{"id":5911,"date":"2022-08-05T19:19:08","date_gmt":"2022-08-05T19:19:08","guid":{"rendered":"https:\/\/commerceiets.com\/?p=5911"},"modified":"2022-08-05T19:19:08","modified_gmt":"2022-08-05T19:19:08","slug":"measures-of-dispersion-in-statistics","status":"publish","type":"post","link":"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/","title":{"rendered":"MEASURES OF DISPERSION IN STATISTICS FREE NOTES"},"content":{"rendered":"\n<h5 class=\"wp-block-heading\">MEASURES OF DISPERSION IN STATISTICS : Dispersion, meaning and definition; Importance; Relative and absolute measures of dispersion.<\/h5>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_65 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title \" >Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/#DISPERSION\" title=\"DISPERSION\">DISPERSION<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/#ACCORDING_TO_BROOK_AND_DICKS\" title=\"ACCORDING TO BROOK AND DICKS\">ACCORDING TO BROOK AND DICKS<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/#ACCORDING_TO_SIMPSON_AND_KAFKA\" title=\"ACCORDING TO SIMPSON AND KAFKA\">ACCORDING TO SIMPSON AND KAFKA<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/#ACCORDING_TO_WI_KING\" title=\"ACCORDING TO W.I. KING\">ACCORDING TO W.I. KING<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/#FEATURES_OF_DISPERSION\" title=\"FEATURES OF DISPERSION\">FEATURES OF DISPERSION<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/#OBJECTIVES_OR_IMPORTANCE_OR_USES_OF_MEASURES_OF_DISPERSION\" title=\"OBJECTIVES OR IMPORTANCE OR USES OF MEASURES OF DISPERSION\">OBJECTIVES OR IMPORTANCE OR USES OF MEASURES OF DISPERSION<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/#CHARACTERISTICS_OF_A_GOOD_MEASURE_OF_DISPERSION\" title=\"CHARACTERISTICS OF A GOOD MEASURE OF DISPERSION\">CHARACTERISTICS OF A GOOD MEASURE OF DISPERSION<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/2\/#ABSOLUTE_MEASURES_OF_DISPERSION\" title=\"ABSOLUTE MEASURES OF DISPERSION\">ABSOLUTE MEASURES OF DISPERSION<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/2\/#RANGE\" title=\"RANGE\">RANGE<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/2\/#QUARTILE_DEVIATION_or_INTER-QUARTILE_RANGE\" title=\"QUARTILE DEVIATION or INTER-QUARTILE RANGE\">QUARTILE DEVIATION or INTER-QUARTILE RANGE<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/4\/#STANDARD_DEVIATION\" title=\"STANDARD DEVIATION\">STANDARD DEVIATION<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/4\/#VARIANCE\" title=\"VARIANCE\">VARIANCE<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/5\/#RELATIVE_MEASURES_OF_DISPERSION\" title=\"RELATIVE MEASURES OF DISPERSION\">RELATIVE MEASURES OF DISPERSION<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/5\/#_COEFFICIENT_OF_RANGE\" title=\"&nbsp;COEFFICIENT OF RANGE:\">&nbsp;COEFFICIENT OF RANGE:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/5\/#COEFFICIENT_OF_QUARTILE_DEVIATION\" title=\"COEFFICIENT OF QUARTILE DEVIATION:\">COEFFICIENT OF QUARTILE DEVIATION:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/5\/#COEFFICIENT_OF_MEAN_DEVIATION\" title=\"COEFFICIENT OF MEAN DEVIATION\">COEFFICIENT OF MEAN DEVIATION<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/5\/#COEFFICIENT_OF_STANDARD_DEVIATION\" title=\"COEFFICIENT OF STANDARD DEVIATION\">COEFFICIENT OF STANDARD DEVIATION<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/5\/#COEFFICIENT_OF_VARIATION\" title=\"COEFFICIENT OF VARIATION\">COEFFICIENT OF VARIATION<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"DISPERSION\"><\/span><strong>DISPERSION<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Dispersion indicates the measure of the extent to which individual items differ. It indicates lack of uniformity in the size of items.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ACCORDING_TO_BROOK_AND_DICKS\"><\/span><strong>ACCORDING TO BROOK AND DICKS<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<p>&#8220;Dispersion is the degree of the variation of the variable about a central value.&#8221;<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ACCORDING_TO_SIMPSON_AND_KAFKA\"><\/span><strong>ACCORDING TO SIMPSON AND KAFKA<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<p>&#8220;The measurement of the scatterness of the mass of figures in a series about an average is called a measure of variation or dispersion.