{"id":140,"date":"2019-03-11T09:14:09","date_gmt":"2019-03-11T09:14:09","guid":{"rendered":"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=140"},"modified":"2019-03-11T09:47:22","modified_gmt":"2019-03-11T09:47:22","slug":"measures-of-dispersion-ii-with-skewness-and-kurtosis","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/chapter\/measures-of-dispersion-ii-with-skewness-and-kurtosis\/","title":{"rendered":"Measures of Dispersion II with Skewness and Kurtosis"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/sPXVfyVuxLA\" target=\"_blank\" rel=\"noopener\"><img src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a>\r\n<\/span><\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Learning Objectives<\/strong>\r\n<div>\r\n<ul>\r\n \t<li><strong>Introduction<\/strong><\/li>\r\n \t<li><strong>Skewness<\/strong><\/li>\r\n \t<li><strong>Kurtosis<\/strong><\/li>\r\n \t<li><strong>Summary<\/strong><\/li>\r\n \t<li><strong>Suggested Readings<\/strong><\/li>\r\n<\/ul>\r\n<strong>\u00a0 \u00a0 1.\u00a0<\/strong><strong>Learning Objectives<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This module is a continuation of the module \u201cMeasures of Dispersion- I\u201d. In this module, introduction to relative measures of dispersion and different measure of dispersion are discussed with examples. Properties of different measures are also discussed with merits and demerits. There are different relative measures like coefficient of range, coefficient of mean deviation, coefficient of variation etc. Through this module, one can easily understand about which method to use under what type of conditions. In this module, another important measure like Skewness and Kurtosis are also discussed. Moments are also introduced here for the derivation of these measures. Questions with detailed solutions are included to give an in-depth knowledge of the topic.<\/p>\r\n&nbsp;\r\n\r\n<strong>2.<\/strong>\u00a0\u00a0\u00a0 <strong>Introduction<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">As measures of dispersion are basically used to discuss about the variation, scatterness of the observations from the central tendency measure. We already covered the one category of measures of dispersion. In this module, second branch of measures of dispersion that is relative measures of dispersion will be discussed. Relative measure of dispersion is the ratio of absolute dispersion with its appropriate average i.e. to find out the relative measure of dispersion from an absolute measure then the quantity that is used in the denominator must be of same units that of absolute measure. Mostly it is considered as the average. The basic purpose of using these measures over absolute measures that we can compare different dataset which is not possible with absolute measure due to its dependency on the units. Hence the relative measures of dispersion have a great importance in statistics due to its property of independence of units.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Also in the modules \u201cCentral Tendency Measures I\u201d, \u201cCentral Tendency Measures \u2013II\u201d and \u201cMeasures of Dispersion \u2013I\u201d, we discussed about different central tendency measures and measures of dispersions. The measures of central tendency and measures of dispersion both together discuss about the characteristics of the dataset but they are not able to demonstrate that to what extent the observations deviate from the central value i.e. whether equal number of observations are dispersed from the central tendency or whether the data is symmetrical about the mean or not. Therefore it does not answers how many observations have their value below the mean value or above the mean value. If one is interested to know about the concentration of the observations around a central tendency measure then it is\u00a0<span style=\"text-align: initial;font-size: 1em\">essential to study two more measures. These are Skewness and Kurtosis. These two <\/span>measure<span style=\"text-align: initial;font-size: 1em\"> are considered as <\/span>a supportive<span style=\"text-align: initial;font-size: 1em\"> measures for better understanding the characteristics of the data.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Skewness is basically used to tell about the shape of the data i.e whether data is symmetric or skew symmetric. Skewness value help in determining the concentration of the observations below and above the average value. If the observations are concentrated in the <\/span>centre<span style=\"text-align: initial;font-size: 1em\"> then it is called symmetric. If the observations lie on either side of <\/span>concentration<span style=\"text-align: initial;font-size: 1em\"> of observations then there are two possibilities either more than average value or less than average value. Hence there are two types of skewness for asymmetric data i.e. positive or negative depending on the concentration of the observations. We will discuss <\/span>about<span style=\"text-align: initial;font-size: 1em\"> it later.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Another important measure is kurtosis which <\/span>refer<span style=\"text-align: initial;font-size: 1em\"> to the peakedness, flaterness of the curve that can be drawn from the dataset. It is basically used to study the concentration of the observations at the central part is whether more or less. If the concentration of observations at <\/span>central<span style=\"text-align: initial;font-size: 1em\"> part is very high then the curve is leptokurtic. On the other hand, if the concentration of the observations at the <\/span>centre<span style=\"text-align: initial;font-size: 1em\"> is less than the curve is platykurtic.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Hence central tendency measures, measures of dispersion, skewness <\/span>and<span style=\"text-align: initial;font-size: 1em\"> kurtosis represent a complete package to understand the data in depth. In <\/span>other word<span style=\"text-align: initial;font-size: 1em\">, it completely <\/span>describe<span style=\"text-align: initial;font-size: 1em\"> the distribution of the data.