{"id":33674,"date":"2021-10-07T13:21:38","date_gmt":"2021-10-07T13:21:38","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=33674"},"modified":"2026-04-06T06:05:12","modified_gmt":"2026-04-06T06:05:12","slug":"calculating-the-t-statistic-for-hypothesis-testing-on-correlation","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/","title":{"rendered":"Calculating the t-statistic for Hypothesis Testing on Correlation"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Hypothesis Testing (2025 Level I CFA\u00ae Exam \u2013 Quantitative Methods \u2013 Module 6)\",\n  \"description\": \"Level I CFA\u00ae Program video lesson offered by AnalystPrep covering Topic 1 \u2013 Quantitative Methods, Reading 6: Hypothesis Testing. This lesson explains the full hypothesis testing framework, including null and alternative hypotheses, one-tailed and two-tailed tests, test statistics, Type I and Type II errors, significance levels, power of a test, decision rules, confidence intervals, p-values, and multiple testing issues. It also covers hypothesis tests for population means, mean differences, variances, equality of variances, parametric vs nonparametric tests, correlation tests, and tests of independence using contingency tables, all aligned with CFA\u00ae Level I Learning Outcome Statements.\",\n  \"uploadDate\": \"2021-11-18\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/twTTWMuEGx8\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=twTTWMuEGx8\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/twTTWMuEGx8\",\n  \"duration\": \"PT58M47S\"\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/twTTWMuEGx8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<p>The Spearman\u2019s rank correlation coefficient is a non-parametric statistical test used to examine whether there is a significant relationship between two sets of data. Like the correlation coefficient, Spearman\u2019s rank correlation can have any value between \u20131 and +1. A value of 0 indicates no relationship, and values of +1 indicate a perfectly positive correlation, while -1 indicates a perfectly negative correlation.<\/p>\n<div style=\"margin:20px 0\">\n<a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\"\nstyle=\"display:block;width:100%;text-align:center;padding:10px;border:2px solid #2f5bea;border-radius:40px;font-size:16px;color:#2f5bea;text-decoration:none\">\nPractice correlation questions with our free trial.\n<\/a>\n<\/div>\n<h3><strong>Steps of Calculating Spearman\u2019s Rank Correlation Coefficient<\/strong><\/h3>\n<ol>\n<li>Rank the values of variables X and Y in descending order.<\/li>\n<li>Find the difference (denoted as \\(d_i\\)) between the rank of the pair values of X and Y.<\/li>\n<li>Square the difference and calculate the sum of the difference, that is \\(\\sum d_i\\).<\/li>\n<li>&nbsp;Calculate the Spearman&#8217;s rank correlation coefficient as follows:<\/li>\n<\/ol>\n<p>$$<br>r_{s}=1-\\frac{6 \\sum d_{i}^{2}}{n\\left(n^{2}-1\\right)}<br>$$<\/p>\n<p><strong>Example: Calculatig Spearman&#8217;s Rank Correlation Coefficient<\/strong><\/p>\n<p>An analyst is studying the relationship between returns for two sectors, steel and cement over past 5 years by using spearman\u2019s rank correlation coefficient. The returns of both sectors are provided as under.<\/p>\n<p>$$<br>\\begin{array}{c|c|c}<br>&nbsp;\\textbf { Year } &amp; \\textbf { Steel sector returns } &amp; \\textbf { Cement sector returns } \\\\<br>\\hline 1 &amp; 2.5 \\% &amp; 3.2 \\% \\\\<br>\\hline 2 &amp; 5 \\% &amp; 4.5 \\% \\\\<br>\\hline 3 &amp; 5.6 \\% &amp; 4.2 \\% \\\\<br>\\hline 4 &amp; -3 \\% &amp; -1.7 \\% \\\\<br>\\hline 5 &amp; 0.5 \\% &amp; 1.1 \\% \\\\<br>\\end{array}<br>$$<\/p>\n<p>The Spearman\u2019s rank correlation coefficient is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$<br>\\begin{array}{c|c|c|c|c|c|c}<br>\\textbf { Year } &amp; \\begin{array}{l}<br>\\textbf { Steel } \\\\<br>\\textbf { sector } \\\\<br>\\textbf { returns } \\\\<br>\\textbf { (X) }<br>\\end{array} &amp; \\begin{array}{l}<br>\\textbf { Cement } \\\\<br>\\textbf { sector } \\\\<br>\\textbf { returns } \\\\<br>\\textbf{ (Y) }<br>\\end{array} &amp; \\begin{array}{l}<br>\\textbf { Rank of } \\\\<br>\\textbf { X }<br>\\end{array} &amp; \\begin{array}{l}<br>\\textbf { Rank of } \\\\<br>\\mathrm{Y}<br>\\end{array} &amp; \\bf{d} &amp; \\bf{d^2} \\\\<br>\\hline 1 &amp; 2.50 \\% &amp; 1.60 \\% &amp; 3 &amp; 4 &amp; -1 &amp; 1 \\\\<br>\\hline 2 &amp; 5 \\% &amp; 4.50 \\% &amp; 2 &amp; 1 &amp; 1 &amp; 1 \\\\<br>\\hline 3 &amp; 5.60 \\% &amp; 4.20 \\% &amp; 1 &amp; 2 &amp; -1 &amp; 1 \\\\<br>\\hline 4 &amp; -3 \\% &amp; -1.70 \\% &amp; 5 &amp; 5 &amp; 0 &amp; 0 \\\\<br>\\hline 5 &amp; 0.50 \\% &amp; 2.20 \\% &amp; 4 &amp; 3 &amp; 1 &amp; 1 \\\\<br>\\hline &amp; &amp; &amp; &amp; &amp; \\textbf { Sum } &amp; 4 \\\\<br>\\end{array}<br>$$<\/p>\n<p>We can now use the formula:<\/p>\n<p>$$ \\begin{align} r_{s}&amp;=1-\\frac{6 \\sum d_{i}^{2}}{n\\left(n^{2}-1\\right)} \\\\ &amp;=1- \\left[\\frac{(6 \\times 4)}{ 5 \\times (5^2 \u2013 1)}\\right] \\\\ &amp;= 0.8 \\end{align} $$<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align:center;margin:40px 0\">\n<a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\"\nstyle=\"display:inline-block;padding:10px 26px;background:#3f78d7;color:#fff;border-radius:40px;font-size:16px;text-decoration:none\">\nStart Free Trial \u2192\n<\/a>\n<p style=\"margin-top:10px;max-width:600px;margin-left:auto;margin-right:auto;font-size:14px\">\nSolve CFA-style questions on Spearman rank correlation, hypothesis testing, and statistical relationships.\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The Spearman\u2019s rank correlation coefficient is a non-parametric statistical test used to examine whether there is a significant relationship between two sets of data. Like the correlation coefficient, Spearman\u2019s rank correlation can have any value between \u20131 and +1. A&#8230;<\/p>\n","protected":false},"author":15,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-33674","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>T-Statistic for Correlation Testing | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how to calculate the t-statistic for hypothesis testing on correlation and assess the strength of relationships between variables.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"T-Statistic for Correlation Testing | CFA Level 1\" \/>\n<meta property=\"og:description\" content=\"Learn how to calculate the t-statistic for hypothesis testing on correlation and assess the strength of relationships between variables.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/\" \/>\n<meta property=\"og:site_name\" content=\"AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-07T13:21:38+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-04-06T06:05:12+00:00\" \/>\n<meta name=\"author\" content=\"Kosikos Tuitoek\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Kosikos Tuitoek\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/\"},\"author\":{\"name\":\"Kosikos Tuitoek\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20\"},\"headline\":\"Calculating the t-statistic for Hypothesis Testing on Correlation\",\"datePublished\":\"2021-10-07T13:21:38+00:00\",\"dateModified\":\"2026-04-06T06:05:12+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/\"},\"wordCount\":400,\"articleSection\":[\"Quantitative Methods\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/\",\"name\":\"T-Statistic for Correlation Testing | CFA Level 1\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\"},\"datePublished\":\"2021-10-07T13:21:38+00:00\",\"dateModified\":\"2026-04-06T06:05:12+00:00\",\"author\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20\"},\"description\":\"Learn how to calculate the t-statistic for hypothesis testing on correlation and assess the strength of relationships between variables.