{"id":1585,"date":"2019-10-01T13:29:00","date_gmt":"2019-10-01T13:29:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1585"},"modified":"2026-01-10T08:21:05","modified_gmt":"2026-01-10T08:21:05","slug":"t-distribution","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/t-distribution\/","title":{"rendered":"T-distribution"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Sampling and Estimation (2021 Level I CFA\u00ae Exam \u2013 Reading 10)\",\n  \"description\": \"This video lesson covers sampling and estimation in quantitative methods for CFA Level 1. It explains population vs. sample data, simple and stratified random sampling, sampling errors, the central limit theorem, confidence intervals, and the t-distribution. Real-world examples illustrate concepts like data mining bias, survivorship bias, and time period bias.\",\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/mgY_3CHHYBw\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=mgY_3CHHYBw\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/mgY_3CHHYBw\",\n  \"duration\": \"PT32M00S\"\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/images\/t-distribution-vs-normal-distribution\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-157.jpg\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-157.jpg\",\n  \"caption\": \"t-Distributions vs. Normal Distributions\",\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\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/mgY_3CHHYBw?si=XZ0q1HBih-QZp04o\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n\n<p>The student\u2019s T-distribution is a bell-shaped probability distribution symmetrical about its mean. It is considered the best distribution to use for the construction of confidence intervals when:<\/p>\n\n\n\n<!--more-->\n\n\n\n<ol class=\"wp-block-list\">\n<li>dealing with small samples of less than 30 elements;<\/li>\n\n\n\n<li>the population variance is unknown; and<\/li>\n\n\n\n<li>the distribution involved is either normal or approximately normal.<\/li>\n<\/ol>\n\n\n\n<p>In the absence of outright normality of a given distribution, the T-distribution may still be appropriate for use if the sample size is large enough to allow the application of the central limit theorem. In this case, the distribution is considered approximately normal.<\/p>\n\n\n\n<div style=\"margin: 0 0 20px 0;\">\n  <a\n    href=\"https:\/\/analystprep.com\/free-trial\/\"\n    target=\"_blank\"\n    rel=\"noopener noreferrer\"\n    style=\"\n      display: inline-block;\n      border: 2px solid #1e63ff;\n      color: #1e63ff;\n      background: #ffffff;\n      padding: 10px 14px;\n      border-radius: 10px;\n      font-weight: 500;\n      line-height: 1.35;\n      text-decoration: none;\n    \"\n  >\n    Want to practice t-distribution concepts and hypothesis testing questions for CFA Level I? Try AnalystPrep\u2019s free trial now.\n  <\/a>\n<\/div>\n\n\n\n<p>The T-statistic, also called the T-score, is given by:<\/p>\n<p>$$ t = \\cfrac {(x &#8211; \\mu)}{\\left(\\cfrac {S}{\\sqrt n} \\right)} $$<\/p>\n<p>Where:<\/p>\n<p>x is the sample mean,<\/p>\n<p>\u03bc is the population mean,<\/p>\n<p>S is the sample standard deviation,<\/p>\n<p>n is the sample size.<\/p>\n<p>The T-distribution allows us to analyze distributions that are not perfectly normal. It has the following properties:<\/p>\n<ol>\n<li>it has a mean of zero;<\/li>\n<li>its \\(\\text {variance}= \\frac {v}{ \\left(\\frac {v}{2} \\right) }\\), where <em>v <\/em>represents the number of degrees of freedom and <em>v <\/em>\u2265 2;<\/li>\n<li>although it\u2019s very close to one when there are many degrees of freedom, the variance is greater than 1 at all times. With a large number of degrees of freedom, the T-distribution resembles the normal distribution; and<\/li>\n<li>its tails are fatter than those of the normal distribution, indicating more probability in the tails.