{"id":11841,"date":"2021-03-02T11:23:59","date_gmt":"2021-03-02T11:23:59","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=11841"},"modified":"2026-04-29T19:14:32","modified_gmt":"2026-04-29T19:14:32","slug":"the-f-statistic","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/quantitative-method\/the-f-statistic\/","title":{"rendered":"The F-statistic"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/cfa-level-2-f-distribution\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-distribution.png\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-distribution.png\",\n  \"caption\": \"F-Distribution Curve for the F-Statistic\",\n  \"width\": 1373,\n  \"height\": 860,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  },\n  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/quantitative-method\/the-f-statistic\/\"\n  }\n}\n<\/script> <script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/cfa-level-2-f-table\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-table-4.7.png\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-table-4.7.png\",\n  \"caption\": \"F-Distribution Table (Table 4.7)\",\n  \"width\": 403,\n  \"height\": 276,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  },\n  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/quantitative-method\/the-f-statistic\/\"\n  }\n}\n<\/script> <script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"What is the F-statistic for testing whether all slope coefficients are equal to zero in the ROC regression?\",\n    \"text\": \"Adil Suleman, CFA, regresses return on capital (ROC) against sales, profit margin, and debt ratio using 25 observations. The total sum of squares is 80.16 and the sum of squared errors is 30.0545. What is the F-statistic for testing the null hypothesis that all slope coefficients are equal to zero?\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The F-statistic is approximately 11.67. It is calculated by first finding the regression sum of squares (RSS = 80.16 \u2212 30.0545 = 50.1055). Dividing RSS by the number of independent variables (3) gives the mean regression sum of squares. Dividing the sum of squared errors by the remaining degrees of freedom (25 \u2212 4 = 21) gives the mean squared error. The ratio of these two values results in an F-statistic of about 11.67.\",\n      \"dateCreated\": \"2026-01-19\"\n    }\n  }\n}\n<\/script><\/p>\n<h3 id=\"mce_22\" class=\"editor-rich-text__tinymce mce-content-body\" data-is-placeholder-visible=\"false\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/8E2AbtAb0a8\" width=\"611\" height=\"344\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/h3>\n<p>The F-test determines whether all the independent variables help explain the dependent variable. It is a test of regression\u2019s overall significance that involves testing the null hypothesis that all the slope coefficients in the regression are jointly equal to zero against the alternative hypothesis that at least one slope coefficient is not equal to 0:<\/p>\n<p style=\"text-align: center;\" data-tadv-p=\"keep\">\\(H_{0}: b_{1}=b_{2}=&#8230;=b_{k}=0\\) versus \\(H_{a}:\\) at least one \\(b_{j}\u22600\\)<\/p>\n<p>It is imperative to understand that we cannot use the t-test to determine whether every slope coefficient is zero. This is because individual tests do not account for interactions among the dependent variables.<\/p>\n<p><!-- TOP CTA \u2013 Full Width Outline Button --><\/p>\n<p>The following inputs are required to determine the test statistic for the null hypothesis.<\/p>\n<p>\u00a0 \u00a0 \u00a0i. The total number of observations, \\(n\\).<\/p>\n<p>\u00a0 \u00a0 ii. The total number of slope coefficients to be estimated, \\(k+1\\), where \\(k\\) = number of slope coefficients.<\/p>\n<p>\u00a0 \u00a0iii. The Residual Sum of Squares (SSE), which is the unexplained variation.<\/p>\n<p>\u00a0 \u00a0iv. The Regression Sum of Squares (RSS), which is the explained variation.