{"id":29858,"date":"2021-09-09T19:12:20","date_gmt":"2021-09-09T19:12:20","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29858"},"modified":"2025-12-18T09:00:25","modified_gmt":"2025-12-18T09:00:25","slug":"chi-square-distribution-and-f-distribution","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/chi-square-distribution-and-f-distribution\/","title":{"rendered":"Chi-square and F-Distributions"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2025 Level I CFA\u00ae Exam \u2013 Quantitative Methods \u2013 Module 4)\",\n  \"description\": \"This video covers the fundamentals of common probability distributions, exploring discrete and continuous variables, normal and lognormal distributions, Bernoulli and binomial properties, safety-first criteria, t-distributions, chi-square, F-distributions, Monte Carlo simulations, and practical applications of probability concepts in financial analysis and risk assessment.\",\n  \"uploadDate\": \"2021-11-09T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/TXO9ODcLWiU\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/TXO9ODcLWiU\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/TXO9ODcLWiU\",\n  \"duration\": \"PT1H47S\"\n}\n<\/script>\n\n\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/Chi-square-Distribution.jpg\",\n  \"caption\": \"Chi-square Distribution\",\n  \"width\": 1590,\n  \"height\": 1119,\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}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/F-Distribution.jpg\",\n  \"caption\": \"F-distribution\",\n  \"width\": 1590,\n  \"height\": 1459,\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}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Which of the following are most likely common characteristics of F-distribution and chi-square distribution?\",\n    \"text\": \"Which of the following are most likely common characteristics of F-distribution and chi-square distribution?\\n\\nA. Both can take only positive value.\\nB. Both are defined by two parameters.\\nC. Both are negatively skewed distributions.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A.\\n\\nBoth F-distribution and chi-square distribution can only take non-negative values.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/TXO9ODcLWiU\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><strong>Chi-square Distribution<\/strong><\/h2>\n<p>A chi-square distribution is an asymmetrical family of distributions. A chi-square distribution with \\(v\\) degrees of freedom is the distribution of the sum of the squares of \u00a0\\(v\\) independent standard normally distributed random variables. Intuitively, chi-square distributions take only non-negative random variables.<\/p>\n<p>A chi-square distribution is used to test the variance of a population that is distributed normally.<\/p>\n<p>In a summary, the following are the properties of a chi-square distribution:<\/p>\n<ul>\n<li>A chi-square distribution is a non-symmetrical distribution (skewed to the right).<\/li>\n<li>A chi-square distribution is defined by one parameter: Degrees of freedom (df), \\(v = n \u2013 1\\).<\/li>\n<li>A chi-square distribution is the sum of the squares of \\(k\\) independent standard normally distributed random variables. Hence, it is a non-negative distribution.<\/li>\n<li>For each degree of freedom, there are different chi-square distributions.<\/li>\n<li>The shape of a chi-square distribution changes with the change in the degrees of freedom. The more the degrees of free increase, the more the distribution assumes the shape of a standard normal distribution.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-39302 size-full\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/Chi-square-Distribution.jpg\" alt=\"Chi-square Distribution\" width=\"1590\" height=\"1119\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/Chi-square-Distribution.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/Chi-square-Distribution-300x211.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/Chi-square-Distribution-1024x721.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/Chi-square-Distribution-768x540.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/Chi-square-Distribution-1536x1081.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/Chi-square-Distribution-400x282.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<h2><strong>F-Distribution<\/strong><\/h2>\n<p>An F-distribution is used to test the equality of variances of two normally distributed populations from two independent random samples.<\/p>\n<p>The following are the properties of an F-distribution:<\/p>\n<ul>\n<li>An F-distribution is an asymmetrical distribution (skewed to the right).<\/li>\n<li>An F-distribution is defined by two parameters, i.e., degrees of freedom of the numerator ( \\(m\\)) and degrees of freedom of the denominator ( \\(n\\)).<\/li>\n<li>Like a chi-square distribution, an F-distribution can only have positive values.<\/li>\n<li>As the\u00a0degrees of freedom\u00a0for the numerator and the denominator\u00a0increase, the F-distribution approximates the\u00a0normal distribution.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-39304 size-full\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/F-Distribution.jpg\" alt=\"F-distribution\" width=\"1590\" height=\"1459\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/F-Distribution.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/F-Distribution-300x275.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/F-Distribution-1024x940.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/F-Distribution-768x705.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/F-Distribution-1536x1409.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/F-Distribution-400x367.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<h3><strong>Relationship between the Chi-square and F-distributions<\/strong><\/h3>\n<p>The F-distribution is the ratio of two chi-square\u00a0distributions with degrees of freedom<em> \\(m\\)<\/em> and \u00a0\\(n\\), respectively, where each chi-square has first been divided by its degrees of freedom, i.e.,<\/p>\n<p>$$<br \/>F=\\frac{\\left(\\frac{\\chi_{1}^{2}}{m}\\right)}{\\left(\\frac{\\chi_{2}^{2}}{n}\\right)}<br \/>$$<\/p>\n<p>Where \u00a0\\(m\\) is the numerator degrees of freedom and \u00a0\\(n\\)\u00a0is the denominator degrees of freedom.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Which of the following are <em>most likely<\/em> common characteristics of F-distribution and chi-square distribution?<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>Both can take only positive value.<\/li>\n<li>Both are defined by two parameters.<\/li>\n<li>Both are negatively skewed distributions.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>Both F-distribution and chi-square distribution can only take non-negative values.<\/p>\n<p><strong>B is incorrect<\/strong>. A chi-square distribution is defined by one parameter (i.e., n-1 degrees of freedom), while an F-distribution is defined by parameters, i.e., degrees of freedom of the numerator (m) and degrees of freedom of the denominator (n).<\/p>\n<p><strong>C is incorrect<\/strong>. Both the F-distribution and the chi-square distribution are positively skewed distributions.<\/p>\n<\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Chi-square Distribution A chi-square distribution is an asymmetrical family of distributions. A chi-square distribution with \\(v\\) degrees of freedom is the distribution of the sum of the squares of \u00a0\\(v\\) independent standard normally distributed random variables. Intuitively, chi-square distributions take&#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-29858","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>Chi-Square &amp; F-Distributions | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn about chi-square and F-distributions, their properties, and how degrees of freedom impact their shapes. 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