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Relative Measures<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">(a)\u00a0\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Coefficient of Range: <\/strong><span style=\"text-align: initial;font-size: 1em\">This measure of dispersion is evaluated from the range of the data set. <\/span>First<span style=\"text-align: initial;font-size: 1em\"> range of the data set is calculated then<\/span><strong style=\"text-align: initial;font-size: 1em\">\u00a0<\/strong><\/p>\r\n\r\n<\/div>\r\n<strong><img class=\"size-full wp-image-145 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-63.png\" alt=\"\" width=\"767\" height=\"251\" \/><\/strong>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>Ques 1<\/strong>. Find out the quartile deviation from the following data set related to the marks obtained by 15 students in statistics.<\/p>\r\n60, 67, 56, 78, 92, 55, 72, 54, 49, 59, 37, 84, 83, 69, 62\r\n\r\n&nbsp;\r\n\r\n<strong>Ans<\/strong>: Arrange the observations in the ascending order\r\n\r\n37, 49, 54, 55, 56, 59, 60, 62, 67,69, 72,78, 83, 84, 92\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Compute the range that is 92-37 = 55<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Coefficient of Range is = 55\/ 129 = 42.63%.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Now compute the first quartile\u00a0 N+1<sub>th<\/sub>\/4\u00a0term i.e. 55<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Similarly third quartile as 12th term i.e. 78.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Quartile deviation is (78 \u2013 53)\/2 = 25\/2 = 12.5.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Coefficient of quartile deviation = 25\/131=19.08%<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Hence <\/span>coefficient<span style=\"text-align: initial;font-size: 1em\"> of range and coefficient of quartile deviation is 42.63% and 19.08% respectively.<\/span>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">(c)\u00a0<\/strong><strong style=\"text-align: initial;font-size: 1em\">Coefficient of Mean Deviation: <\/strong><span style=\"text-align: initial;font-size: 1em\">Thi<\/span><strong style=\"text-align: initial;font-size: 1em\">s <\/strong><span style=\"text-align: initial;font-size: 1em\">relative measure of dispersion is derived from the mean deviation. Mean deviation is computed as the absolute value of <\/span>difference<span style=\"text-align: initial;font-size: 1em\"> of observations from a central tendency i.e. mean, median and mode. Mostly, <\/span>mean<span style=\"text-align: initial;font-size: 1em\"> deviation is calculated by taking deviation of observations from mean and median. So in order to convert the mean deviation measure into independent of <\/span>unit<span style=\"text-align: initial;font-size: 1em\"> coefficient of mean deviation is computed.<\/span><strong style=\"text-align: initial;font-size: 1em\">\u00a0<\/strong><\/p>\r\n\r\n<\/div>\r\n<strong><img class=\"aligncenter size-full wp-image-146\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-64.png\" alt=\"\" width=\"473\" height=\"50\" \/><\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Central tendency measure can be mean or median but it is divided by the measure that is used for the derivation of mean deviation or from which the mean deviation is derived.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Ques 2. <\/strong><span style=\"text-align: initial;font-size: 1em\">Calculate the coefficient of mean deviation about mean for the following dataset.<\/span><\/p>\r\n\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-147\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-65.png\" alt=\"\" width=\"769\" height=\"421\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-148 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-66.png\" alt=\"\" width=\"736\" height=\"512\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>(d)\u00a0 <\/strong><strong>Coefficient of Standard Deviation<\/strong>\r\n\r\n<strong>\u00a0<\/strong>\r\n<p style=\"text-align: justify\">As standard deviation is evaluated in terms of the observations units and is considered as a absolute measure of dispersion. It is essential for the comparison purpose that the measure must be independent of units. The relative measure based on standard deviation that is independent of units is called coefficient of standard deviation. It is defined as<\/p>\r\n<img class=\"aligncenter size-full wp-image-149\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-67.png\" alt=\"\" width=\"312\" height=\"47\" \/>\r\n<p style=\"text-align: justify\">As coefficient of standard deviation would be given in fraction. So if we want to express our coefficient value in term of percentage by multiplying the coefficient by 100. Then this relative measure is called coefficient of variation (C.V.). It is defined as<\/p>\r\n<img class=\"aligncenter size-full wp-image-150\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-68.png\" alt=\"\" width=\"277\" height=\"44\" \/>\r\n<p style=\"text-align: justify\">The coefficient of variation is among the most popular relative measure of dispersion. It is basically used to compare the variability among two or more dataset. The dataset that has more value of coefficient of variation among two is said to be more variable and vice versa.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Ques 3. <\/strong><span style=\"text-align: initial;font-size: 1em\">Calculate the coefficient of standard deviation and coefficient of variation for the following dataset.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-151\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-69.png\" alt=\"\" width=\"470\" height=\"239\" \/>\r\n<p style=\"text-align: justify\"><strong>Ans<\/strong>. Calculate the mean of the observations as shown in Table 3. Subtract the mean from the observations as shown in 4th column. Now take square of these observations as shown in the 5th column in Table 4. Column 6 shows the product of frequencies with values from column 5. Now take sum of the observations in column 5 and divide it by i.e. total frequency. Take square root of the value 125.22 and the standard deviation value is 11.190.<\/p>\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-152\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-70.png\" alt=\"\" width=\"727\" height=\"396\" \/>\r\n<div>\r\n\r\n&nbsp;\r\n\r\nNow for the coefficient of standard deviation, the formula is\r\n\r\n<img class=\"aligncenter size-full wp-image-153\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-71.png\" alt=\"\" width=\"757\" height=\"158\" \/>\r\n\r\n<\/div>\r\n<div>\r\n<p style=\"text-align: justify\">Hence, we discussed about different relative measures of dispersion. As these measures are derived from the absolute measures to make them independent of units. So one should know about absolute measures of dispersion in depth and the properties before applying these measures on the dataset.