{"id":31826,"date":"2021-09-27T01:40:38","date_gmt":"2021-09-27T01:40:38","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=31826"},"modified":"2026-02-05T12:05:45","modified_gmt":"2026-02-05T12:05:45","slug":"measures-of-central-tendency-2","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/measures-of-central-tendency-2\/","title":{"rendered":"Measures of Central Tendency"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Organizing, Visualizing, and Describing Data \u2013 Part II: Measures of Central Tendency (CFA\u00ae Level I Quantitative Methods)\",\n  \"description\": \"CFA\u00ae Level I Quantitative Methods lesson covering measures of central tendency and descriptive statistics. This session explains how to calculate and interpret mean, median, and mode, evaluate alternative definitions of mean for investment problems, calculate quantiles, and interpret dispersion measures. The lesson also covers target downside deviation, skewness, kurtosis, and correlation, all of which are essential for analyzing financial data and distributions.\",\n  \"uploadDate\": \"2021-10-08\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/M0gKgPztSoM\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=M0gKgPztSoM\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/M0gKgPztSoM\",\n  \"duration\": \"PT41M38S\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/repeat.jpg#image\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/repeat.jpg\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/repeat.jpg\",\n  \"name\": \"Central Tendency Illustration\",\n  \"caption\": \"Visual illustrating key concepts in measures of central tendency.\",\n  \"description\": \"Image from the CFA Level 1 Quantitative Methods *Measures of Central Tendency* page, supporting discussion of the mean, median, and mode as measures used to describe the center of a distribution.\",\n  \"encodingFormat\": \"image\/jpeg\",\n  \"width\": 1024,\n  \"height\": 673,\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  },\n  \"creditText\": \"AnalystPrep\",\n  \"copyrightNotice\": \"\u00a9 AnalystPrep\",\n  \"mainEntityOfPage\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/measures-of-central-tendency-2\/\"\n  },\n  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/measures-of-central-tendency-2\/\"\n  }\n}\n<\/script>\n\n\n\n<p>\n  <iframe loading=\"lazy\"\n    src=\"\/\/www.youtube.com\/embed\/M0gKgPztSoM\"\n    width=\"611\"\n    height=\"343\"\n    allowfullscreen=\"allowfullscreen\">\n  <\/iframe>\n<\/p>\n\n\n\n\n<p>Measures of central tendency are values that tend to occur at the center of a well-ordered data set. As such, some analysts call them measures of central location. Mean, median, and mode are all measures of central tendency. Even then, there are situations in which, compared to the others, one is the most appropriate. The mean is the most common among the three measures. It can also be subdivided into smaller sub-types, as we shall see shortly.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Population vs. Sample<\/strong><\/h2>\n\n\n\n<p>A population includes all of the elements from a set of data. A sample, on the other hand, consists of a few observations drawn from the population. For example, all domestic equity mutual funds available in a country\u2019s market qualify to be called a population. If 15 domestic equity mutual funds are selected from among all the domestic equity mutual funds, then this is a sample.<\/p>\n\n\n\n<a href=\"https:\/\/analystprep.com\/free-trial\/\"\n   target=\"_blank\"\n   rel=\"noopener noreferrer\"\n   style=\"display:block;margin:20px 0 28px;padding:14px 18px;border:2px solid #2563eb;border-radius:12px;text-align:center;color:#2563eb;text-decoration:none;font-weight:500;font-size:15px;background-color:#ffffff;\">\n   Practice measures of central tendency with a free trial.\n<\/a>\n\n\n\n<h3><strong>Uses of Sample<\/strong><\/h3>\n<ul>\n<li>Since collecting data from every element in a population is very difficult, samples can be used to represent the population.<\/li>\n<li>The use of a sample saves time and makes the analysis of a large population manageable.<\/li>\n<\/ul>\n<p>A <strong>parameter<\/strong> refers to a measure that is used to describe a characteristic of the\u00a0<strong>population<\/strong>. It\u2019s a numerical quantity that describes a given aspect of the population as a whole.