\u201d<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ACCORDING_TO_WI_KING\"><\/span><strong>ACCORDING TO W.I. KING<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<p>&#8221; The term dispersion is based to indicate the facts that within a given group, the items differ from one another in size or in other words, there is lack of uniformity in their size.&#8221;<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"FEATURES_OF_DISPERSION\"><\/span><strong>FEATURES OF DISPERSION<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>1. It involves different methods through which variations can be measured in quantitative manner.<\/p>\n\n\n\n<p>2. It deals with statistical series.<\/p>\n\n\n\n<p>3. It indicates the degree to which various items of a series deviate from its central value.<\/p>\n\n\n\n<p>4. It supplements the measures of central tendency in revealing the characteristics of a frequency distribution.<\/p>\n\n\n\n<p>5. It speaks of the reliability of the average value of a series.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"OBJECTIVES_OR_IMPORTANCE_OR_USES_OF_MEASURES_OF_DISPERSION\"><\/span><strong>OBJECTIVES OR IMPORTANCE OR USES OF MEASURES OF DISPERSION<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p><strong>1. To Measure the Reliability of an Average: <\/strong>Dispersion tells us how far an average is representative of the mass of the data. When the dispersion is small, the average is a typical value in the sense that it is a good estimate of the average in the universe from which the data have been taken.<\/p>\n\n\n\n<p><strong>2. To Serve as a Basis for Control of the Variability: <\/strong>In order to control the variation or dispersion of a phenomenon it is necessary to determine the nature and cause of variation. The measurement of inequality in the distribution of income and wealth requires the measures of variation Similarly variations in body temperature, blood pressure etc., are noted for proper diagnosis.<\/p>\n\n\n\n<p><strong>3. To Compare two or more Series with Regard to their Variability: <\/strong>The study of dispersion is essential for determining the degree of consistency, uniformity, reliability etc.. A low degree of variation means more uniformity, consistency, reliability of data, whereas a high degree of variation lacks uniformity, consistency, reliability etc.<\/p>\n\n\n\n<p><strong>4. To Facilitate the use of Other Statistical Techniques:<\/strong> The study of dispersion helps in the application of various statistical tools like correlation, regression, statistical quality control etc.<\/p>\n\n\n\n<p><strong>5. To establish trend in time series:<\/strong> Dispersion helps to find out the trend and remove seasonal, cyclical and random fluctuations.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"CHARACTERISTICS_OF_A_GOOD_MEASURE_OF_DISPERSION\"><\/span><strong>CHARACTERISTICS OF A GOOD MEASURE OF DISPERSION<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>A good measure of dispersion should possess the following characteristics:<\/p>\n\n\n\n<ul><li>It should be simple to understand.<\/li><li>It should be easy to calculate.<\/li><li>It should be rigidly defined.<\/li><li>It should be based on each and every item of the distribution.<\/li><li>It should be suitable for algebraic and arithmetical manipulation.<\/li><li>It should have sampling stability.<\/li><li>It should not be unduly affected by extreme items.<\/li><\/ul>\n\n\n\n<!--nextpage-->\n\n\n\n<h2 class=\"wp-block-heading\">MEASURES OF DISPERSION IN STATISTICS <\/h2>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"941\" height=\"868\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/image.png\" alt=\"MEASURES OF DISPERSION IN STATISTICS \" class=\"wp-image-5912\" srcset=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/image.png 941w, https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/image-300x277.png 300w, https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/image-768x708.png 768w\" sizes=\"(max-width: 941px) 100vw, 941px\" \/><figcaption>MEASURES OF DISPERSION IN STATISTICS <\/figcaption><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>ABSOLUTE MEASURES OF DISPERSION<\/strong><\/h3>\n\n\n\n<p>The absolute measures of dispersion can be compared with one another only if the two belong to the same population and are expressed in the same units