<\/p>\n<p>The F-statistic (which is a <strong>one-tailed test<\/strong>) is computed as:<\/p>\n<p>$$\\begin{align*}F&amp;=\\frac{\\bigg(\\frac{RSS}{k}\\bigg)}{\\bigg(\\frac{SSE}{n-(k+1)}\\bigg)}\\\\&amp;=\\frac{\\text{Mean Regression Sum of Squares (MSR)}}{\\text{Mean Squared Error (MSE)}}\\end{align*}$$<\/p>\n<p>Where:<\/p>\n<ul>\n<li>RSS = Regression Sum of Squares<\/li>\n<li>SSE = Sum of Squared Errors<\/li>\n<li>n = number of observations<\/li>\n<li>k = number of independent variables<\/li>\n<\/ul>\n<p>A large value of F implies that the regression model does a good job in explaining variation in the dependent variable. On the other hand, the value of F will be zero when the independent variables do not explain the dependent variable at all.<\/p>\n<p>The F-test is denoted as \\(F_{k, (n-(k+1))}\\). The test should have \\(k\\) degrees of freedom in the numerator and \\(n-(k+1)\\_ degrees of freedom in the denominator.<\/p>\n<div style=\"text-align: center; margin: 22px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 10px 18px; border: 2px solid #1e5bd8; color: #1e5bd8; border-radius: 9999px; text-decoration: none; font-weight: 600;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\">  Access our CFA free trial to practice F-statistics and regression testing<\/a><\/div>\n<h2>Decision Rule<\/h2>\n<p>We reject the null hypothesis at a given significance level, \\(\u03b1\\), if the calculated value of \\(F\\) is greater than the upper critical value of the one-tailed \\(F\\) distribution with the specified degrees of freedom.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12787\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-distribution.png\" alt=\"\" width=\"1373\" height=\"860\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-distribution.png 1373w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-distribution-300x188.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-distribution-1024x641.png 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-distribution-768x481.png 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-distribution-400x251.png 400w\" sizes=\"auto, (max-width: 1373px) 100vw, 1373px\" \/><\/p>\n<p>Reject \\(H_0\\) if F-statistic \\(&gt;F_{c}\\) (critical value)<\/p>\n<h3>Example: Calculating and Interpreting the F-statistic<\/h3>\n<p>The following ANOVA table presents the output of the multiple regression analysis of the price of the US Dollar index on the inflation rate and real rate of interest.<\/p>\n<p>$$\\textbf{Exhibit 1: ANOVA Table} $$<\/p>\n<p>$$\\small{\\begin{array}{c|c|c|c|c|c}\\hline{}&amp; \\textbf{df} &amp; \\textbf{SS} &amp; \\textbf{MS} &amp; \\textbf{F} &amp; \\textbf{Significance F}\\\\ \\hline\\text{Regression} &amp; 2 &amp; 432.2520 &amp; 216.1260 &amp; 7.5405 &amp; 0.0179\\\\ \\hline\\text{Residual} &amp; 7 &amp; 200.6349 &amp; 28.6621 &amp; {}&amp; {}\\\\ \\hline\\text{Total} &amp; 9 &amp; 632.8869 &amp; &amp; &amp;{}\\\\ \\hline\\end{array}}$$<\/p>\n<p>Test the null hypothesis that all of the independent variables are equal to zero at the 5% significance level.<\/p>\n<h4>Solution<\/h4>\n<p>We test the null hypothesis:<\/p>\n<p>\\(H_{0}: \\beta_{1}=\\beta_{2}=0\\) versus \\(H_a\\): at least one \\(b_j\u22600\\).<\/p>\n<p>Number of slope coefficients \\((k)=2\\).<\/p>\n<p>Degrees of freedom in the denominator = \\(n-(k+1)=10-(2+1)=7\\).<\/p>\n<p>RSS = 432.2520<\/p>\n<p>SSE = 200. 6349<\/p>\n<p>Thus,<\/p>\n<p>$$\\begin{align*}F&amp;=\\frac{\\bigg(\\frac{RSS}{k}\\bigg)}{\\bigg(\\frac{SSE}{n-(k+1)}\\bigg)}\\\\&amp;=\\frac{\\frac{432.2520}{2}}{\\frac{200.6349}{7}}\\\\&amp;=7.5405\\end{align*}$$<\/p>\n<p>For\u00a0 \\(\\alpha=0.05\\), the critical value of F with \\(k= 2\\) and \\((n &#8211; k &#8211; 1) = 7\\) degrees of freedom \\(F_{0.05,2,7}\\) is approximately 4.74.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12815 aligncenter\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-table-4.7.png\" alt=\"\" width=\"403\" height=\"276\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-table-4.7.png 403w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-table-4.7-300x205.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/03\/cfa-level-2-f-table-4.7-400x274.png 400w\" sizes=\"auto, (max-width: 403px) 100vw, 403px\" \/><\/p>\n<p>The calculated value of the F-test statistic, 7.5405, is greater than the critical value of F, 4. 