_Fry" title="Hannah Fry">Fry, Hannah</a> (25 May 2018). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=hHG8io5qIU8&t=480s">"Weber's Law"</a> <span class="cs1-format">(video)</span>. <i>YouTube</i>. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211212/hHG8io5qIU8">Archived</a> from the original on 2021-12-12<span class="reference-accessdate">. Retrieved <span class="nowrap">28 May</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=YouTube&rft.atitle=Weber%27s+Law&rft.date=2018-05-25&rft.aulast=Fry&rft.aufirst=Hannah&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DhHG8io5qIU8%26t%3D480s&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWeber%E2%80%93Fechner+law" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Weber–Fechner_law&oldid=1247920468">https://en.wikipedia.org/w/index.php?title=Weber–Fechner_law&oldid=1247920468</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Perception" title="Category:Perception">Perception</a></li><li><a href="/wiki/Category:Behavioral_concepts" title="Category:Behavioral concepts">Behavioral concepts</a></li><li><a href="/wiki/Category:Psychophysics" title="Category:Psychophysics">Psychophysics</a></li><li><a href="/wiki/Category:Mathematical_psychology" title="Category:Mathematical psychology">Mathematical psychology</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">CS1 maint: location missing publisher</a></li><li><a href="/wiki/Category:CS1:_long_volume_value" title="Category:CS1: long volume value">CS1: long volume value</a></li><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li><li><a href="/wiki/Category:Wikipedia_articles_incorporating_a_citation_from_the_Encyclopedia_Americana_with_a_Wikisource_reference" title="Category:Wikipedia articles incorporating a citation from the Encyclopedia Americana with a Wikisource reference">Wikipedia articles incorporating a citation from the Encyclopedia Americana with a Wikisource reference</a></li><li><a href="/wiki/Category:Wikipedia_articles_incorporating_a_citation_from_the_1911_Encyclopaedia_Britannica_with_Wikisource_reference" title="Category:Wikipedia articles incorporating a citation from the 1911 Encyclopaedia Britannica with Wikisource reference">Wikipedia articles incorporating a citation from the 1911 Encyclopaedia Britannica with Wikisource reference</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 26 September 2024, at 19:05<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-07-20T19:22:16Z","dateModified":"2024-09-26T19:05:07Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/f\/fe\/Weber-Fechner_law_demo_-_dots.svg","headline":"psychophysics law that the minimum perceptible increase of stimulus is proportional to the stimulus and that intensity of our sensation increases logarithmically"}</script> </body> </html>