like inches, kilograms, rupees etc. The various absolute measures of dispersion are:<\/p>\n\n\n\n<ul><li>Range<\/li><li>Quartile deviation<\/li><li>Mean deviation<\/li><li>Standard deviation<\/li><li>Variance<\/li><\/ul>\n\n\n\n<p>The detailed explanation of all the terms is as follows:<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>RANGE<\/strong><\/h4>\n\n\n\n<p>Range is the simplest measure of dispersion. The difference between the highest value and lowest value of a series is known as range. It is defined as the difference between the two extreme items of the distribution. In other words, range is the difference between the highest and lowest values of the distribution.<\/p>\n\n\n\n<p>Absolute Range = Highest Value-Lowest Value<\/p>\n\n\n\n<p>R=H.V.-L.V.<\/p>\n\n\n\n<p><strong>MERITS OF RANGE: <\/strong>Range can be easily calculated as we require only two extreme values. If one is interested in getting a quick, but not very accurate picture of variability, one can compute range Range is useful for the following reasons:<\/p>\n\n\n\n<p>1. Quality Control: Quality control is done to check the quality of the product without 100% inspection. Range plays an important role, when statistical methods of quality control are used. In preparing R-charts, range plays an important role. Thus range is useful in determining and maintaining the quality of the products.<\/p>\n\n\n\n<p>2. Variations in Stock and Shares Prices: Range is useful in studying the variations in the prices of shares and stock. By computing range, we can get an idea about the range of fluctuation, say, gold, jewellery prices etc..<\/p>\n\n\n\n<p>3. Forecasts: The meteorology does make use of the range in determining the difference between the minimum and maximum temperature. This information is useful for general public.<\/p>\n\n\n\n<p><strong>DEMERITS OF RANGE: <\/strong>Range suffers from the following limitations:<\/p>\n\n\n\n<p>1. Range is not based on all the items of the series: We require only the extreme items largest and smallest. The items in-between are not given any weightage. Under the circumstances, range cannot be representative and hence it is not suitable for mathematical treatment.<\/p>\n\n\n\n<p>According to W.I.King who states &#8220;Range is too indefinite to be used as a practical measure of dispersion.&#8221;<\/p>\n\n\n\n<p>2. Range is affected by unusual items: Range is affected by unusual items of the series as it takes into account only items-largest and smallest. But what about the items which are very close to the smallest and the largest Under such circumstances, range cannot be considered as a reliable measure of Dispersion.<\/p>\n\n\n\n<p>3. Not fruitful in open end series: In open-end classes, range cannot be determined, because the lower and upper classes are not given.<\/p>\n\n\n\n<p>4. Affected by fluctuations of sample: Range is very much affected by the fluctuations of sampling. The value of range i e is never stable, it varies from sample to sample. As the size of the sample increases, the range tends to increase though not in the same ratio.<\/p>\n\n\n\n<p>5. Do not represent all values of the series: If the two values ie, the smallest and the largest items of a distribution are not changed whereas other items are replaced by a set of observations within these two limits, the range will not change. But the value of range is considerably changed if any item is added or deleted on either side of the extreme value. It means range does not take into account the range composition of the series. Its concern is mainly with smallest and largest items.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>QUARTILE DEVIATION or INTER-QUARTILE RANGE<\/strong><\/h4>\n\n\n\n<p>Quartile Deviation or Inter-quartile range is the difference between First quartile and third quartiles. By using quartile deviation, the dependence on extreme items can be avoided.<\/p>\n\n\n\n<p>QD= (Q<sub>3<\/sub>-Q<sub>1 <\/sub>)\/2<\/p>\n\n\n\n<p><strong>MERITS OF QUARTILE DEVIATION:<\/strong><\/p>\n\n\n\n<ol type=\"1\"><li>Quartile deviation is easy to calculate. It is based on two values only i.e. lower quartile and upper quartile. It does not involve any mathematical complications. Since quartile deviation makes use of 50% of the data, it is a better measure than range which takes into account just two items ie., the largest and the smallest.