{"id":25925,"date":"2022-02-14T11:00:29","date_gmt":"2022-02-14T04:00:29","guid":{"rendered":"https:\/\/tom.ji42.com\/?p=25925"},"modified":"2022-02-14T11:00:29","modified_gmt":"2022-02-14T04:00:29","slug":"degrees-of-freedom","status":"publish","type":"post","link":"https:\/\/tom.tomwork.net\/?p=25925","title":{"rendered":"Degrees of Freedom"},"content":{"rendered":"<div class=\"loc grid-content\">\n<div id=\"article-body_1-0\" class=\"comp right-rail article-body mntl-block\">\n<div id=\"mntl-sc-page_1-0\" class=\"comp article-body-content mntl-sc-page mntl-block\" data-sc-sticky-offset=\"80\" data-sc-ad-label-height=\"0\" data-sc-ad-track-spacing=\"90\" data-sc-min-track-height=\"250\" data-sc-max-track-height=\"600\" data-sc-breakpoint=\"40em\" data-sc-load-immediate=\"2\" data-sc-content-positions=\"[300, 1, 1, 1250, 1, 1]\" data-bind-scroll-on-start=\"true\">\n<h2 id=\"mntl-sc-block_1-0\" class=\"comp mntl-sc-block finance-sc-block-heading mntl-sc-block-heading\"><span class=\"mntl-sc-block-heading__text\">What Are Degrees of Freedom?<\/span><\/h2>\n<p id=\"mntl-sc-block_1-0-1\" class=\"comp mntl-sc-block finance-sc-block-html mntl-sc-block-html\">Degrees of freedom refers to the maximum number of logically independent values, which are values that have the freedom to vary, in the data sample.<\/p>\n<div id=\"mntl-sc-block_1-0-2\" class=\"comp mntl-sc-block mntl-sc-block-adslot mntl-block\">\n<div id=\"mntl-block_21-0\" class=\"comp mntl-block\"><\/div>\n<\/div>\n<div id=\"mntl-sc-block_1-0-3\" class=\"comp mntl-sc-block finance-sc-block-callout mntl-block\">\n<div id=\"mntl-sc-block_1-0-4\" class=\"comp theme-whatyouneedtoknow mntl-sc-block mntl-sc-block-callout mntl-block\" data-tracking-id=\"mntl-sc-block-callout\" data-tracking-container=\"true\">\n<h3 id=\"mntl-sc-block-callout-heading_1-0\" class=\"comp mntl-sc-block-callout-heading mntl-text-block\">KEY TAKEAWAYS<\/h3>\n<div id=\"mntl-sc-block-callout-body_1-0\" class=\"comp mntl-sc-block-callout-body mntl-text-block\">\n<ul>\n<li>Degrees of freedom refers to the maximum number of logically independent values, which are values that have the freedom to vary, in the data sample.<\/li>\n<li>Degrees of freedom are commonly discussed in relation to various forms of hypothesis testing in statistics, such as a chi-square.<\/li>\n<li>Calculating degrees of freedom is key when trying to understand the importance of a chi-square statistic and the validity of the null hypothesis.<!--more--><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p><span id=\"toc-understanding-degrees-of-freedom\" class=\"heading-toc\"><\/span><\/p>\n<h2 id=\"mntl-sc-block_1-0-5\" class=\"comp mntl-sc-block finance-sc-block-heading mntl-sc-block-heading\"><span class=\"mntl-sc-block-heading__text\">Understanding Degrees of Freedom<\/span><\/h2>\n<p id=\"mntl-sc-block_1-0-6\" class=\"comp mntl-sc-block finance-sc-block-html mntl-sc-block-html\">The easiest way to understand degrees of freedom conceptually is through an example:<\/p>\n<div id=\"mntl-sc-block_1-0-7\" class=\"comp mntl-sc-block mntl-sc-block-adslot mntl-block\">\n<div id=\"mntl-block_22-0\" class=\"comp mntl-block\"><\/div>\n<\/div>\n<ul id=\"mntl-sc-block_1-0-8\" class=\"comp mntl-sc-block finance-sc-block-html mntl-sc-block-html\">\n<li>Consider a data sample consisting of, for the sake of simplicity, five positive integers. The values could be any number with no known relationship between them. This data sample would, theoretically, have five degrees of freedom.<\/li>\n<li>Four of the numbers in the sample are {3, 8, 5, and 4} and the average of the entire data sample is revealed to be 6.<\/li>\n<li>This must mean that the fifth number has to be 10. It can be nothing else. It does not have the freedom to vary.<\/li>\n<li>So the degrees of freedom for this data sample is 4.