{"id":25162,"date":"2021-08-12T12:19:10","date_gmt":"2021-08-12T12:19:10","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=25162"},"modified":"2026-03-19T20:30:52","modified_gmt":"2026-03-19T20:30:52","slug":"quantiles-and-related-visualizations","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/quantiles-and-related-visualizations\/","title":{"rendered":"Quantiles and Related Visualizations"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Determine the 5th decile from a dataset\",\n    \"text\": \"A mutual fund achieved the following rates of growth over an 11-month period:\\n\\n{3%, 2%, 7%, 8%, 2%, 4%, 3%, 7.5%, 7.2%, 2.7%, 2.09%}\\n\\nThe 5th decile from the data is closest to:\\n\\nA. 2%\\n\\nB. 3%\\n\\nC. 4%\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B.\\n\\nFirst, arrange the data in ascending order:\\n{2%, 2%, 2.09%, 2.7%, 3%, 3%, 4%, 7%, 7.2%, 7.5%, 8%}\\n\\nThe 5th decile is the 50th percentile (median).\\n\\nPosition = (n + 1) \u00d7 0.5 = (11 + 1) \u00d7 0.5 = 6th data point.\\n\\nThe 6th value is 3%.\"\n    }\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\/08\/cfa-level-1-box-plot.png\",\n  \"caption\": \"Box Plot\",\n  \"width\": 1536,\n  \"height\": 1132,\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\/08\/Reading-2i.jpg\",\n  \"caption\": \"Example: Box and Whisker Plot\",\n  \"width\": 1615,\n  \"height\": 542,\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<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/M0gKgPztSoM\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><strong>Quantiles<\/strong><\/h2>\n<p>Quartiles, quintiles, deciles, and percentiles are values or cut points that partition a finite number of observations into nearly equal-sized subsets. The number of partitions depends on the type of cut point involved.<\/p>\n<p><!--more--><\/p>\n<h3><strong>Quartiles<\/strong><\/h3>\n<p>They divide data into <strong>four<\/strong> parts. The first quartile, Q<sub>1<\/sub>, is referred to as the lower quartile, and the last quartile, Q<sub>4,<\/sub> is known as the upper quartile. Q<sub>1<\/sub> splits data into the lower 25% of the values and the upper 75%. Similarly, the upper quartile subdivides data into the lower 75% of the values and the upper 25%. The difference between the upper and lower quartile is known as the <em><strong>interquartile range<\/strong><\/em>, which indicates the spread of the middle 50% of the data under review.<\/p>\n<h3><strong>Quintiles<\/strong><\/h3>\n<p>Though rarely used in practice, quintiles split a set of data into <strong>five<\/strong> equal parts, i.e., fifths. Therefore, the second quintile splits data into the lower 40% of the values and the upper 60%.<\/p>\n<h3><strong>Deciles<\/strong><\/h3>\n<p>The deciles subdivide data into<strong> ten<\/strong> equal parts. There are 10 deciles in any data set. For example, the fourth decile splits data into the lower 40% of the values and the upper 60%.<\/p>\n<h3><strong>Percentiles<\/strong><\/h3>\n<p>Percentiles split data into <strong>100<\/strong> equal parts, i.e., hundredths. So, for instance, the 77<sup>th <\/sup>percentile splits the data into the lower 77% of the values and the upper 23%.<\/p>\n<p>Financial analysts commonly use the four types of subdivisions to rank investment performance. You should note that quartiles, quintiles, and deciles can all be expressed as percentiles.\u00a0For instance, the first quartile is just the 25<sup>th <\/sup>percentile. Similarly, the fourth decile is simply the 40<sup>th <\/sup>percentile. This enables the application of the formula below:<\/p>\n<p>$$ \\text{Position of percentile} =\\cfrac {(n + 1) y}{100} $$<\/p>\n<p>Where \\(n\\) is the number of observations and \\(y\\) is the percentile.<\/p>\n<p><em><strong>Note:<\/strong> You must always order the data set, usually in ascending order, before calculating any of these values.<\/em><\/p>\n<h4><strong>Example: Quartiles <\/strong><\/h4>\n<p>Given the following distribution of returns, calculate the lower quartile.<\/p>\n<p>{10%\u00a0\u00a0 23%\u00a0\u00a0 12%\u00a0\u00a0 21%\u00a0\u00a0 14%\u00a0\u00a0 17%\u00a0\u00a0 16%\u00a0\u00a0 11%\u00a0\u00a0 15%\u00a0\u00a0 19%}<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>First, we have to arrange the values in ascending order:<\/p>\n<p>{10%\u00a0\u00a0 11%\u00a0\u00a0 12%\u00a0\u00a0 14%\u00a0\u00a0 15%\u00a0\u00a0 16%\u00a0\u00a0 17%\u00a0\u00a0 19%\u00a0\u00a0\u00a0 21%\u00a0\u00a0 23%}<\/p>\n<p>Next, we establish the position of the first quartile. This is simply the 25<sup>th <\/sup>percentile. Therefore:<\/p>\n<p>$$ \\begin{align*} P_{25} &amp; =\\cfrac {(10 + 1)25}{100} \\\\ &amp; = 2.75^{\\text{th}} \\text{ value}\\\\ \\end{align*} $$<\/p>\n<p>Since the value is not straightforward, we have to extrapolate between the 2<sup>nd <\/sup>and the 3<sup>rd<\/sup> data points. The 25<sup>th<\/sup> percentile is three-fourths (0.75) of the way from the 2<sup>nd<\/sup> data point (11%) to the 3<sup>rd<\/sup> data point (12%):<\/p>\n<p>$$ \\begin{align*} &amp; 11\\% + 0.75 * (12 \u2013 11) \\\\ &amp; = 11.75\\% \\end{align*} $$<\/p>\n<h2><strong>Box and Whisker Plot<\/strong><\/h2>\n<p>Box and whisker plot is used to display the dispersion of data across quartiles. A box and whisker plot consists of a \u201cbox\u201d with \u201cwhiskers\u201d connected to the box. It shows the following five-number summary of a set of data.