<\/li><li>Quartile deviation is superior to range because quartiles are generally located where there is big concentration of items The values of quartiles will not change much it one more value of any size is added to or taken away from the distribution.<\/li><li>The quartile deviation like range is not influenced by the extreme items. It can be used in highly skewed distribution where other measures of dispersion do not give reliable results. The quartile deviation is influenced by the position in an array of each item in the series.<\/li><li>In open-end distribution, quartile deviation is suitable. The quartile deviation is also useful where data is ranked but not measured qualitatively, say in case of beauty, intelligence, tastes etc.<\/li><li>The quartile deviation provides wonderful short-cut for estimating standard deviation<\/li><li>It is quite satisfactory when only the middle half of the group is dealt with as it takes into account 50% of the middle data, i.e., between lower quartile and upper quartile.<\/li><\/ol>\n\n\n\n<p><strong>DEMERITS OF QUARTILE DEVIATION:<\/strong><\/p>\n\n\n\n<ol type=\"1\"><li>Quartile deviation ignores 50% of the items i.c., the first 25% and the last 25% The value of quartile d\u00e9viation does not depend on each and every item of the series. So it cannot be called a good method of measuring dispersion. An ideal method should be based on each and every item.<\/li><li>It is not a measure of dispersion, because it does not show the scatter around an average but rather a distance on a scale. It means quartile deviation is not itself measured from an average. But it is a positional average. Some Statisticians are of the view that quartile deviation is a measure of partition rather than a measure of dispersion.<\/li><li>Quartile deviation is not capable of further algebraic treatment because it ignores the first and the last 25% items on both sides.<\/li><li>Quartile deviation depends on the values of central items. If these values are irregular, then the results are affected adversely.<\/li><li>Quartile deviation is highly influenced by fluctuations of sampling.<\/li><li>Quartile deviation is not affected by the distribution of items between lower quartile and upper quartile or by the distribution outside the quartiles. That means the values of the quartiles may be the same for two dissimilar series.<\/li><\/ol>\n\n\n\n<p>Thus quartile deviation has limited reliability, particularly, in case of those distributions where variation is substantial.<\/p>\n\n\n\n<!--nextpage-->\n\n\n\n<h2 class=\"wp-block-heading\">MEASURES OF DISPERSION IN STATISTICS <\/h2>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>MEAN DEVIATION<\/strong><\/h4>\n\n\n\n<p>Average deviation or mean deviation is the average difference between the items in a series from the mean or median or mode of that series. Theoretically, it is beneficial to take deviations from median, because the sum of deviations of items from Median, is minimum when plus and minus signs are ignored. But, often mean is used for its calculation, so it is called mean deviation. The mean deviation is also called as the first moment of dispersion.<\/p>\n\n\n\n<p><strong>ACCORDING TO CLARK AND SCHKADE<\/strong><\/p>\n\n\n\n<p>&#8220;Average deviation is the average amount of scatter of the items in a distribution from either the mean or the median, ignoring the signs of the deviations. The average that is taken of the scatter is an arithmetic mean, which accounts for the fact that this measure is often called the mean deviation.&#8221;<\/p>\n\n\n\n<p>The formula for calculating mean deviation is<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>FORMULAE OF MEAN DEVIATION<\/strong><\/td><\/tr><tr><td>Individual series<\/td><td>MD= \u03a3 | d |\/N<\/td><\/tr><tr><td>Discrete series<\/td><td>MD= \u03a3 f| d |\/N<\/td><\/tr><tr><td>Continuous Series<\/td><td>MD= \u03a3 f| d |\/N<\/td><\/tr><\/tbody><\/table><figcaption>MEASURES OF DISPERSION IN STATISTICS <\/figcaption><\/figure>\n\n\n\n<p><strong>MERITS OF MEAN DEVIATION:<\/strong><\/p>\n\n\n\n<p>1 It is relatively simple to calculate and easy to understand. It is very close to arithmetic mean. In order to give information to persons having no knowledge of statistics the measure of mean deviation is more useful.