737. We, therefore, <strong>reject the null hypothesis <\/strong>that coefficients of both independent variables equal 0.<\/p>\n<p>Additionally, notice that from the ANOVA table, the column \u201cSignificance F\u201d reports a p-value of 0.0179, which is less than 0.05. The p-value implies that the smallest level of significance at which the null hypothesis can be rejected is 0.0179.<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Adil Suleman, CFA, wishes to establish the possible drivers of a company\u2019s percentage return on capital (ROC). Suleman identifies performance measures such as the profit margin (%), sales, and debt ratio as possible drivers of ROC.<\/p>\n<p>He then regresses the ROC against the three independent variables: Sales, profit margin, and debt ratio. The output from a statistical software package is presented below:<\/p>\n<ul>\n<li data-tadv-p=\"keep\">Observations: 25.<\/li>\n<li data-tadv-p=\"keep\">The total sum of squares: 80.16.<\/li>\n<li data-tadv-p=\"keep\">Sum of squared errors: 30.0545.<\/li>\n<\/ul>\n<p>Suleman proceeds to test the null hypothesis at the 5% significance level that all three of the independent variables are equal to zero.<\/p>\n<p>The F-statistic for testing whether the slope coefficient is equal to zero is <em>closest to:\u00a0<\/em><\/p>\n<p><em>\u00a0 \u00a0 \u00a0<\/em>A<em>.\u00a0<\/em>11.67.<\/p>\n<p>\u00a0 \u00a0 \u00a0B. 18.67.<\/p>\n<p>\u00a0 \u00a0 \u00a0C. 19.56.<\/p>\n<h3>Solution<\/h3>\n<p><strong>The correct answer is A.<\/strong><\/p>\n<p>$$\\begin{align*}F&amp;=\\frac{\\text{(Mean Regression Sum of Squares)}}{\\text{Mean squared error}}\\\\&amp;=\\frac{(\\frac{RSS}{k})}{(\\frac{SSE}{n-(k+1)})}\\end{align*}$$<\/p>\n<p>$$\\text{RSS}=\\text{SST}-\\text{SSE}$$<\/p>\n<p>$$\\begin{align*}\\text{RSS}&amp;=80.16-30.0545\\\\&amp;=50.1055\\end{align*}$$<\/p>\n<p>$$\\begin{align*}\\text{Mean Regression Sum of Squares}&amp;=\\frac{50.1055}{3}\\\\&amp;=16.7018\\end{align*}$$<\/p>\n<p>$$\\begin{align*}\\text{Mean Squared Error}&amp;= \\frac{30.0545}{25-3-1}\\\\&amp;=1.4312\\end{align*}$$<\/p>\n<p>$$\\begin{align*}F&amp;=\\frac{16.7018}{1.4312}\\\\&amp;=11.67\\end{align*}$$<\/p>\n<\/blockquote>\n<p>Reading 2: Multiple Regression<\/p>\n<p><em>LOS 2 (g) Calculate and interpret the <\/em>F<strong><em>&#8211;<\/em><\/strong><em>statistic, and describe how it is used in regression analysis.<\/em><\/p>\n<h2>Why the F-Statistic Matters in CFA Level II<\/h2>\n<p>The F-statistic is a core CFA Level II Quantitative Methods concept. Candidates may be tested on overall regression significance, ANOVA interpretation, joint hypothesis testing, and whether independent variables explain variation in the dependent variable.<\/p>\n<p>Build confidence with the <a href=\"https:\/\/analystprep.com\/cfa-level-2\/\" target=\"_blank\">CFA Level II prep course<\/a> featuring guided lessons, practice questions, and full mock exams.<\/p>\n<div style=\"text-align: center; margin: 40px 0;\"><a style=\"background: #1a73e8; color: #ffffff; padding: 14px 26px; border-radius: 999px; text-decoration: none; font-weight: bold; font-size: 16px; display: inline-block;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Start Free Trial \u2192 <\/a><\/p>\n<p style=\"margin-top: 10px; font-size: 14px; color: #555;\">Access CFA Level II practice questions, video lessons, and study notes covering regression analysis, F-tests, model significance, and quantitative methods.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The F-test determines whether all the independent variables help explain the dependent variable. It is a test of regression\u2019s overall significance that involves testing the null hypothesis that all the slope coefficients in the regression are jointly equal to zero&#8230;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,229],"tags":[216,230,244],"class_list":["post-11841","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-quantitative-method","tag-the-f-statistic","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>The F-Statistic | CFA Level II Notes<\/title>\n<meta name=\"description\" content=\"Learn how the F-statistic is calculated and interpreted, including its role in regression 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