{"id":29845,"date":"2021-09-09T14:17:42","date_gmt":"2021-09-09T14:17:42","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29845"},"modified":"2026-04-06T09:14:10","modified_gmt":"2026-04-06T09:14:10","slug":"t-distribution-and-degrees-of-freedom","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/t-distribution-and-degrees-of-freedom\/","title":{"rendered":"t-Distribution and Degrees of Freedom"},"content":{"rendered":"\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\/2019\/10\/page-157.jpg\",\n  \"caption\": \"T-distribution vs normal distribution\",\n  \"width\": 2235,\n  \"height\": 1057,\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\/2022\/08\/mock-400x284.png\",\n  \"caption\": \"The table represents one-tailed confidence intervals\",\n  \"width\": 400,\n  \"height\": 284,\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 statements regarding a t-distribution is most likely correct?\",\n    \"text\": \"Which of the following statements regarding a t-distribution is most likely correct?\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. A t-distribution is a symmetrical, bell-shaped distribution centered at zero. Compared with the standard normal distribution, it has heavier tails for low degrees of freedom, but it remains symmetric about zero. As the degrees of freedom increase, the t-distribution becomes narrower and approaches the standard normal distribution.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<p>A student\u2019s t-distribution is a bell-shaped probability distribution symmetrical about its mean. It is regarded as the most suitable distribution to use in the construction of confidence intervals in the following instances:<\/p>\n\n\n\n<!--more-->\n\n\n\n<ol class=\"wp-block-list\">\n<li>When dealing with small samples of less than 30 elements.<\/li>\n\n\n\n<li>When the population variance is unknown.<\/li>\n\n\n\n<li>When the distribution involved is either normal or approximately normal.<\/li>\n<\/ol>\n\n\n\n<p>Apart from being used in the construction of confidence intervals, a t-distribution is used to test the following:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Single population mean.<\/li>\n\n\n\n<li>The differences between two population means.<\/li>\n\n\n\n<li>The mean difference between paired (dependent) populations.<\/li>\n\n\n\n<li>The population correlation coefficient.<\/li>\n<\/ul>\n\n\n\n<p>In the absence of explicit normality of a given distribution, a t-distribution may still be appropriate for use if the sample size is large enough for the central limit theorem to be applied. In such a case, the distribution is considered approximately normal.<\/p>\n\n\n\n<p>A t-statistic, also called the t-score, is given by:<\/p>\n\n\n\n<p>$$ t = \\cfrac {(x &#8211; \\mu)}{\\left(\\cfrac {S}{\\sqrt n} \\right)} $$<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<p>\\(x\\) = Sample mean.<\/p>\n\n\n\n<p>\\(\u03bc\\) = Population mean.<\/p>\n\n\n\n<p>\\(S\\) = Sample standard deviation.<\/p>\n\n\n\n<p>\\(n\\) = Sample size.<\/p>\n\n\n\n<p>Relationship between the t-distribution and the Normal Distribution<\/p>\n\n\n\n<p>A t-distribution allows us to analyze distributions that are not perfectly normal. A t-distribution has thicker tails relative to a normal distribution.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"2235\" height=\"1057\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-157.jpg\" alt=\"T-distribution vs normal distribution\" class=\"wp-image-17044\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-157.jpg 2235w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-157-300x142.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-157-768x363.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-157-1024x484.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-157-400x189.jpg 400w\" sizes=\"auto, (max-width: 2235px) 100vw, 2235px\" \/><\/figure>\n<\/div>\n\n\n<p>The shape of a t-distribution is dependent on the number of degrees of freedom. It, therefore, follows that as the number of d.f. increases, the distribution becomes more \u2018spiked,\u2019 and its tails become thinner, closer to those of the normal distribution.<\/p>\n\n\n\n<p>The t-distribution has the following properties:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A t-distribution is symmetrical. It is a bell-shaped distribution that assumes the shape of a normal distribution and has a mean of zero.<\/li>\n\n\n\n<li>A t-distribution is defined by one parameter, that is, degrees of freedom (df) \\(v= n \u2013 1\\), where \\(n\\) is the sample size. Its \\(\\text {variance} = \\frac {v}{ \\left(\\frac {v}{2} \\right) }\\), where \\(v\\) represents the number of degrees of freedom and \\(v \u2265 2\\).