<\/p>\n\n\n\n<p>2. It is based on all the items of the series. Any small change in the series would affect the values of mean deviation.<\/p>\n\n\n\n<p>3. It is less affected by the extreme items as compared to standard deviation.<\/p>\n\n\n\n<p>4. Mean deviation is useful for comparison because the deviations are taken from actual mean or median or mode.<\/p>\n\n\n\n<p>5. It is based on measurement rather than estimation.<\/p>\n\n\n\n<p>6. It is widely used in economics, socio-economic field and business.<\/p>\n\n\n\n<p>7. Mean deviation is rigidly defined like that of an ideal measure of dispersion.<\/p>\n\n\n\n<p>8. As mean deviation is based on the deviation about an average, it provides us a better measure for comparison.<\/p>\n\n\n\n<p>9. It gives us accurate results. The averaging of the absolute deviation from an average removes the irregularities in the distributions.<\/p>\n\n\n\n<p><strong>DEMERITS OF MEAN DEVIATION<\/strong><\/p>\n\n\n\n<ol type=\"1\"><li>In mean deviation plus minus signs are ignored, which is not justified mathematically. This limitation makes mean deviation useless for further mathematical treatment.<\/li><li>The mean deviation calculated from mode is not reliable, because mode in many cases is indeterminate. Even the mean deviation calculated from median is not reliable If we use arithmetic mean, then it loses scientific character because the sum of deviations from mean is greater than the sum of deviations from median when plus minus signs are ignored.<\/li><li>If mean, median and mode are in fractions, then the calculation of mean deviation becomes cumbersome<\/li><li>It is generally not useful for statistical inferences as it is not a satisfactory measure when taken from mode or dealing with a skewed distribution. Theoretically, mean deviation gives us the best results when deviations are taken from Median. But median is not a satisfactory measure when the distribution has large variations.<\/li><li>From mean deviations of different groups of series, it is not possible to find out the combined mean deviation of all groups taken together. It means, mean deviation is not capable of further algebraic treatment.<\/li><li>It is rarely used for sociological studies.<\/li><li>In case of open-end classes mean deviation cannot be calculated. 8. It has a tendency to increase with the sample size though not in the same ratio.<\/li><li>statistical techniques. It is useful for small samples also where no detailed analysis is required.<\/li><\/ol>\n\n\n\n<!--nextpage-->\n\n\n\n<h2 class=\"wp-block-heading\">MEASURES OF DISPERSION IN STATISTICS <\/h2>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>STANDARD DEVIATION<\/strong><\/h4>\n\n\n\n<p>The concept of standard deviation was introduced by Karl Pearson in 1893.<\/p>\n\n\n\n<p>Standard deviation is &#8220;the square root of the mean of the deviations squared, or the root mean square deviation from the mean&#8221;. Standard deviation is denoted by the Greek letter sigma (\u03c3).<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"757\" height=\"142\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/2.png\" alt=\"MEASURES OF DISPERSION IN STATISTICS \" class=\"wp-image-5913\" srcset=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/2.png 757w, https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/2-300x56.png 300w\" sizes=\"(max-width: 757px) 100vw, 757px\" \/><figcaption>MEASURES OF DISPERSION IN STATISTICS <\/figcaption><\/figure>\n\n\n\n<p>Instead of ignoring the signs as in the calculation of mean deviation, we can make them all positive by squaring them. But by the squaring operation, the deviations from the mean will not give a zero sum but a positive number, and each deviation will contribute to the sum of squares regardless of sign. Then in order to compensate for squaring the deviations the square root is taken.