<\/p>\n<p>A <strong>statistic<\/strong>, on the contrary, is a measure that describes a characteristic of a <strong>sample<\/strong>. This could be the average value or the sample standard deviation of the sampled items. Researchers use sample statistics to estimate the unknown population parameters. For example, we often estimate the actual population mean using the sample mean.<\/p>\n<h2><strong>Arithmetic Mean<\/strong><\/h2>\n<p>The\u00a0<strong>population mean<\/strong> is the summation of all the observed values in a population, \\(\\sum{X_i}\\)<sub>\u00a0<\/sub>divided by the total number of observations, \\(N\\). The population mean differs from the <strong>sample mean<\/strong>, which is based on a few observed values \u2018\\(n\\)\u2019 that are chosen from the population. Thus:<\/p>\n<p>$$\\begin{align} \\text{Population mean} &amp;=\\cfrac { \\sum { { X }_{ i } } }{ N }\\\\ \\text{Sample mean} &amp;=\\cfrac { \\sum { { X }_{ i } } }{ n } \\end{align}$$<\/p>\n<p>Analysts use the sample mean to\u00a0<strong><em>estimate<\/em>\u00a0<\/strong>the actual population mean.<\/p>\n<ul>\n<li>The population mean and the sample mean are both arithmetic means. The arithmetic mean for any data set is unique and is computed using all the data values. Among all the measures of central tendency, it is the only measure for which the sum of the deviations from the mean is\u00a0<strong>zero<\/strong>.<\/li>\n<\/ul>\n<h4><strong>Example: Calculating the Arithmetic Mean<\/strong><\/h4>\n<p>The following are the annual returns realized from a given asset between 2005 and 2015.<\/p>\n<p>{ 12% \u00a0\u00a013%\u00a0 \u00a011.5%\u00a0\u00a0 14%\u00a0\u00a0 9.8%\u00a0\u00a0 17%\u00a0\u00a0 16.1%\u00a0\u00a0 13%\u00a0\u00a0 11%\u00a0\u00a0 14% }<\/p>\n<p>1. Calculate the population mean.<\/p>\n<p>2. Compute the sample mean assuming the returns for the first 7 years are unknown, i.e., we only have 13%, 11%, and 14%.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\begin{align*} \\text{Population mean} &amp; =\\cfrac {(0.12 + 0.13 + 0.115 + 0.14 + 0.098 + 0.17 + 0.161 + 0.13 + 0.11 + 0.14)}{10} \\\\ &amp; = 0.1314 \\text{ or } 13.14\\% \\\\ \\\\ \\text {Sample mean} &amp; = \\cfrac {(0.13 + 0.11 + 0.14)}{3} \\\\ &amp; = 0.1267 \\text{ or } 12.67\\% \\\\ \\end{align*} $$<\/p>\n<p>A commonly-cited demerit of the arithmetic mean is that it\u2019s not resistant to the effects of extreme observations or what we call \u2018outsider values.\u2019 For instance, consider the following data set:<\/p>\n<p>{1\u00a0\u00a0 3\u00a0\u00a0 4\u00a0\u00a0 5\u00a0\u00a0 34}<\/p>\n<p>The arithmetic mean is 9.4, which is greater than most of the values. This is due to the last extreme observation, i.e., 34.<\/p>\n<h3><strong>Properties of the Arithmetic Mean<\/strong><strong>\u00a0<\/strong><\/h3>\n<ul>\n<li>The sum of the deviations around the mean always equals 0.<\/li>\n<li>The arithmetic mean is highly sensitive to extremely large or small observations (outliers).<\/li>\n<\/ul>\n<h2><strong>Trimmed Mean<\/strong><\/h2>\n<p>The trimmed mean is a measure of central tendency in which we calculate the mean by excluding a small percentage of the lowest and highest values. For example, we calculate the mean in a 5% trimmed mean by removing the lowest 2.5% and the highest 2.5% of values.<\/p>\n<h2><strong>Winsorized mean<\/strong><\/h2>\n<p>The Winsorized mean is a measure of central tendency. It is calculated by assigning a stated percentage of the lowest values equal to one specified low value and a stated percentage of the highest values equal to one specified high value. In the same way, as the trimmed mean, this approach removes a significant number of outliers from a data set.