<\/li>\n\n\n\n<li>The variance is greater than 1 at all times. Note, however, that it gets very close to one when there are many degrees of freedom. With a large number of degrees of freedom, a t-distribution resembles a normal distribution.<\/li>\n\n\n\n<li>The tails of a t-distribution are fatter and less peaked than those of a normal distribution, indicating more probability in the tails.<\/li>\n\n\n\n<li>The shape of a t-distribution changes with the change in the degrees of freedom. The more the d.f. increase, the more the shape of a t-distribution looks like a standard normal distribution.<\/li>\n<\/ul>\n\n\n\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 t-distribution questions with our free trial.\n<\/a>\n<\/div>\n\n\n<h2><strong>The Degrees of Freedom<\/strong><\/h2>\n<p>A t-distribution, just like several other distributions, has only one parameter: The degrees of freedom (d.f.). The number of degrees of freedom refers to the number of independent observations (total number of observations less 1):<\/p>\n<p>$$ v = n-1 $$<\/p>\n<p>Hence, a sample of 10 observations or elements would be analyzed using a t-distribution with 9 degrees of freedom. Similarly, a 6 d.f. distribution would be used for a sample size of 7 observations.<\/p>\n<h2><strong>Notations<\/strong><\/h2>\n<p>It is the standard practice for statisticians to use \\(t_\u03b1\\)<sub>\u00a0<\/sub>to represent the t-score with a cumulative probability of \\((1 &#8211; \u03b1)\\). Therefore, if we were interested in a t-score with a 0.9 cumulative probability, \u03b1 would be equal to 1 \u2013 0.9 = 0.1. We would denote the statistic as \\(t_0.1\\).<\/p>\n<p>However, the value of \\(t_\u03b1\\)<sub>\u00a0<\/sub>depends on the number of degrees of freedom and is often written as \\(t_{\u03b1,n-1}\\). For example, we could write \\(t_{0.05,2}= 2.92\\) where the second subscript (2) represents the number of d.f.<\/p>\n<h2><strong>Important Relationships<\/strong><\/h2>\n<p>Just like the normal distribution, the t-distribution is symmetrical about the mean. As such,<\/p>\n<p>$$ t_{\\alpha}= -t_{1 &#8211; \\alpha} \\text{\u00a0and } t_{1 \u2013 \\alpha}\u00a0= -t_{\\alpha} $$<\/p>\n<p>The table below represents one-tailed confidence intervals and various probabilities for a range of degrees of freedom.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-40363 size-medium\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/08\/mock-300x213.png\" alt=\"The table represents one-tailed confidence intervals\" width=\"300\" height=\"213\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/08\/mock-300x213.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/08\/mock-768x545.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/08\/mock-400x284.png 400w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2022\/08\/mock.png 887w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Which of the following statements regarding a t-distribution is <em>most likely<\/em> correct?<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>A t distribution is less spread out than a standard normal distribution.<\/li>\n<li>A t-distribution is symmetric about zero.<\/li>\n<li>As the degrees-of-freedom increase, a t-distribution becomes wider and flat.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>A t-distribution is a symmetrical, bell-shaped distribution that looks like a normal distribution and has a mean of zero.<\/p>\n<p><strong>A is incorrect<\/strong>. A t-distribution is more spread out than a standard normal distribution.<\/p>\n<p><strong>C is incorrect<\/strong>. As the degrees-of-freedom increase, a t-distribution becomes narrower, taller, and approaches a standard normal distribution.<\/p>\n<\/blockquote>\n\n\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 t-distribution, degrees of freedom, and hypothesis testing techniques.\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>A student\u2019s t-distribution is a bell-shaped probability distribution symmetrical about its mean. It is regarded as the most suitable distribution to use in the construction of confidence intervals in the following instances:<\/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-29845","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-Distribution and Degrees of Freedom | CFA Level 1<\/title>\n<meta name=\"description\" content=\"The t-distribution is a bell-shaped, symmetrical probability distribution. Its shape depends on the degrees of freedom. 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