<\/p>\n\n\n\n<p>The standard deviation is calculated as follows:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"473\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/3.png\" alt=\"MEASURES OF DISPERSION IN STATISTICS \" class=\"wp-image-5914\" srcset=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/3.png 852w, https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/3-300x167.png 300w, https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/3-768x426.png 768w\" sizes=\"(max-width: 852px) 100vw, 852px\" \/><figcaption>MEASURES OF DISPERSION IN STATISTICS <\/figcaption><\/figure>\n\n\n\n<p><strong>PROPERTIES OF STANDARD DEVIATION<\/strong><\/p>\n\n\n\n<ol type=\"1\"><li>Standard deviation is independent of change of origin. In other words, if any constant \u201cK\u201d is added or subtracted from all the values of variable \u2018X\u2019 it does not affect the standard deviation. Mathematically,<\/li><\/ol>\n\n\n\n<p>S.D. (X\u00b1K)= SD (X)<\/p>\n\n\n\n<ul><li>Standard deviation is not independent of charge of scale. In other words, if all values of a variable X are multiplied or divided by some constant \u201cK\u201d the standard deviation changes accordingly. Mathematically,<\/li><\/ul>\n\n\n\n<p>SD= (KX)= K SD (X)<\/p>\n\n\n\n<p>SD= (X\/K)= (1\/K)SD X<\/p>\n\n\n\n<ul><li>If X is a variable and K1 and K2 are constants, SD (K1X\u00b1K2)= K1 SD (X)<\/li><li>Standard deviation has some units as that of a variable.<\/li><li>Standard deviation has following relationship with the mean deviation and quartile deviation.<\/li><\/ul>\n\n\n\n<p>4SD=5MD=6QD<\/p>\n\n\n\n<ul><li>If arithmetic mean and standard deviation of two or more sectors are given, combined standard deviation can be calculated by using the following methods:<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"722\" height=\"158\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/4.png\" alt=\"MEASURES OF DISPERSION IN STATISTICS \" class=\"wp-image-5915\" srcset=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/4.png 722w, https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/4-300x66.png 300w\" sizes=\"(max-width: 722px) 100vw, 722px\" \/><figcaption>MEASURES OF DISPERSION IN STATISTICS <\/figcaption><\/figure>\n\n\n\n<ul><li>Standard deviation is always positive.<\/li><li>Standard deviation is always calculated by taking deviations from Arithmetic Mean. Therefore, sum of squares of deviations from Assumed Mean is minimum.<\/li><li>In case of symmetrical distribution or Normal Distribution, following area relationship holds good:<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"590\" height=\"148\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/5.png\" alt=\"MEASURES OF DISPERSION IN STATISTICS \" class=\"wp-image-5916\" srcset=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/5.png 590w, https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/5-300x75.png 300w\" sizes=\"(max-width: 590px) 100vw, 590px\" \/><figcaption>MEASURES OF DISPERSION IN STATISTICS <\/figcaption><\/figure>\n\n\n\n<p><strong>MERITS OF STANDARD DEVIATION<\/strong><\/p>\n\n\n\n<p>Standard deviation is the most important and extensively applied measure of dispersion. This measure of dispersion satisfies all the essentials of an ideal measure of dispersion.<\/p>\n\n\n\n<p>1. When samples are combined in different ways, the standard deviation of the combined data can be calculated from means and standard deviations of the original data.<\/p>\n\n\n\n<p>2. In the normal distribution, standard deviation plays a pivotal role.<\/p>\n\n\n\n<p>3. The sum of squares of the deviations from the actual mean, is the minimum. Thus, standard deviation is the best measure of dispersion<\/p>\n\n\n\n<p>4. Standard deviation is liable to further algebraic treatment.<\/p>\n\n\n\n<p>5. The standard errors of various methods are based on their standard deviations.<\/p>\n\n\n\n<p><strong>DEMERITS OF STANDARD DEVIATION<\/strong><\/p>\n\n\n\n<p>1. Standard deviation does not give any idea about the exactness of the measurement. The standard deviation can be higher in a particular period as compared to earlier period. It is just possible that there may be higher value of standard deviation due to which there is greater variability in the distribution. But it may be due to the reason that in the first year, the mean may be low than in the second year. This result can be true if the two means are uniform. Hence it is clear that standard deviation depends on arithmetic mean and so it is likely to change with the change in the size of mean.<\/p>\n\n\n\n<p>2. Standard deviation is referred to only as an absolute measure of dispersion and thus it cannot be used for comparing the two phenomena.<\/p>\n\n\n\n<p>3. The extreme items gain great importance. Those items which are close to mean, do not gain much importance. It is due to this reason that the of big deviations would be proportionately higher than the square of small deviations.