<\/p>\n<h2><strong>Weighted Mean<\/strong><\/h2>\n<p>The weighted mean takes the weight of every observation into account. It recognizes that different observations may have <strong>disproportionate effects<\/strong> on the arithmetic mean. Thus:<\/p>\n<p>$$ \\text{Weighted mean} = \\sum { { X }_{ i }{ W }_{ i } } $$<\/p>\n<h4><strong>Example: Calculating the Weighted Mean<\/strong><\/h4>\n<p>A portfolio consists of 30% ordinary shares, 25% T-bills, and 45% preference shares with returns of 7%, 4%, and 6%, respectively. The portfolio return is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The return of any portfolio is always the weighted average of the returns of individual assets. Thus:<\/p>\n<p>$$ \\text{Portfolio return} = (0.07 \u00d7 0.3) + (0.04 \u00d7 0.25) + (0.06 \u00d7 0.45) = 5.8\\% $$<\/p>\n<h2><strong>Geometric Mean<\/strong><\/h2>\n<p>The geometric mean is a measure of central tendency, mainly used to measure growth rates. We define it as the n<sup>th<\/sup>\u00a0root of the product of n observations:<\/p>\n<p>$$ \\text{GM} ={ \\left( { X }_{ 1 }\\ast { X }_{ 2 }\\ast { X }_{ 3 }\\ast { X }_{ 4 }\\ast &#8230;\\ast { X }_{ n-1 }\\ast { X }_{ n } \\right) }^{ \\frac { 1 }{ n } } $$<\/p>\n<p>The formula above only works when we have <strong>non-negative<\/strong> values. To solve this problem, especially when dealing with percentage returns, we add 1 to every value and then subtract 1 from the final result.<\/p>\n<h4><strong>Example: Calculating the Geometric Mean<\/strong><\/h4>\n<p>An ordinary share from a certain company registered the following rates of return over a 6-year period:<\/p>\n<p>{ -4%\u00a0\u00a0 2%\u00a0\u00a0 8%\u00a0\u00a0 12%\u00a0\u00a0 14%\u00a0\u00a0 15% }<\/p>\n<p>The compound annual rate of return for the period is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\text{Geometric Mean} = (0.96 \u00d7 1.02 \u00d71.08 \u00d7 1.12 \u00d7 1.14 \u00d7 1.15)^{\\frac{1}{6}} = 1.0761 \u2013 1 = 0.0761 \\text{ or } 7.61 % $$<\/p>\n<p><strong>Computing Geometric Mean Using BA II Plus\u2122 Financial Calculator<\/strong>:<\/p>\n<p>Enter 1.55280 [y<sup>x<\/sup>] 6 [1\/x] [=]<\/p>\n<p>Where 1.55280 = 0.96 \u00d7 1.02 \u00d71.08 \u00d7 1.12 \u00d7 1.14 \u00d7 1.15<\/p>\n<h3><strong>Important Points Related to the Geometric Mean and the Arithmetic Mean<\/strong><\/h3>\n<ul>\n<li>The geometric mean is always less than or equal to the arithmetic mean.<\/li>\n<li>The geometric mean is equal to the arithmetic mean when there is no variability in the observations (when all the observations in the series are the same).<\/li>\n<li><span style=\"font-size: revert; color: initial;\">The gap between the geometric mean and the arithmetic widens as the variability of values increases.<\/span><\/li>\n<li>The arithmetic mean should be used for estimating the average return over a one-period horizon.<\/li>\n<li>For estimating the average returns over more than one period, the geometric mean should be used.<\/li>\n<\/ul>\n<h3><strong>Relationship between Arithmetic Mean and Geometric Mean<\/strong><\/h3>\n<p>The relationship between the arithmetic mean and geometric mean is given by:<\/p>\n<p>$$\\bar{X}_G \\approx \\bar{X}-\\frac{s^2}{2}$$<\/p>\n<p>Where:<\/p>\n<p>\\(\\bar{X}_G\\) = Geometric mean.<\/p>\n<p>\\(\\bar{X}\\) = Arithmetic mean.<\/p>\n<p>\\(s^2\\) = Sample variance.<\/p>\n<p>The above equation shows that the larger the variance of the sample, the wider the difference between the geometric mean and the arithmetic mean.<\/p>\n<h2><strong>Harmonic Mean<\/strong><\/h2>\n<p>Analysts use the harmonic mean to determine the average growth rates of economies or assets. If we have \\(N\\) observations:<\/p>\n<p>$$ \\text{HM} = \\cfrac {N}{ \\left(\\sum { \\frac { 1 }{ { X }_{ i } } } \\right)} $$<\/p>\n<h4><strong>Example: Calculating the Harmonic Mean<\/strong><\/h4>\n<p>For the last three months of 2015, the price of a stock was $4, $5, and $7, respectively. The average cost per share is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$\\text{Harmonic Mean}=\\frac{3}{\\left(\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{7}\\right)}=\\$ 5.06$$<\/p>\n<h3><strong>Important Points Related to the Harmonic Mean<\/strong><\/h3>\n<ul>\n<li>\\(\\text{Harmonic mean &lt; Geometric mean &lt; Arithmetic mean when returns are variable.}\\)<\/li>\n<li>\\(\\text{Harmonic mean = Geometric mean = Arithmetic mean when returns are constant.}\\)<\/li>\n<li>\\(\\text{Arithmetic mean \u00d7 Harmonic mean = Geometric mean.}\\)<\/li>\n<\/ul>\n<h2><strong>Median<\/strong><\/h2>\n<p>The median is the statistical value located at the center of a data set organized in the order of magnitude. For an odd number of observations, the median is simply the <strong>middle value<\/strong>. If the number of observations is even, the median is the\u00a0<strong>middle point<\/strong>\u00a0(average) of the two middle values. Unlike the arithmetic mean,\u00a0<strong>the median is resistant to the effects of extreme observations<\/strong>.