<\/p>\r\n&nbsp;\r\n\r\nNow, in the next session, we will discuss about the skewness and kurtosis.\r\n\r\n&nbsp;\r\n\r\n<strong>3.\u00a0\u00a0 Skewness<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">A skewness is basically to see tendency of the shape of the distribution. If the frequency distribution of the data is not equally distributed about the mean i.e. the frequency distribution is not symmetric then the term that is used to refer this situation is called skewness. Skewness has many synonyms like asymmetry and lack of symmetrical. Some authors give definitions of skewness as:<\/p>\r\n&nbsp;\r\n\r\n\u201cWhen a series is not symmetrical, it is said to be asymmetrical or skewed\u201d by Croxton and Cowden.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">\u201cMeasure of skewness tell us the direction and the extent of skewness. In symmetrical distribution, the mean, median and mode are identical. The more the mean moves away from the mode, the larger the asymmetry or skewness\u201d by Simpson and Kafka.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Hence skewness means that the data is not symmetrical about the mean. It is also be defined in term of normal distribution. Normal distribution is the distribution which has mean, median and mode all are equal. Hence the shape of the frequency of this distribution is like bell shape.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-154\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-72.png\" alt=\"\" width=\"600\" height=\"231\" \/>\r\n\r\n<\/div>\r\n<div>\r\n<p style=\"text-align: center\"><strong>Figure 1<\/strong><\/p>\r\n&nbsp;\r\n\r\n<strong>Link for the image<\/strong>\r\n\r\n&nbsp;\r\n\r\n<a href=\"https:\/\/www.kullabs.com\/classes\/subjects\/units\/lessons\/notes\/note-detail\/9958\"><strong>https:\/\/www.kullabs.com\/classes\/subjects\/units\/lessons\/notes\/note-detail\/9958<\/strong><\/a>\r\n\r\n&nbsp;\r\n\r\nHence from the Figure 1, one can observe the shape of the frequency distribution.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">A frequency distribution is said to be positive skewed when the mean (\u03bc) &gt; Median &gt; Mode. In this case, the value of mean is more than the value of median and mode. Also median value is more than the value of mode.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">A frequency distribution is said to be symmetric distribution when the mean (\u03bc) = Median = Mode. In this case, the values of mean, median and mode all are same.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">A frequency distribution is said to be negative skewed distribution when the mean (\u03bc) &lt; Median &lt; Mode. In this case, the value of mode is more than the median and mean. Also the median is more than the mean.<\/p>\r\n&nbsp;\r\n\r\n<strong>Difference between skewness and measures of dispersion<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">As we discuss above different measures of dispersion, as dispersion measures are basically used to know about the variation in the dataset while skewness is concerned with the concentration of the observations around the central part of the data.<\/p>\r\n&nbsp;\r\n\r\nThere are some important differences between measure of dispersion and skewness. These are:\r\n<p style=\"text-align: justify\">(i)\u00a0 Skewness is basically concerned about the shape of the frequency distribution while measures of dispersion are more concerned about the amount of variations.<\/p>\r\n<p style=\"text-align: justify\">(ii)\u00a0 Skewness shows the nature of data about its central value while dispersion try to measure up to what extent the central tendency value represent the whole data set.<\/p>\r\n<p style=\"text-align: justify\">(iii) It is possible that the data that is more dispersed but has symmetric frequency distribution. Hence in that case one can say that symmetric does not mean that variation is less.<\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iv) Measures of dispersion are based on first and second order moments while skewness is based on first, second and third order moments.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">This is the reason that both skewness and measures of dispersion are studied together in literature. As both measures help in understanding the features of the frequency distribution in depth.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There are many methods available in the literature to find out the skewness. Some of them are discussed here.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Measures of skewness are used to detect whether the frequency distribution is symmetric or skew. As the values of these measures depend on the units of the observations. So there are two categories of measures of skewness. These are:<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(a)\u00a0\u00a0 <\/span>Absolute<span style=\"text-align: initial;font-size: 1em\"> measure of skewness<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(b)\u00a0\u00a0 <\/span>Relative<span style=\"text-align: initial;font-size: 1em\"> measure of skewness<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Absolute measures of skewness<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">These measures of skewness are basically used to check the asymmetry of the data. Hence these measures assume that the data is not symmetric otherwise the values of these measures will be zero. Some of the measures of skewness are given below :<\/p>\r\n(a)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Mean(\u03bc) \u2212 Mode (Md)\r\n\r\n(b)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0Mean (\u03bc) \u2013 Median (M)\r\n\r\n(c)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0Q3 +\u00a0 Q1 \u2212 2M\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Hence by using the above three measures one can check the skewness of the distribution. As we already discussed that in skewed distribution either the mean value is greater than median or mode or mode is greater than median and mean. Hence, these measure just give you an indication about the presence of skewness in the data expect when the value of these measure is zero in that case data is symmetric.