\",\"breadcrumb\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Calculating the t-statistic for Hypothesis Testing on Correlation\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\",\"name\":\"AnalystPrep | CFA\u00ae Exam Study Notes\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20\",\"name\":\"Kosikos Tuitoek\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g\",\"caption\":\"Kosikos Tuitoek\"},\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/kosikos-tuitoek-enockanalystprep-com\/\"}]}<\/script>\n<meta property=\"og:video\" content=\"https:\/\/www.youtube.com\/embed\/twTTWMuEGx8\" \/>\n<meta property=\"og:video:type\" content=\"text\/html\" \/>\n<meta property=\"og:video:duration\" content=\"3528\" \/>\n<meta property=\"og:video:width\" content=\"480\" \/>\n<meta property=\"og:video:height\" content=\"270\" \/>\n<meta property=\"ya:ovs:adult\" content=\"false\" \/>\n<meta property=\"ya:ovs:upload_date\" content=\"2021-10-07T13:21:38+00:00\" \/>\n<meta property=\"ya:ovs:allow_embed\" content=\"true\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"T-Statistic for Correlation Testing | CFA Level 1","description":"Learn how to calculate the t-statistic for hypothesis testing on correlation and assess the strength of relationships between variables.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/","og_locale":"en_US","og_type":"article","og_title":"T-Statistic for Correlation Testing | CFA Level 1","og_description":"Learn how to calculate the t-statistic for hypothesis testing on correlation and assess the strength of relationships between variables.","og_url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/","og_site_name":"AnalystPrep | CFA\u00ae Exam Study Notes","article_published_time":"2021-10-07T13:21:38+00:00","article_modified_time":"2026-04-06T06:05:12+00:00","author":"Kosikos Tuitoek","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Kosikos Tuitoek","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/"},"author":{"name":"Kosikos Tuitoek","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20"},"headline":"Calculating the t-statistic for Hypothesis Testing on Correlation","datePublished":"2021-10-07T13:21:38+00:00","dateModified":"2026-04-06T06:05:12+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/"},"wordCount":400,"articleSection":["Quantitative Methods"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/","name":"T-Statistic for Correlation Testing | CFA Level 1","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website"},"datePublished":"2021-10-07T13:21:38+00:00","dateModified":"2026-04-06T06:05:12+00:00","author":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20"},"description":"Learn how to calculate the t-statistic for hypothesis testing on correlation and assess the strength of relationships between variables.","breadcrumb":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-the-t-statistic-for-hypothesis-testing-on-correlation\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/cfa-level-1-exam\/"},{"@type":"ListItem","position":2,"name":"Calculating the t-statistic for Hypothesis Testing on Correlation"}]},{"@type":"WebSite","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/","name":"AnalystPrep | CFA\u00ae Exam Study Notes","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20","name":"Kosikos Tuitoek","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g","caption":"Kosikos Tuitoek"},"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/kosikos-tuitoek-enockanalystprep-com\/"}]},"og_video":"https:\/\/www.youtube.com\/embed\/twTTWMuEGx8","og_video_type":"text\/html","og_video_duration":"3528","og_video_width":"480","og_video_height":"270","ya_ovs_adult":"false","ya_ovs_upload_date":"2021-10-07T13:21:38+00:00","ya_ovs_allow_embed":"true"},"_links":{"self":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/33674","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/users\/15"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/comments?post=33674"}],"version-history":[{"count":13,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/33674\/revisions"}],"predecessor-version":[{"id":60302,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/33674\/revisions\/60302"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/media?parent=33674"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/categories?post=33674"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/tags?post=33674"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}