{"id":1503,"date":"2012-12-28T10:16:06","date_gmt":"2012-12-28T01:16:06","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=1503"},"modified":"2012-12-28T12:57:13","modified_gmt":"2012-12-28T03:57:13","slug":"students-t-test","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=1503","title":{"rendered":"Student&#8217;s t-test"},"content":{"rendered":"<div class=\"theContentWrap-ccc\"><p>When you would like to compare two averages between samples, you could use such formulas of Student&#8217;s t-test as following chart according to the number and the variances between dependent or independent samples.<\/p>\n<p><a href=\"\/\/fujiitoshiki.com\/improvesociety\/wp-content\/uploads\/t-testEN.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"\/\/fujiitoshiki.com\/improvesociety\/wp-content\/uploads\/t-testEN-300x102.jpg\" alt=\"t-test(EN)\" width=\"300\" height=\"102\" class=\"alignnone size-medium wp-image-1510\" srcset=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/wp-content\/uploads\/t-testEN-300x102.jpg 300w, https:\/\/www.fujiitoshiki.com\/improvesociety\/wp-content\/uploads\/t-testEN.jpg 720w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><\/a><\/p>\n<p>Paired t-test as below;<\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+t+%3D+%5Cfrac%7B%5Cbar+X_D+-+%5Cmu_0%7D%7BSD%2F%5Csqrt%7Bn%7D%7D+%5Ccdot%5Ccdot%5Ccdot%281%29&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle t = \\frac{\\bar X_D - \\mu_0}{SD\/\\sqrt{n}} \\cdot\\cdot\\cdot(1)' title='\\displaystyle t = \\frac{\\bar X_D - \\mu_0}{SD\/\\sqrt{n}} \\cdot\\cdot\\cdot(1)' class='latex' \/><\/p>\n<p>Independent two samples t-test with unequal sample sizes and unequal variances, called as Welch&#8217;s t-test, as below;<\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+t+%3D+%5Cfrac%7B%5Cbar+X_1+-+%5Cbar+X_2%7D%7BSD_p%7D+%5Ccdot%5Ccdot%5Ccdot%282%29+%5Cvspace%7B0.2in%7D%5C%5C+SD_p+%3D+%5Csqrt%7B%5Cfrac%7BSD_1%5E2%7D%7Bn_1%7D+%2B+%5Cfrac%7BSD_2%5E2%7D%7Bn_2%7D%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle t = \\frac{\\bar X_1 - \\bar X_2}{SD_p} \\cdot\\cdot\\cdot(2) \\vspace{0.2in}\\\\ SD_p = \\sqrt{\\frac{SD_1^2}{n_1} + \\frac{SD_2^2}{n_2}}' title='\\displaystyle t = \\frac{\\bar X_1 - \\bar X_2}{SD_p} \\cdot\\cdot\\cdot(2) \\vspace{0.2in}\\\\ SD_p = \\sqrt{\\frac{SD_1^2}{n_1} + \\frac{SD_2^2}{n_2}}' class='latex' \/><\/p>\n<p>Independent two samples t-test with equal sample sizes and unequal variances as below;<\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+t+%3D+%5Cfrac%7B%5Cbar+X_1+-+%5Cbar+X_2%7D%7BSD_p+%5Csqrt%7B%5Cfrac%7B2%7D%7Bn%7D%7D%7D+%5Ccdot%5Ccdot%5Ccdot%283%29+%5Cvspace%7B0.2in%7D%5C%5CSD_p+%3D+%5Csqrt%7B%5Cfrac%7BSD_1%5E2+%2B+SD_2%5E2%7D%7B2%7D%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle t = \\frac{\\bar X_1 - \\bar X_2}{SD_p \\sqrt{\\frac{2}{n}}} \\cdot\\cdot\\cdot(3) \\vspace{0.2in}\\\\SD_p = \\sqrt{\\frac{SD_1^2 + SD_2^2}{2}}' title='\\displaystyle t = \\frac{\\bar X_1 - \\bar X_2}{SD_p \\sqrt{\\frac{2}{n}}} \\cdot\\cdot\\cdot(3) \\vspace{0.2in}\\\\SD_p = \\sqrt{\\frac{SD_1^2 + SD_2^2}{2}}' class='latex' \/><\/p>\n<p>Independent two samples t-test with unequal sample sizes and equal variances as below;<\/p>\n<p><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cdisplaystyle+t+%3D+%5Cfrac%7B%5Cbar+X_1+-+%5Cbar+X_2%7D%7BSD_p+%5Csqrt%7B%5Cfrac%7B1%7D%7Bn_1%7D+%2B+%5Cfrac%7B1%7D%7Bn_2%7D%7D%7D+%5Ccdot%5Ccdot%5Ccdot%284%29+%5Cvspace%7B0.2in%7D%5C%5C+SD_p+%3D+%5Csqrt%7B%5Cfrac%7B%28n_1+-1%29SD_1%5E2+%2B+%28n_2+-+1%29SD_2%5E2%7D%7Bn_1+%2B+n_2+-+2%7D%7D&#038;bg=T&#038;fg=000000&#038;s=0' alt='\\displaystyle t = \\frac{\\bar X_1 - \\bar X_2}{SD_p \\sqrt{\\frac{1}{n_1} + \\frac{1}{n_2}}} \\cdot\\cdot\\cdot(4) \\vspace{0.2in}\\\\ SD_p = \\sqrt{\\frac{(n_1 -1)SD_1^2 + (n_2 - 1)SD_2^2}{n_1 + n_2 - 2}}' title='\\displaystyle t = \\frac{\\bar X_1 - \\bar X_2}{SD_p \\sqrt{\\frac{1}{n_1} + \\frac{1}{n_2}}} \\cdot\\cdot\\cdot(4) \\vspace{0.2in}\\\\ SD_p = \\sqrt{\\frac{(n_1 -1)SD_1^2 + (n_2 - 1)SD_2^2}{n_1 + n_2 - 2}}' class='latex' \/><\/p>\n<p>Reference:<a href=\"\/\/en.wikipedia.org\/wiki\/T-test\" target=\"_blank\">Student&#8217;s t-test<\/a><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>When you would like to compare two averages between samples, you could use such formulas of Student&#8217;s t- &hellip; <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=1503\" class=\"more-link\"><span class=\"screen-reader-text\">&#8220;Student&#8217;s t-test&#8221; \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[7],"tags":[],"class_list":["post-1503","post","type-post","status-publish","format-standard","hentry","category-statistics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/1503","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1503"}],"version-history":[{"count":11,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/1503\/revisions"}],"predecessor-version":[{"id":1516,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/1503\/revisions\/1516"}],"wp:attachment":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1503"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1503"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1503"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}