<\/p>\r\n&nbsp;\r\n\r\nHowever, these measures of skewness have limited utilization in practice due to these reasons. These are:\r\n<p style=\"text-align: justify\">(i)\u00a0 The first and the most important thing is that these measures are based on the units of the observations. Hence the values that are derived from these measures cannot be used for comparison purposes.<\/p>\r\n<p style=\"text-align: justify\">(ii) If absolute measure of skewness values of two data sets are same. It does not mean that data set are always same as it may be possible that there may be variation between distributions in terms of mean and dispersion.<\/p>\r\n&nbsp;\r\n\r\nNow to overcome these limitations, another measure that is independent from units is used is called relative measure of skewness or coefficient of skewness.\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Relative Measure of Skewness<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">In <\/span>these measure<span style=\"text-align: initial;font-size: 1em\">, the limitations of absolute measures have been removed by dividing the absolute measure by the suitable measure or quantity. The following are some coefficient of skewness which <\/span>are<span style=\"text-align: initial;font-size: 1em\"> commonly used.<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">(a) Karl Peason Coefficient of Skewness<\/strong>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-156 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-73.png\" alt=\"\" width=\"767\" height=\"477\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nwhere\u00a0 u3 and\u00a0\u00a0 are the third order moment and standard deviation of the distribution.\r\n\r\n&nbsp;\r\n\r\n<strong>Ques 4 <\/strong>Calculate the absolute measure of skewness and coefficient of skewness (a) Karl Pearson\r\n\r\n&nbsp;\r\n\r\n(b)\u00a0\u00a0 Bowley (c) Kelly (d) Based on moments of the marks of 15 students in statistics given below:\r\n\r\n54, 63, 78, 59, 69, 74, 85, 46, 63, 51, 58, 73, 86, 88, 93\r\n\r\n&nbsp;\r\n\r\n<strong>Ans <\/strong>Arrange the series in ascending order\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">46, 51, 54, 58, 59, 63, 63, 69, 73, 74, 78, 85, 86, 88, 93 <\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">Now compute arithmetic mean, median and mode from the series. <\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">A.M. is 69.33, Median is 69, Mode is 63<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Now 1 is 58 , 3 is 85 and 1 is 46+0.6 (51-46) = 49 and 9 is 88 + 0.4 (93-88) = 90<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Absolute measures values are<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Mean(\u03bc) \u2212 Mode (Md) = 69.33 \u2212 63 = 6.33<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Mean (\u03bc)\u2013 Median (M) = 69.33 \u2212 69 = 0.33<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">3 +\u00a0 1 \u2212 2\u00a0 = 85 + 58 \u2212 2 \u2217 69 = 5<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Hence all absolute measures show that the data is <\/span>positive<span style=\"text-align: initial;font-size: 1em\"> skewed.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Now calculate the relative measures of skewness. We need <\/span>standard<span style=\"text-align: initial;font-size: 1em\"> deviation of the data.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Let 1, 2, \u2026 , be the observations and n is the number of observations. S.D. is evaluated by using the formula<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-157 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-74.png\" alt=\"\" width=\"761\" height=\"447\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Hence from all the absolute and relative measure of skewness, we conclude that data is positively skewed. Although we also notice that different measures have different values for same dataset that is considered as the limitation of the measures of skewness.<\/p>\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">4.<\/strong><span style=\"text-align: initial;font-size: 1em\">\u00a0\u00a0\u00a0 <\/span><strong style=\"text-align: initial;font-size: 1em\">Kurtosis<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Kurtosis word comes from the Greek language with a meaning curved arching. Kurtosis is basically used to measure the peakedness of the frequency distribution. It is possible that two data set have <\/span>same<span style=\"text-align: initial;font-size: 1em\"> arithmetic mean, standard deviation and coefficient of <\/span>skewnss<span style=\"text-align: initial;font-size: 1em\"> but still one has different concentration of values near the mode value. So the distribution can have more peakedness than the usual normal distribution, less peakedness than the usual normal curve and equal to the normal distribution curve. So basically kurtosis is a measure that <\/span>compare<span style=\"text-align: initial;font-size: 1em\"> the peakedness of the curve relative to the peakedness of a normal curve. So kurtosis is basically used to measure the extent how the distribution is more peaked or less peaked than the normal distribution curve.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Many authors give the definitions of the kurtosis as<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u201cA measure of kurtosis indicated the degree to which a curve of a frequency distribution is peaked or <\/span>flat topped<span style=\"text-align: initial;font-size: 1em\">\u201d by Croxton and Cowden<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">\u201c<\/strong><span style=\"text-align: initial;font-size: 1em\">Kurtosis is the degree of peakedness of <\/span>a distribution<span style=\"text-align: initial;font-size: 1em\">, usually taken relative to a normal distribution\u201d by Spiegel.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-158\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-75.png\" alt=\"\" width=\"580\" height=\"444\" \/>\r\n<p style=\"text-align: center\"><strong>Figure 2<\/strong><\/p>\r\n&nbsp;\r\n\r\n<strong>Courtesy for image is<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>whatilearned.wikia.com<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">So if the distribution curve (blue curve) is more peaked than the normal distribution as shown with blue curve in Figure 2, then the distribution is called <strong>Leptokurtic<\/strong>. If the distribution curve (red curve) is more flat than the normal distribution curve then the distribution is called <strong>Platykurtic.