<\/p>\n<h4><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-31494\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/cfa-level-1-box-plot.png\" sizes=\"auto, (max-width: 1536px) 100vw, 1536px\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/cfa-level-1-box-plot.png 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/cfa-level-1-box-plot-300x221.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/cfa-level-1-box-plot-1024x755.png 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/cfa-level-1-box-plot-768x566.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/cfa-level-1-box-plot-400x295.png 400w\" alt=\"\" width=\"1536\" height=\"1132\" \/><\/h4>\n<h4><strong>Example: Box and Whisker Plot<\/strong><\/h4>\n<p>Consider the following box and whisker plot:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-39294\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Reading-2i.jpg\" sizes=\"auto, (max-width: 1615px) 100vw, 1615px\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Reading-2i.jpg 1615w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Reading-2i-300x101.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Reading-2i-1024x344.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Reading-2i-768x258.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Reading-2i-1536x515.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/08\/Reading-2i-400x134.jpg 400w\" alt=\"Example: Box and Whisker Plot\" width=\"1615\" height=\"542\" \/><\/p>\n<ol>\n<li>Which of the following is <em>most likely<\/em> the median?<\/li>\n<\/ol>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol style=\"list-style-type: upper-alpha;\">\n<li>10.<\/li>\n<li>10.5.<\/li>\n<li>11.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<ol start=\"2\">\n<li>Which of the following is <em>most likely<\/em> the interquartile range?<\/li>\n<\/ol>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol style=\"list-style-type: upper-alpha;\">\n<li>4.5.<\/li>\n<li>6.5.<\/li>\n<li>11.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<ol start=\"3\">\n<li>Which of the following is <em>most likely<\/em> the 3<sup>rd<\/sup> Quartile?<\/li>\n<\/ol>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol style=\"list-style-type: upper-alpha;\">\n<li>4.<\/li>\n<li>15.<\/li>\n<li>18.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><strong>Solutions<\/strong><\/p>\n<ol>\n<li><strong>B is correct<\/strong>. Median or Quartile 2 (Q2) = \\(\\frac{(10+11)}{2} = 10.5\\)<\/li>\n<li><strong>C is correct<\/strong>. Interquartile Range is \\(Q_3 \u2212 Q_1 = 15 \u2212 4 = 11\\)<\/li>\n<li><strong>B is correct<\/strong>. Quartile 3 (Q3) is \\(\\frac{3\\times (n + 1)}{ 4} = \\frac{3(7+1)}{ 4 }\\)= 6<sup>th<\/sup> term which is 15.<\/li>\n<\/ol>\n<h2><strong>Quantiles in Investment Practice<\/strong><\/h2>\n<ul>\n<li>Quantiles are used to rank performance.<\/li>\n<li>Quantiles can be used in investment research for comparison purposes. For example, companies can be clustered into deciles to compare the performance of small companies with the large ones. In this case, the first decile will contain the portfolio of companies with the smallest market values, while the tenth decile will contain the companies with the largest market values.<\/li>\n<\/ul>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>A mutual fund achieved the following rates of growth over an 11-month period:<\/p>\n<p>{3%\u00a0\u00a0 2%\u00a0\u00a0 7%\u00a0\u00a0 8%\u00a0\u00a0 2% \u00a0\u00a04%\u00a0\u00a0 3%\u00a0\u00a0 7.5%\u00a0\u00a0 7.2%\u00a0\u00a0 2.7%\u00a0\u00a0 2.09%}<\/p>\n<p>The 5<sup>th<\/sup> decile from the data is\u00a0<em>closest <\/em>to:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>2%.<\/li>\n<li>3%.<\/li>\n<li>4%.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>First, you should re-arrange the data in ascending order:<\/p>\n<p>{2%\u00a0\u00a0 2%\u00a0 2.09%\u00a0 2.7%\u00a0 \u00a0 3%\u00a0\u00a0 3%\u00a0\u00a0 4%\u00a0\u00a0 7%\u00a0\u00a0 7.2%\u00a0\u00a0 7.5%\u00a0\u00a0 8%}<\/p>\n<p>Secondly, you should establish the 5<sup>th<\/sup> decile. This is simply the 50<sup>th<\/sup> percentile and is actually the median:<\/p>\n<p>$$ \\begin{align*} P_{50} &amp; =\\frac {(1 + 11) 50}{100} \\\\ &amp; = 12 \\times 0.5 \\\\ &amp; = 6 \\text{ i}.\\text{e}. \\text{ the }6^{\\text{th}} \\text{ data point}. \\\\ \\end{align*} $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\text{the } 5^{\\text{th}} \\text{ decile} = 50^{\\text{th}} \\text { percentile} = \\text{median} = 3\\% $$<\/p><\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Quantiles Quartiles, quintiles, deciles, and percentiles are values or cut points that partition a finite number of observations into nearly equal-sized subsets. 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