<\/li>\n<\/ul>\n<div id=\"mntl-sc-block_1-0-9\" class=\"comp mntl-sc-block mntl-sc-block-adslot mntl-block\">\n<div id=\"mntl-block_23-0\" class=\"comp mntl-block\"><\/div>\n<\/div>\n<p id=\"mntl-sc-block_1-0-10\" class=\"comp mntl-sc-block finance-sc-block-html mntl-sc-block-html\">The formula for degrees of freedom equals the size of the data sample minus one:<\/p>\n<p class=\"comp mntl-sc-block finance-sc-block-html mntl-sc-block-html\"><span data-value=\"\\begin{aligned} &amp;\\text{D}_\\text{f} = N - 1 \\\\ &amp;\\textbf{where:} \\\\ &amp;\\text{D}_\\text{f} = \\text{degrees of freedom} \\\\ &amp;N = \\text{sample size} \\\\ \\end{aligned}\"><span class=\"katex katex--loaded\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"\"><span class=\"mord text\">D<\/span><span class=\"msupsub\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">f<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"mrel\">=<\/span><span class=\"mord mathit\">N<\/span><span class=\"mbin\">\u2212<\/span>1<br \/>\n<\/span><span class=\"\"><span class=\"mord text\"><span class=\"mord textbf\">where:<br \/>\n<\/span><\/span><\/span><span class=\"\"><span class=\"mord text\">D<\/span><span class=\"msupsub\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">f<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"mrel\">=<\/span><span class=\"mord text\">degrees\u00a0of\u00a0freedom<br \/>\n<\/span><\/span><span class=\"\"><span class=\"mord mathit\">N<\/span><span class=\"mrel\">=<\/span><span class=\"mord text\">sample\u00a0size<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<div id=\"mntl-sc-block_1-0-13\" class=\"comp mntl-sc-block mntl-sc-block-adslot mntl-block\"><\/div>\n<p id=\"mntl-sc-block_1-0-14\" class=\"comp mntl-sc-block finance-sc-block-html mntl-sc-block-html\">Degrees of freedom are commonly discussed in relation to various forms of hypothesis testing in statistics, such as a chi-square. It is essential to calculate degrees of freedom when trying to understand the importance of a chi-square statistic and the validity of the null hypothesis.<\/p>\n<div id=\"mntl-sc-block_1-0-15\" class=\"comp mntl-sc-block mntl-sc-block-adslot mntl-block\">\n<div id=\"mntl-block_24-0\" class=\"comp mntl-block\">\n<div id=\"native_1-0\" class=\"comp native mntl-native\"><\/div>\n<\/div>\n<\/div>\n<p><span id=\"toc-chi-square-tests\" class=\"heading-toc\"><\/span><\/p>\n<h2 id=\"mntl-sc-block_1-0-16\" class=\"comp mntl-sc-block finance-sc-block-heading mntl-sc-block-heading\"><span class=\"mntl-sc-block-heading__text\">Chi-Square Tests<\/span><\/h2>\n<p id=\"mntl-sc-block_1-0-17\" class=\"comp mntl-sc-block finance-sc-block-html mntl-sc-block-html\">There are two different kinds of\u00a0<a href=\"https:\/\/www.investopedia.com\/terms\/c\/chi-square-statistic.asp\" data-component=\"link\" data-source=\"inlineLink\" data-type=\"internalLink\" data-ordinal=\"1\">chi-square tests<\/a>: the test of independence, which asks a question of relationship, such as, &#8220;Is there a relationship between gender and SAT scores?&#8221;; and the\u00a0<a href=\"https:\/\/www.investopedia.com\/terms\/g\/goodness-of-fit.asp\" data-component=\"link\" data-source=\"inlineLink\" data-type=\"internalLink\" data-ordinal=\"2\">goodness-of-fit test<\/a>, which asks something like &#8220;If a coin is tossed 100 times, will it come up heads 50 times and tails 50 times?