<\/strong> Hence<strong>,<\/strong> the black curve represent the normal curve is also known as <strong>Mesokurtic<\/strong>.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Measure of Kurtosis<\/strong>\r\n\r\n&nbsp;\r\n\r\nKurtosis is defined as\r\n\r\n<img class=\"size-full wp-image-159 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-76.png\" alt=\"\" width=\"637\" height=\"520\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-160 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-77.png\" alt=\"\" width=\"387\" height=\"143\" \/>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<ol start=\"5\">\r\n \t<li><strong>Summary<\/strong><\/li>\r\n<\/ol>\r\n<p style=\"text-align: justify\">\u00a0 \u00a0 In this module, first we introduced the relative measures of dispersion. There are different relative measures like coefficient of range, coefficient of mean deviation, coefficient of variation etc. Properties of different measures are also discussed with merits and demerits. Difference between dispersion and skewness are also discussed. Through this module, based on the merits and demerits one can easily understand about which method to use under what type of conditions. In this module, another important measure like Skewness and Kurtosis are discussed.<\/p>\r\n\r\n<ol start=\"6\">\r\n \t<li><strong> Suggested Readings<\/strong><\/li>\r\n<\/ol>\r\nAgresti, A. and B. Finlay, Statistical Methods for the Social Science, 3rd Edition, Prentice Hall, 1997.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Daniel, W. W. and C. L. Cross, C. L., Biostatistics<em>:<\/em> A Foundation for Analysis in the Health Sciences<em>,<\/em> 10th Edition<em>,<\/em> John Wiley &amp; Sons, 2013.<\/p>\r\n&nbsp;\r\n\r\nHogg, R. V., J. Mckean and A. Craig, Introduction to Mathematical Statistics, Macmillan Pub. Co. Inc., 1978.\r\n\r\n&nbsp;\r\n\r\nMeyer, P. L., Introductory Probability and Statistical Applications, Oxford &amp; IBH Pub, 1975.\r\n\r\n&nbsp;\r\n\r\nTriola, M. F., Elementary Statistics, 13th\u00a0 Edition, Pearson, 2017.\r\n\r\n&nbsp;\r\n\r\nWeiss, N. A., Introductory Statistics, 10th Edition, Pearson, 2017.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Measures of Dispersion II with Skewness and Kurtosis<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/sPXVfyVuxLA\" target=\"_blank\" rel=\"noopener\"><img class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nOne can refer to the following links for further understanding of the statistics terms.\r\n\r\n&nbsp;\r\n\r\n<a href=\"http:\/\/biostat.mc.vanderbilt.edu\/wiki\/pub\/Main\/ClinStat\/glossary.pdf\">http:\/\/biostat.mc.vanderbilt.edu\/wiki\/pub\/Main\/ClinStat\/glossary.pdf<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"http:\/\/www.stats.gla.ac.uk\/steps\/glossary\/alphabet.html\">http:\/\/www.stats.gla.ac.uk\/steps\/glossary\/alphabet.html<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"http:\/\/www.reading.ac.uk\/ssc\/resources\/Docs\/Statistical_Glossary.pdf\">http:\/\/www.reading.ac.uk\/ssc\/resources\/Docs\/Statistical_Glossary.pdf<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"https:\/\/stats.oecd.org\/glossary\/\">https:\/\/stats.oecd.org\/glossary\/<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"http:\/\/www.statsoft.com\/Textbook\/Statistics-Glossary\">http:\/\/www.statsoft.com\/Textbook\/Statistics-Glossary<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"https:\/\/www.stat.berkeley.edu\/~stark\/SticiGui\/Text\/gloss.htm\">https:\/\/www.stat.berkeley.edu\/~stark\/SticiGui\/Text\/gloss.htm<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"https:\/\/stats.oecd.org\/glossary\/alpha.asp?Let=A\">https:\/\/stats.oecd.org\/glossary\/alpha.asp?Let=A<\/a>","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/sPXVfyVuxLA\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a><br \/>\n<\/span><\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Learning Objectives<\/strong><\/p>\n<div>\n<ul>\n<li><strong>Introduction<\/strong><\/li>\n<li><strong>Skewness<\/strong><\/li>\n<li><strong>Kurtosis<\/strong><\/li>\n<li><strong>Summary<\/strong><\/li>\n<li><strong>Suggested Readings<\/strong><\/li>\n<\/ul>\n<p><strong>\u00a0 \u00a0 1.\u00a0<\/strong><strong>Learning Objectives<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This module is a continuation of the module \u201cMeasures of Dispersion- I\u201d. In this module, introduction to relative measures of dispersion and different measure of dispersion are discussed with examples. Properties of different measures are also discussed with merits and demerits. There are different relative measures like coefficient of range, coefficient of mean deviation, coefficient of variation etc. Through this module, one can easily understand about which method to use under what type of conditions. In this module, another important measure like Skewness and Kurtosis are also discussed. Moments are also introduced here for the derivation of these measures. Questions with detailed solutions are included to give an in-depth knowledge of the topic.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>2.<\/strong>\u00a0\u00a0\u00a0 <strong>Introduction<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As measures of dispersion are basically used to discuss about the variation, scatterness of the observations from the central tendency measure. We already covered the one category of measures of dispersion. In this module, second branch of measures of dispersion that is relative measures of dispersion will be discussed. Relative measure of dispersion is the ratio of absolute dispersion with its appropriate average i.e. to find out the relative measure of dispersion from an absolute measure then the quantity that is used in the denominator must be of same units that of absolute measure. Mostly it is considered as the average. The basic purpose of using these measures over absolute measures that we can compare different dataset which is not possible with absolute measure due to its dependency on the units. Hence the relative measures of dispersion have a great importance in statistics due to its property of independence of units.