<\/li>\n<\/ol>\n<h2><strong>The Degrees of Freedom<\/strong><\/h2>\n<p>The 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<h3><strong>Notations<\/strong><\/h3>\n<p>It is standard practice for statisticians to use t<sub>\u03b1 <\/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<sub>0.1<\/sub>.<\/p>\n<p>However, the value of t<sub>\u03b1 <\/sub>depends on the number of degrees of freedom. For example,<\/p>\n<p>\\(t_{0.05,2}= 2.92\\) where the second subscript (2) represents the number of d.f., and<\/p>\n<p>$$ t_{0.05,20} = 1.725 $$<\/p>\n<h3><strong>Important Relationships<\/strong><\/h3>\n<p>$$ t_{\\alpha}= -t_{1 &#8211; \\alpha} \\text{\u00a0and } t_{1 \u2013 \\alpha}\u00a0= -t_{\\alpha} $$<\/p>\n<p>The above relationships are true because the T-distribution is symmetrical about the mean.<\/p>\n<p>The T-distribution has thicker tails relative to the normal distribution.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-17044\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-157.jpg\" alt=\"t-distribution-vs-normal-distribution\" width=\"2235\" height=\"1057\" 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\" \/><\/p>\n<p>The shape of the T-distribution is dependent on the number of degrees of freedom so that as the number of d.f. increases, the distribution becomes more \u2018spiked,\u2019 and its tails become thinner.<\/p>\n<p>The table below represents one-tailed confidence intervals and various probabilities for a range of degrees of freedom.<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c} \\textbf{r} &amp; \\textbf{90%} &amp; \\textbf{95%} &amp; \\textbf{97.5%} &amp; \\textbf{99.5%} \\\\ \\hline {1} &amp; {3.07768} &amp; {6.31375} &amp; {12.7062} &amp; {63.6567} \\\\ \\hline {2} &amp; {1.88562} &amp; {2.91999} &amp; {4.30265} &amp; {9.92484} \\\\ \\hline {3} &amp; {1.63774} &amp; {2.35336} &amp; {3.18245} &amp; {5.84091} \\\\ \\hline {4} &amp; {1.53321} &amp; {2.13185} &amp; {2.77645} &amp; {4.60409} \\\\ \\hline {5} &amp; {1.47588} &amp; {2.01505} &amp; {2.57058} &amp; {4.03214} \\\\ \\hline {10} &amp; {1.37218} &amp; {1.81246} &amp; {2.22814} &amp; {3.16927} \\\\ \\hline {30} &amp; {1.31042} &amp; {1.69726} &amp; {2.04227} &amp; {2.75000} \\\\ \\hline {100} &amp; {1.29007} &amp; {1.66023} &amp; {1.98397} &amp; {2.62589} \\\\ \\hline {\\infty} &amp; {1.29007} &amp; {1.66023} &amp; {1.98397} &amp; {2.62589} \\\\ \\end{array} $$<\/p>\n<div class=\"notes_inv\">\u00a0<\/div>\n\n\n<div style=\"text-align: center; margin: 32px 0;\">\n  <a\n    href=\"https:\/\/analystprep.com\/free-trial\/\"\n    target=\"_blank\"\n    rel=\"noopener noreferrer\"\n    style=\"\n      display: inline-block;\n      background-color: #1e63ff;\n      color: #ffffff;\n      padding: 12px 26px;\n      border-radius: 12px;\n      font-weight: 600;\n      font-size: 16px;\n      text-decoration: none;\n    \"\n  >\n    Start Free Trial \u2192\n  <\/a>\n\n  <div style=\"margin-top: 10px; font-size: 14px; color: #374151;\">\n    Practice CFA Level I quantitative methods questions on the t-distribution, degrees of freedom, confidence intervals, and hypothesis tests with clear solutions.\n  <\/div>\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>The student\u2019s T-distribution is a bell-shaped probability distribution symmetrical about its mean. It is considered the best distribution to use for the construction of confidence intervals when:<\/p>\n","protected":false},"author":2,"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-1585","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 Overview | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Learn about the t-distribution, its shape, and when to use it for small or non-normal samples in statistical analysis.\" \/>\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\/t-distribution\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"T-Distribution Overview | CFA Level 1 - 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