&#8221;<\/p>\n<div id=\"mntl-sc-block_1-0-18\" class=\"comp mntl-sc-block mntl-sc-block-adslot mntl-block\">\n<div id=\"mntl-block_25-0\" class=\"comp mntl-block\"><\/div>\n<\/div>\n<p id=\"mntl-sc-block_1-0-19\" class=\"comp mntl-sc-block finance-sc-block-html mntl-sc-block-html\">For these tests, degrees of freedom are utilized to determine if a certain\u00a0<a href=\"https:\/\/www.investopedia.com\/terms\/n\/null_hypothesis.asp\" data-component=\"link\" data-source=\"inlineLink\" data-type=\"internalLink\" data-ordinal=\"1\">null hypothesis<\/a>\u00a0can be rejected based on the total number of variables and samples within the experiment. For example, when considering students and course choice, a sample size of 30 or 40 students is likely not large enough to generate significant data. Getting the same or similar results from a study using a sample size of 400 or 500 students is more valid.<\/p>\n<div id=\"mntl-sc-block_1-0-20\" class=\"comp mntl-sc-block mntl-sc-block-adslot mntl-block\">\n<div id=\"mntl-block_26-0\" class=\"comp mntl-block\"><\/div>\n<\/div>\n<p><span id=\"toc-history-of-degrees-of-freedom\" class=\"heading-toc\"><\/span><\/p>\n<h2 id=\"mntl-sc-block_1-0-21\" class=\"comp mntl-sc-block finance-sc-block-heading mntl-sc-block-heading\"><span class=\"mntl-sc-block-heading__text\">History of Degrees of Freedom<\/span><\/h2>\n<p id=\"mntl-sc-block_1-0-22\" class=\"comp mntl-sc-block finance-sc-block-html mntl-sc-block-html\">The earliest and most basic concept of degrees of freedom was noted in the early 1800s, intertwined in the works of mathematician and astronomer Carl Friedrich Gauss. The modern usage and understanding of the term were expounded upon first by William Sealy Gosset, an English statistician, in his article &#8220;The Probable Error of a Mean,&#8221; published in Biometrika in 1908 under a pen name to preserve his anonymity.<span class=\"mntl-inline-citation mntl-dynamic-tooltip--trigger\" data-id=\"#citation-1\">1<\/span><\/p>\n<div id=\"mntl-sc-block_1-0-23\" class=\"comp mntl-sc-block mntl-sc-block-adslot mntl-block\"><\/div>\n<p id=\"mntl-sc-block_1-0-24\" class=\"comp mntl-sc-block finance-sc-block-html mntl-sc-block-html\">In his writings, Gosset did not specifically use the term &#8220;degrees of freedom.&#8221; He did, however, give an explanation for the concept throughout the course of developing what would eventually be known as Student\u2019s\u00a0<a href=\"https:\/\/www.investopedia.com\/terms\/t\/tdistribution.asp\" data-component=\"link\" data-source=\"inlineLink\" data-type=\"internalLink\" data-ordinal=\"1\">T-distribution<\/a>. The actual term was not made popular until 1922. English biologist and statistician Ronald Fisher began using the term &#8220;degrees of freedom&#8221; when he started publishing reports and data on his work developing chi-squares.<\/p>\n<div id=\"mntl-block_20-0\" class=\"comp js-scads-inline-content mntl-block\"><\/div>\n<\/div>\n<div id=\"textnote-placeholder_4-0\" class=\"comp textnote-placeholder mntl-block\"><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"loc grid-post-content\">\n<div id=\"article-sources_1-0\" class=\"comp article-sources mntl-block\">\n<div id=\"mntl-article-sources_1-0\" class=\"comp mntl-article-sources mntl-expandable-block\" data-scroll-offset=\"100\">\n<div class=\"loc toggle-content\">\n<div id=\"mntl-article-sources__wrapper_1-0\" class=\"comp article-sources__wrapper mntl-article-sources__wrapper mntl-block\" data-click-tracked=\"true\">\n<h3 id=\"mntl-article-sources__heading_1-0\" class=\"comp article-sources__heading mntl-article-sources__heading mntl-text-block\">ARTICLE SOURCES<\/h3>\n<div id=\"mntl-article-sources__icon-wrapper_1-0\" class=\"comp article-sources__icon-wrapper mntl-article-sources__icon-wrapper mntl-block\">\n<div id=\"article-sources__show-icon_1-0\" class=\"comp article-sources__show-icon mntl-block\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"loc grid-pre-footer\">\n<div id=\"performance-marketing_1-0\" class=\"comp performance-marketing mntl-block\">\n<div id=\"lazy-ad-tables_1-0\" class=\"comp lazy-ad-tables mntl-block defer-hidden\" data-scroll-defer-offset=\"600\" data-defer=\"scroll\" data-defer-params=\"\"><\/div>\n<\/div>\n<div id=\"related-recirc-section_1-0\" class=\"comp