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Also in the modules \u201cCentral Tendency Measures I\u201d, \u201cCentral Tendency Measures \u2013II\u201d and \u201cMeasures of Dispersion \u2013I\u201d, we discussed about different central tendency measures and measures of dispersions. The measures of central tendency and measures of dispersion both together discuss about the characteristics of the dataset but they are not able to demonstrate that to what extent the observations deviate from the central value i.e. whether equal number of observations are dispersed from the central tendency or whether the data is symmetrical about the mean or not. Therefore it does not answers how many observations have their value below the mean value or above the mean value. If one is interested to know about the concentration of the observations around a central tendency measure then it is\u00a0<span style=\"text-align: initial;font-size: 1em\">essential to study two more measures. These are Skewness and Kurtosis. These two <\/span>measure<span style=\"text-align: initial;font-size: 1em\"> are considered as <\/span>a supportive<span style=\"text-align: initial;font-size: 1em\"> measures for better understanding the characteristics of the data.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Skewness is basically used to tell about the shape of the data i.e whether data is symmetric or skew symmetric. Skewness value help in determining the concentration of the observations below and above the average value. If the observations are concentrated in the <\/span>centre<span style=\"text-align: initial;font-size: 1em\"> then it is called symmetric. If the observations lie on either side of <\/span>concentration<span style=\"text-align: initial;font-size: 1em\"> of observations then there are two possibilities either more than average value or less than average value. Hence there are two types of skewness for asymmetric data i.e. positive or negative depending on the concentration of the observations. We will discuss <\/span>about<span style=\"text-align: initial;font-size: 1em\"> it later.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Another important measure is kurtosis which <\/span>refer<span style=\"text-align: initial;font-size: 1em\"> to the peakedness, flaterness of the curve that can be drawn from the dataset. It is basically used to study the concentration of the observations at the central part is whether more or less. If the concentration of observations at <\/span>central<span style=\"text-align: initial;font-size: 1em\"> part is very high then the curve is leptokurtic. On the other hand, if the concentration of the observations at the <\/span>centre<span style=\"text-align: initial;font-size: 1em\"> is less than the curve is platykurtic.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Hence central tendency measures, measures of dispersion, skewness <\/span>and<span style=\"text-align: initial;font-size: 1em\"> kurtosis represent a complete package to understand the data in depth. In <\/span>other word<span style=\"text-align: initial;font-size: 1em\">, it completely <\/span>describe<span style=\"text-align: initial;font-size: 1em\"> the distribution of the data.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Relative Measures<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">(a)\u00a0\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Coefficient of Range: <\/strong><span style=\"text-align: initial;font-size: 1em\">This measure of dispersion is evaluated from the range of the data set. <\/span>First<span style=\"text-align: initial;font-size: 1em\"> range of the data set is calculated then<\/span><strong style=\"text-align: initial;font-size: 1em\">\u00a0<\/strong><\/p>\n<\/div>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-145 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-63.png\" alt=\"\" width=\"767\" height=\"251\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-63.png 767w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-63-300x98.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-63-65x21.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-63-225x74.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-63-350x115.png 350w\" sizes=\"auto, (max-width: 767px) 100vw, 767px\" \/><\/strong><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>Ques 1<\/strong>. Find out the quartile deviation from the following data set related to the marks obtained by 15 students in statistics.<\/p>\n<p>60, 67, 56, 78, 92, 55, 72, 54, 49, 59, 37, 84, 83, 69, 62<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Ans<\/strong>: Arrange the observations in the ascending order<\/p>\n<p>37, 49, 54, 55, 56, 59, 60, 62, 67,69, 72,78, 83, 84, 92<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Compute the range that is 92-37 = 55<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Coefficient of Range is = 55\/ 129 = 42.63%.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Now compute the first quartile\u00a0 N+1<sub>th<\/sub>\/4\u00a0term i.e. 55<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Similarly third quartile as 12th term i.e. 78.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Quartile deviation is (78 \u2013 53)\/2 = 25\/2 = 12.5.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Coefficient of quartile deviation = 25\/131=19.08%<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Hence <\/span>coefficient<span style=\"text-align: initial;font-size: 1em\"> of range and coefficient of quartile deviation is 42.63% and 19.08% respectively.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">(c)\u00a0<\/strong><strong style=\"text-align: initial;font-size: 1em\">Coefficient of Mean Deviation: <\/strong><span style=\"text-align: initial;font-size: 1em\">Thi<\/span><strong style=\"text-align: initial;font-size: 1em\">s <\/strong><span style=\"text-align: initial;font-size: 1em\">relative measure of dispersion is derived from the mean deviation. Mean deviation is computed as the absolute value of <\/span>difference<span style=\"text-align: initial;font-size: 1em\"> of observations from a central tendency i.e. mean, median and mode. Mostly, <\/span>mean<span style=\"text-align: initial;font-size: 1em\"> deviation is calculated by taking deviation of observations from mean and median. So in order to convert the mean deviation measure into independent of <\/span>unit<span style=\"text-align: initial;font-size: 1em\"> coefficient of mean deviation is computed.<\/span><strong style=\"text-align: initial;font-size: 1em\">\u00a0<\/strong><\/p>\n<\/div>\n<p><strong><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-146\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-64.png\" alt=\"\" width=\"473\" height=\"50\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-64.png 473w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-64-300x32.