<\/p>\n<h4><strong>Example: Calculating the Median<\/strong><\/h4>\n<p>The following are the annual returns on a given asset realized between 2005 and 2015.<\/p>\n<p>{ 12% \u00a0\u00a013%\u00a0 \u00a011.5%\u00a0\u00a0 14%\u00a0\u00a0 9.8%\u00a0\u00a0 17%\u00a0\u00a0 16.1%\u00a0\u00a0 13%\u00a0\u00a0 11%\u00a0\u00a0 14% }<\/p>\n<p>The median is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>First, we arrange the returns in ascending order:<\/p>\n<p>{ 9.8%\u00a0\u00a0 11%\u00a0\u00a0 11.5%\u00a0\u00a0 12%\u00a0\u00a0 13%\u00a0\u00a0 13%\u00a0\u00a0 14%\u00a0\u00a0 14%\u00a0\u00a0 16.1%\u00a0\u00a0 17% }<\/p>\n<p>Since the number of observations is even, the median return will be the middle point of the two middle values:<\/p>\n<p>$$\\frac{13\\%+13\\%}{2}=13\\%$$<\/p>\n<p>The main advantage of the median is that the median is less affected by outliers than the mean. Therefore, the median is useful in describing data that follow a non-symmetric distribution, such as a skewed distribution, which we will see later in this reading.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"673\" class=\"aligncenter size-full wp-image-31827\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/repeat.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/repeat.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/repeat-300x197.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/repeat-768x505.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/repeat-400x263.jpg 400w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<h3><strong>Limitations of the Median<\/strong><\/h3>\n<ul>\n<li>The median only focuses on the relative position of the ranked observation and ignores the rest of the information about the size of the obser\u00advations.<\/li>\n<\/ul>\n<h2><strong>Mode<\/strong><\/h2>\n<p>The mode is the value that occurs most frequently in a data set. On a histogram, it is the highest bar. A data set may have a mode or none, e.g., the returns in the example above. One of its major merits is that it can be determined from incomplete data, provided we know the observations with the highest frequency.<\/p>\n<p>If a distribution has two modes, it is called bimodal. If the distribution has the three most frequently occurring values, then it is called trimodal.<\/p>\n<p>An interval with the highest frequency is called the modal interval (or intervals) in a frequency distribution. In a histogram, the modal interval always has the highest bar. The mode is the only measure of central tendency that can be used with nominal data.<\/p>\n<h4><strong>Example: Calculating the Mode<\/strong><\/h4>\n<p>Determine the mode from the following data set:<\/p>\n<p>{ 20%\u00a0\u00a0 23%\u00a0\u00a0 20%\u00a0\u00a0 16%\u00a0\u00a0 21%\u00a0\u00a0 20%\u00a0\u00a0 16%\u00a0\u00a0 23%\u00a0\u00a0 25%\u00a0\u00a0 27%\u00a0\u00a0 20% }<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The mode is 20%. It occurs 4 times, a frequency higher than that of any other value in the data set.<\/p>\n\n\n<div style=\"text-align:center; margin-top:32px;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"display:inline-block;padding:16px 40px;background-color:#2563eb;color:#ffffff;text-decoration:none;border-radius:14px;font-weight:700;font-size:17px;\">\n     Start Free Trial \u2192\n  <\/a>\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Measures of central tendency are values that tend to occur at the center of a well-ordered data set. As such, some analysts call them measures of central location. Mean, median, and mode are all measures of central tendency. Even then,&#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-31826","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>Central Tendency Measures | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Learn about mean, median, and mode as key measures of central tendency in statistical analysis and data interpretation.\" \/>\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\/measures-of-central-tendency-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Central Tendency Measures | CFA Level 1 - 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