related-recirc-section mntl-block\" data-tracking-container=\"true\">\n<div id=\"slot-1_11-0\" class=\"comp slot-1 related-recirc-section--terms mntl-block\" data-tracking-container=\"true\">\n<div id=\"related-terms_11-0\" class=\"comp related-terms mntl-block\">\n<h3 id=\"related-terms__heading_11-0\" class=\"comp related-terms__heading mntl-text-block\">Related Terms<\/h3>\n<div id=\"related-terms__list_11-0\" class=\"comp related-terms__list mntl-block\">\n<div id=\"relates-terms__content_11-0\" class=\"comp relates-terms__content mntl-block\"><a id=\"related-terms__title_11-0\" class=\" related-terms__title mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/c\/chi-square-statistic.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">Chi-Square (\u03c72) Statistic Definition<\/span><\/a><\/p>\n<div id=\"related-terms__description_11-0\" class=\"comp related-terms__description mntl-text-block\">A chi-square (\u03c72) statistic is a test that measures how expectations compare to actual observed data (or model results).<\/div>\n<p><a id=\"related-terms__read-more_11-0\" class=\" related-terms__read-more mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/c\/chi-square-statistic.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">more<\/span><\/a><\/div>\n<\/div>\n<div id=\"related-terms__list_11-0-1\" class=\"comp related-terms__list mntl-block\">\n<div id=\"relates-terms__content_11-0-1\" class=\"comp relates-terms__content mntl-block\"><a id=\"related-terms__title_11-0-1\" class=\" related-terms__title mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/t\/t-test.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">T-Test Definition<\/span><\/a><\/p>\n<div id=\"related-terms__description_11-0-1\" class=\"comp related-terms__description mntl-text-block\">A t-test is a type of inferential statistic used to determine if there is a significant difference between the means of two groups, which may be related in certain features.<\/div>\n<p><a id=\"related-terms__read-more_11-0-1\" class=\" related-terms__read-more mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/t\/t-test.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">more<\/span><\/a><\/div>\n<\/div>\n<div id=\"related-terms__list_11-0-2\" class=\"comp related-terms__list mntl-block\">\n<div id=\"relates-terms__content_11-0-2\" class=\"comp relates-terms__content mntl-block\"><a id=\"related-terms__title_11-0-2\" class=\" related-terms__title mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/s\/statistical-significance.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">Why Statistical Significance Matters<\/span><\/a><\/p>\n<div id=\"related-terms__description_11-0-2\" class=\"comp related-terms__description mntl-text-block\">Statistical significance refers to a result that is not likely to occur randomly but rather is likely to be attributable to a specific cause.<\/div>\n<p><a id=\"related-terms__read-more_11-0-2\" class=\" related-terms__read-more mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/s\/statistical-significance.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">more<\/span><\/a><\/div>\n<\/div>\n<div id=\"related-terms__list_11-0-3\" class=\"comp related-terms__list mntl-block\">\n<div id=\"relates-terms__content_11-0-3\" class=\"comp relates-terms__content mntl-block\"><a id=\"related-terms__title_11-0-3\" class=\" related-terms__title mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/g\/goodness-of-fit.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">What Is Goodness-Of-Fit?<\/span><\/a><\/p>\n<div id=\"related-terms__description_11-0-3\" class=\"comp related-terms__description mntl-text-block\">A goodness-of-fit test helps you see if your sample data is accurate or somehow skewed. Discover how the popular chi-square goodness-of-fit test works.