<\/p>\n\n\n\n<p>4. Standard deviation is more difficult to be measured as compared to other measures of dispersion because square root is involved.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>VARIANCE<\/strong><\/h4>\n\n\n\n<p>Variance is square of Standard Deviation. The term was first used by RA. Fisher in 1913. The measure of variation is liable for further quantitative analysis. If we are dealing with a phenomenon affected by a number of variables, in that case variance helps us in separating the effects of different factors.<\/p>\n\n\n\n<p>Smaller the values of <img loading=\"lazy\" decoding=\"async\" width=\"9\" height=\"18\" src=\"\">, the lesser the variability or greater the consistency and vice- versa:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/6-1.png\" alt=\"MEASURES OF DISPERSION IN STATISTICS \" class=\"wp-image-5917\"\/><figcaption>MEASURES OF DISPERSION IN STATISTICS <\/figcaption><\/figure>\n\n\n\n<!--nextpage-->\n\n\n\n<h2 class=\"wp-block-heading\">MEASURES OF DISPERSION IN STATISTICS <\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>RELATIVE MEASURES OF DISPERSION<\/strong><\/h3>\n\n\n\n<p>Relative measures of dispersion is needed to make the absolute measures of dispersion comparable. The relative measures are calculated by dividing the absolute measures of dispersion by a measure of central tendency i.e. Mean, Median or Mode. The following are the relative measures of dispersion.<\/p>\n\n\n\n<ul><li>Coefficient of range<\/li><li>Coefficient of Quartile Deviation<\/li><li>Coefficient of mean deviation<\/li><li>Coefficient of standard deviation<\/li><li>Coefficient of variation.<\/li><\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">&nbsp;<strong>COEFFICIENT OF RANGE:<\/strong><\/h4>\n\n\n\n<p>The relative measure corresponding to the range is called as Coefficient of Range. It is calculated as follows:<\/p>\n\n\n\n<p>(HV-LV)\/ (HV +LV)<\/p>\n\n\n\n<p>Whereas, HV = Highest Value and LV= Lowest Value<\/p>\n\n\n\n<p>Coefficient of Range is also known as Coefficient of Scatterness.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>COEFFICIENT OF QUARTILE DEVIATION:<\/strong><\/h4>\n\n\n\n<p>The relative measure of Quartile deviation is Coefficient of Quartile deviation. It is calculated as follows:<\/p>\n\n\n\n<p>Coefficient of Quartile Deviation= (Q<sub>3<\/sub>-Q<sub>1<\/sub>)\/ (Q<sub>3<\/sub>+Q<sub>1<\/sub>) &nbsp;<\/p>\n\n\n\n<p>Coefficient of quartile deviation is studied to compare the degree of variation in the series.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>COEFFICIENT OF MEAN DEVIATION<\/strong><\/h4>\n\n\n\n<p>The coefficient of mean deviation is calculated with the objective of comparison. The coefficient of mean deviation is calculated by dividing the mean deviation by the average used. The formulae to calculated mean deviation are as follows:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>If deviations are taken from mean<\/td><td>MD<sub>Mean<\/sub>\/ Mean<\/td><\/tr><tr><td>If deviations are taken from Mode<\/td><td>MD<sub>Mode<\/sub>\/ Mode<\/td><\/tr><tr><td>If deviations are taken from median<\/td><td>MD<sub>Median<\/sub>\/ Median<\/td><\/tr><\/tbody><\/table><figcaption>MEASURES OF DISPERSION IN STATISTICS <\/figcaption><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>COEFFICIENT OF STANDARD DEVIATION<\/strong><\/h4>\n\n\n\n<p>In order to compare the variability in two or more series, relative measure of standard deviation is calculated. It is called &#8220;Coefficient of Standard Deviation&#8221;, which is calculated by dividing standard deviation (\u03c3) by the mean of the data. Symbolically<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/7-1.png\" alt=\"MEASURES OF DISPERSION IN STATISTICS \" class=\"wp-image-5918\"\/><figcaption>MEASURES OF DISPERSION IN STATISTICS <\/figcaption><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>COEFFICIENT OF VARIATION<\/strong><\/h4>\n\n\n\n<p>Coefficient of variation or coefficient of variability is a relative measure of dispersion. It has been developed by Karl Pearson. The coefficient of variation is used in such problems where we want to compare the variability of two or more than two series. A group which has more variability as compared to the other or has more coefficient of variation, the consistency would be less and vice versa.