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-64-65x7.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-64-225x24.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-64-350x37.png 350w\" sizes=\"auto, (max-width: 473px) 100vw, 473px\" \/><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Central tendency measure can be mean or median but it is divided by the measure that is used for the derivation of mean deviation or from which the mean deviation is derived.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Ques 2. <\/strong><span style=\"text-align: initial;font-size: 1em\">Calculate the coefficient of mean deviation about mean for the following dataset.<\/span><\/p>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-147\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-65.png\" alt=\"\" width=\"769\" height=\"421\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-65.png 769w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-65-300x164.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-65-768x420.png 768w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-65-65x36.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-65-225x123.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-65-350x192.png 350w\" sizes=\"auto, (max-width: 769px) 100vw, 769px\" \/><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-148 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-66.png\" alt=\"\" width=\"736\" height=\"512\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-66.png 736w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-66-300x209.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-66-65x45.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-66-225x157.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-66-350x243.png 350w\" sizes=\"auto, (max-width: 736px) 100vw, 736px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>(d)\u00a0 <\/strong><strong>Coefficient of Standard Deviation<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify\">As standard deviation is evaluated in terms of the observations units and is considered as a absolute measure of dispersion. It is essential for the comparison purpose that the measure must be independent of units. The relative measure based on standard deviation that is independent of units is called coefficient of standard deviation. It is defined as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-149\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-67.png\" alt=\"\" width=\"312\" height=\"47\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-67.png 312w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-67-300x45.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-67-65x10.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-67-225x34.png 225w\" sizes=\"auto, (max-width: 312px) 100vw, 312px\" \/><\/p>\n<p style=\"text-align: justify\">As coefficient of standard deviation would be given in fraction. So if we want to express our coefficient value in term of percentage by multiplying the coefficient by 100. Then this relative measure is called coefficient of variation (C.V.). It is defined as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-150\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-68.png\" alt=\"\" width=\"277\" height=\"44\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-68.png 277w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-68-65x10.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-68-225x36.png 225w\" sizes=\"auto, (max-width: 277px) 100vw, 277px\" \/><\/p>\n<p style=\"text-align: justify\">The coefficient of variation is among the most popular relative measure of dispersion. It is basically used to compare the variability among two or more dataset. The dataset that has more value of coefficient of variation among two is said to be more variable and vice versa.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Ques 3. <\/strong><span style=\"text-align: initial;font-size: 1em\">Calculate the coefficient of standard deviation and coefficient of variation for the following dataset.<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-151\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-69.png\" alt=\"\" width=\"470\" height=\"239\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-69.png 470w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-69-300x153.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-69-65x33.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-69-225x114.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-69-350x178.png 350w\" sizes=\"auto, (max-width: 470px) 100vw, 470px\" \/><\/p>\n<p style=\"text-align: justify\"><strong>Ans<\/strong>. Calculate the mean of the observations as shown in Table 3. Subtract the mean from the observations as shown in 4th column. Now take square of these observations as shown in the 5th column in Table 4. Column 6 shows the product of frequencies with values from column 5. Now take sum of the observations in column 5 and divide it by i.e. total frequency. Take square root of the value 125.22 and the standard deviation value is 11.190.<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-152\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-70.png\" alt=\"\" width=\"727\" height=\"396\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-70.png 727w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-70-300x163.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-70-65x35.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-70-225x123.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-70-350x191.png 350w\" sizes=\"auto, (max-width: 727px) 100vw, 727px\" \/><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p>Now for the coefficient of standard deviation, the formula is<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-153\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-71.png\" alt=\"\" width=\"757\" height=\"158\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-71.png 757w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-71-300x63.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-71-65x14.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-71-225x47.