<\/div>\n<p><a id=\"related-terms__read-more_11-0-3\" class=\" related-terms__read-more mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/g\/goodness-of-fit.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">more<\/span><\/a><\/div>\n<\/div>\n<div id=\"related-terms__list_11-0-4\" class=\"comp related-terms__list mntl-block\">\n<div id=\"relates-terms__content_11-0-4\" class=\"comp relates-terms__content mntl-block\"><a id=\"related-terms__title_11-0-4\" class=\" related-terms__title mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/a\/anova.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">How Analysis of Variance (ANOVA) Works<\/span><\/a><\/p>\n<div id=\"related-terms__description_11-0-4\" class=\"comp related-terms__description mntl-text-block\">Analysis of variance (ANOVA) is a statistical analysis tool that separates the total variability found within a data set into two components: random and systematic factors.<\/div>\n<p><a id=\"related-terms__read-more_11-0-4\" class=\" related-terms__read-more mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/a\/anova.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">more<\/span><\/a><\/div>\n<\/div>\n<div id=\"related-terms__list_11-0-5\" class=\"comp related-terms__list mntl-block\">\n<div id=\"relates-terms__content_11-0-5\" class=\"comp relates-terms__content mntl-block\"><a id=\"related-terms__title_11-0-5\" class=\" related-terms__title mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/w\/wilcoxon-test.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">How the Wilcoxon Test Is Used<\/span><\/a><\/p>\n<div id=\"related-terms__description_11-0-5\" class=\"comp related-terms__description mntl-text-block\">The Wilcoxon test, which refers to either the rank sum test or the signed rank test, is a nonparametric test that compares two paired groups.<\/div>\n<p><a id=\"related-terms__read-more_11-0-5\" class=\" related-terms__read-more mntl-text-link\" href=\"https:\/\/www.investopedia.com\/terms\/w\/wilcoxon-test.asp\" data-tracking-container=\"true\"><span class=\"link__wrapper\">more<\/span><\/a><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>What Are Degrees of Freedom? Degrees of freedom refers to the maximum number of logically independent values, which are values that have the freedom to vary, in the data sample. KEY TAKEAWAYS Degrees of freedom refers to the maximum number of logically independent values, which are values that have the freedom to vary, in the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6],"tags":[],"class_list":["post-25925","post","type-post","status-publish","format-standard","hentry","category-6"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p6cOVM-6K9","_links":{"self":[{"href":"https:\/\/tom.tomwork.net\/index.php?rest_route=\/wp\/v2\/posts\/25925","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tom.tomwork.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tom.tomwork.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tom.tomwork.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tom.tomwork.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=25925"}],"version-history":[{"count":1,"href":"https:\/\/tom.tomwork.net\/index.php?rest_route=\/wp\/v2\/posts\/25925\/revisions"}],"predecessor-version":[{"id":25926,"href":"https:\/\/tom.tomwork.net\/index.php?rest_route=\/wp\/v2\/posts\/25925\/revisions\/25926"}],"wp:attachment":[{"href":"https:\/\/tom.tomwork.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=25925"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tom.tomwork.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=25925"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tom.tomwork.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=25925"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}