<\/p>\n\n\n\n<p>Higher C.V. Lower consistency, reliability, uniformity.<\/p>\n\n\n\n<p>Lower C.V. Higher consistency, reliability, uniformity<\/p>\n\n\n\n<p>According to Prof. Karl Pearson, &#8220;coefficient of variation is the percentage variation in mean, standard deviation being considered as the total variation in the mean.&#8221;<\/p>\n\n\n\n<p>If we wish to compare the variability of distributions, we have to compute coefficient of variation, <\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"602\" height=\"62\" src=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/8.png\" alt=\"MEASURES OF DISPERSION IN STATISTICS \" class=\"wp-image-5919\" srcset=\"https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/8.png 602w, https:\/\/commerceiets.com\/wp-content\/uploads\/2022\/08\/8-300x31.png 300w\" sizes=\"(max-width: 602px) 100vw, 602px\" \/><figcaption>MEASURES OF DISPERSION IN STATISTICS <\/figcaption><\/figure>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link\" href=\"https:\/\/www.linkedin.com\/in\/kajal-mahajan-7549b2197\/\" target=\"_blank\" rel=\"noreferrer noopener\">CONNECT ON LINKEDIN<\/a><\/div>\n<\/div>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td><strong>Also Study:<\/strong><\/td><td><strong>Also Study:<\/strong><\/td><td><strong>Also Study:<\/strong><\/td><td><strong>Also Study:<\/strong><\/td><td><strong>Also Study:<\/strong><\/td><\/tr><tr><td><a href=\"https:\/\/commerceiets.com\/measures-of-central-tendency\/\">Measures of central tendency<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/arithmetic-mean\/\">Arithmetic Mean<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/median\/\">Median<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/mode\/\">Mode<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/statistics\/\">Statistics<\/a><\/td><\/tr><tr><td><a href=\"https:\/\/commerceiets.com\/functions-of-statistics\/\">Functions of statistics<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/scope-of-statistics\/\">Scope of statistics<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/limitations-of-statistics\/\">Limitations of Statistics<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/distrust-of-statistics\/\">Distrust of statistics<\/a><\/td><td><a href=\"https:\/\/commerceiets.com\/business-statistics-notes-bcom-1st-year\/\">INTRODUCTION TO STATISTICS<\/a><\/td><\/tr><tr><td><a href=\"https:\/\/commerceiets.com\/measures-of-central-tendency-bcom-1st-year-notes-for-free\/\">MEASURES OF CENTRAL TENDENCY BCOM NOTES<\/a><\/td><td><\/td><td><\/td><td><\/td><td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Measure of dispersion, Measures of dispersion pdf, relative measures of dispersion, absolute measures of dispersion, definition of dispersion, measures of dispersion formula, importance of measures of dispersion, range in measures of dispersion.<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>MEASURES OF DISPERSION IN STATISTICS : Dispersion, meaning and definition; Importance; Relative and absolute measures of dispersion. DISPERSION Dispersion indicates the measure of the extent to which individual items differ. It indicates lack of uniformity in the size of items. ACCORDING TO BROOK AND DICKS &#8220;Dispersion is the degree of the variation of the variable&#8230;<\/p>\n<p class=\"more-link-wrap\"><a href=\"https:\/\/commerceiets.com\/measures-of-dispersion-in-statistics\/\" class=\"more-link\">Read More<span class=\"screen-reader-text\"> &ldquo;MEASURES OF DISPERSION IN STATISTICS FREE NOTES&rdquo;<\/span> &raquo;<\/a><\/p>\n","protected":false},"author":2,"featured_media":5920,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[67],"tags":[327,328,329,330,331,332,221,333,334,335,336],"_links":{"self":[{"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/posts\/5911"}],"collection":[{"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/comments?post=5911"}],"version-history":[{"count":0,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/posts\/5911\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/media\/5920"}],"wp:attachment":[{"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/media?parent=5911"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/categories?post=5911"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/commerceiets.com\/wp-json\/wp\/v2\/tags?post=5911"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}