{"id":31297,"date":"2021-09-22T06:36:51","date_gmt":"2021-09-22T06:36:51","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=31297"},"modified":"2026-03-31T12:31:49","modified_gmt":"2026-03-31T12:31:49","slug":"standard-error-of-the-sample-mean","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/standard-error-of-the-sample-mean\/","title":{"rendered":"Standard Error of the Sample Mean"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Sampling and Estimation (2025 Level I CFA\u00ae Exam \u2013 Quantitative Methods \u2013 Module 5)\",\n  \"description\": \"CFA\u00ae Level I Quantitative Methods video lesson covering Sampling and Estimation. This module explains sampling methods (simple random, stratified, cluster, convenience, and judgmental), sampling error, the sampling distribution, and the central limit theorem. It also covers standard error of the sample mean, properties of estimators, point estimates vs confidence intervals, confidence intervals for a population mean under different variance assumptions, resampling methods (bootstrap and jackknife), and key sample selection issues such as data mining bias, survivorship bias, look-ahead bias, and time-period bias.\",\n  \"uploadDate\": \"2021-11-11\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/QDmc4Pa92bs\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=QDmc4Pa92bs\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/QDmc4Pa92bs\",\n  \"duration\": \"PT37M40S\"\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\": \"What happens to the standard error of the sample mean when the sample size increases to 80?\",\n    \"text\": \"Suppose we increased the sample size to 80 in the example above and derived similar values for the mean and standard deviation of returns. The standard error of the sample mean would be closest to which of the following? A. 0.01. B. 0.02. C. 0.08.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The standard error of the sample mean is closest to 0.01. The standard error is calculated as the sample standard deviation divided by the square root of the sample size. Using S = 0.13 and n = 80, the standard error equals 0.13 \/ \u221a80 \u2248 0.01. This demonstrates that increasing the sample size reduces the standard error of the mean.\",\n      \"dateCreated\": \"2026-01-21\"\n    }\n  }\n}\n<\/script>\n\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/QDmc4Pa92bs\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The standard error (SE) of the sample mean refers to the standard deviation of the distribution of the sample means. It gives analysts an estimate of the variability they would expect if they were to draw multiple samples from the same population. While the standard deviation measures the variability obtained within one sample, the standard error gives an estimate of the variability among many samples.<\/p>\n<p><!--more--><\/p>\n<div style=\"text-align: center; margin: 22px 0;\">\n  <div style=\"max-width: 680px; margin: 0 auto;\">\n    <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n       style=\"display: flex; align-items: center; justify-content: center;\n       width: 100%; padding: 10px 18px;\n       border: 2px solid #1e5bd8; color: #1e5bd8;\n       border-radius: 9999px; text-decoration: none; font-weight: 600;\">\n      Practice standard error concepts with our free trial\n    <\/a>\n  <\/div>\n<\/div>\n<h2><strong>Standard Error of the Sample Mean Formula<\/strong><\/h2>\n<p>Provided the population standard deviation, \u03c3, is known, analysts use the following formula to estimate the standard error of the sample mean, denoted as \u03c3<sub>x<\/sub>:<\/p>\n<p>$$ \\sigma_x=\\cfrac {\\sigma}{\\sqrt n} $$<\/p>\n<p>Where <em>n<\/em> is the sample size.<\/p>\n<p>It is noteworthy that the population standard deviation, \u03c3, is usually unknown.&nbsp; When the population standard deviation, \u03c3, is unknown, the following formula is used to estimate the standard error of the sample mean, also denoted as S<sub>x<\/sub>:<\/p>\n<p>$$ S_x =\\cfrac {S}{\\sqrt n} $$<\/p>\n<p>Where:<\/p>\n<p>\\(S\\) = The sample standard deviation; and<\/p>\n<p>\\(S^2 =\\cfrac {\\sum \\left(X_i- X \\right)^2}{n- 1}&nbsp; \\).<\/p>\n<h2><strong>Breaking Down the Standard Error of the Sample Mean<\/strong><\/h2>\n<p>The standard error of the sample mean gives analysts an idea of how <strong>precisely <\/strong>the sample mean estimates the population mean. A lower value of the standard error indicates a more precise estimation of the population mean. On the other hand, a larger value of the standard error indicates a less precise estimate of the population mean.<\/p>\n<p>It is also important to note that the standard error becomes smaller as the sample size increases. This happens because increasing the sample size ultimately brings the sample mean closer to the true value of the population mean.<\/p>\n<h4><strong>Example: Calculating the Standard Error of the Sample Mean When the Population Standard Deviation is Known<\/strong><\/h4>\n<p>In a certain property investment company with an international presence, workers have a mean hourly wage of $12 with a population standard deviation of $3. Given a sample size of 30, the standard error of the sample mean is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\begin{align*} \\sigma_x &amp; =\\cfrac {\\sigma}{\\sqrt n} \\\\ &amp; =\\cfrac {3}{\\sqrt {30}} \\\\ &amp; = $0.55 \\\\ \\end{align*} $$<\/p>\n<p><em><strong>Interpretation:<\/strong> <\/em>If we were to draw several samples of size 30 from the employee population and construct a sampling distribution of the sample means, we would end up with a mean of $12 and a standard error of $0.55.<\/p>\n<h4><strong>Example: Calculating the Standard Error of the Sample Mean When \u03c3 is Unknown<\/strong><\/h4>\n<p>A sample of 30 latest returns on XYZ stock reveals a mean return of $4 with a sample standard deviation of $0.13. The standard error of the sample mean is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\begin{align*} S_x &amp; =\\cfrac {S}{\\sqrt n} \\\\ &amp; =\\cfrac {0.13}{\\sqrt {30}} \\\\ &amp; = $0.02 \\\\ \\end{align*} $$<\/p>\n<p><em><strong>Interpretation:<\/strong> <\/em>If we were to draw more samples from the population of yearly returns on XYZ stock and construct a sample mean distribution, we would end up with a mean of $4 and a standard error of $0.02.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Suppose we increased the sample size to 80 in the example above and derived similar values for the mean and standard deviation of returns, then the standard error of the sample mean would be <em>closest<\/em> to:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>0.01.<\/li>\n<li>0.02.<\/li>\n<li>0.08.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>$$ \\begin{align*} S_x &amp; =\\cfrac {S}{\\sqrt n} \\\\ &amp; = \\cfrac {0.13}{\\sqrt {80}} \\\\ &amp; = $0.01 \\\\ \\end{align*} $$<\/p>\n<p>This clearly proves that increasing the sample size reduces the SE of the sample mean.<\/p>\n<\/blockquote>\n<div style=\"text-align:center; margin: 40px 0;\">\n  <a href=\"https:\/\/analystprep.com\" target=\"_blank\" style=\"display:inline-flex; align-items:center; justify-content:center; padding:12px 20px; border-radius:999px; background-color:#1a73e8; color:#ffffff; text-decoration:none; font-weight:600;\">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"font-size:15px; margin-top:12px; color:#555;\">\n    Practice standard error formulas, sampling distributions, and interpretation with exam-style questions.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The standard error (SE) of the sample mean refers to the standard deviation of the distribution of the sample means. It gives analysts an estimate of the variability they would expect if they were to draw multiple samples from the&#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-31297","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>Standard Error of the Mean | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Understand the standard error of the sample mean, its formula, and how it measures variability in repeated sampling.\" \/>\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\/standard-error-of-the-sample-mean\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Standard Error of the Mean | CFA Level 1 - 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