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-71-350x73.png 350w\" sizes=\"auto, (max-width: 757px) 100vw, 757px\" \/><\/p>\n<\/div>\n<div>\n<p style=\"text-align: justify\">Hence, we discussed about different relative measures of dispersion. As these measures are derived from the absolute measures to make them independent of units. So one should know about absolute measures of dispersion in depth and the properties before applying these measures on the dataset.<\/p>\n<p>&nbsp;<\/p>\n<p>Now, in the next session, we will discuss about the skewness and kurtosis.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>3.\u00a0\u00a0 Skewness<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A skewness is basically to see tendency of the shape of the distribution. If the frequency distribution of the data is not equally distributed about the mean i.e. the frequency distribution is not symmetric then the term that is used to refer this situation is called skewness. Skewness has many synonyms like asymmetry and lack of symmetrical. Some authors give definitions of skewness as:<\/p>\n<p>&nbsp;<\/p>\n<p>\u201cWhen a series is not symmetrical, it is said to be asymmetrical or skewed\u201d by Croxton and Cowden.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">\u201cMeasure of skewness tell us the direction and the extent of skewness. In symmetrical distribution, the mean, median and mode are identical. The more the mean moves away from the mode, the larger the asymmetry or skewness\u201d by Simpson and Kafka.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Hence skewness means that the data is not symmetrical about the mean. It is also be defined in term of normal distribution. Normal distribution is the distribution which has mean, median and mode all are equal. Hence the shape of the frequency of this distribution is like bell shape.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-154\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-72.png\" alt=\"\" width=\"600\" height=\"231\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-72.png 600w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-72-300x116.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-72-65x25.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-72-225x87.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-72-350x135.png 350w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/p>\n<\/div>\n<div>\n<p style=\"text-align: center\"><strong>Figure 1<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Link for the image<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/www.kullabs.com\/classes\/subjects\/units\/lessons\/notes\/note-detail\/9958\"><strong>https:\/\/www.kullabs.com\/classes\/subjects\/units\/lessons\/notes\/note-detail\/9958<\/strong><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>Hence from the Figure 1, one can observe the shape of the frequency distribution.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A frequency distribution is said to be positive skewed when the mean (\u03bc) &gt; Median &gt; Mode. In this case, the value of mean is more than the value of median and mode. Also median value is more than the value of mode.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A frequency distribution is said to be symmetric distribution when the mean (\u03bc) = Median = Mode. In this case, the values of mean, median and mode all are same.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A frequency distribution is said to be negative skewed distribution when the mean (\u03bc) &lt; Median &lt; Mode. In this case, the value of mode is more than the median and mean. Also the median is more than the mean.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Difference between skewness and measures of dispersion<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As we discuss above different measures of dispersion, as dispersion measures are basically used to know about the variation in the dataset while skewness is concerned with the concentration of the observations around the central part of the data.<\/p>\n<p>&nbsp;<\/p>\n<p>There are some important differences between measure of dispersion and skewness. These are:<\/p>\n<p style=\"text-align: justify\">(i)\u00a0 Skewness is basically concerned about the shape of the frequency distribution while measures of dispersion are more concerned about the amount of variations.<\/p>\n<p style=\"text-align: justify\">(ii)\u00a0 Skewness shows the nature of data about its central value while dispersion try to measure up to what extent the central tendency value represent the whole data set.<\/p>\n<p style=\"text-align: justify\">(iii) It is possible that the data that is more dispersed but has symmetric frequency distribution. Hence in that case one can say that symmetric does not mean that variation is less.<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iv) Measures of dispersion are based on first and second order moments while skewness is based on first, second and third order moments.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">This is the reason that both skewness and measures of dispersion are studied together in literature. As both measures help in understanding the features of the frequency distribution in depth.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There are many methods available in the literature to find out the skewness. Some of them are discussed here.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Measures of skewness are used to detect whether the frequency distribution is symmetric or skew. As the values of these measures depend on the units of the observations. So there are two categories of measures of skewness. These are:<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(a)\u00a0\u00a0 <\/span>Absolute<span style=\"text-align: initial;font-size: 1em\"> measure of skewness<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(b)\u00a0\u00a0 <\/span>Relative<span style=\"text-align: initial;font-size: 1em\"> measure of skewness<\/span><\/p>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0 Absolute measures of skewness<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">These measures of skewness are basically used to check the asymmetry of the data. Hence these measures assume that the data is not symmetric otherwise the values of these measures will be zero. Some of the measures of skewness are given below :<\/p>\n<p>(a)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Mean(\u03bc) \u2212 Mode (Md)<\/p>\n<p>(b)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0Mean (\u03bc) \u2013 Median (M)<\/p>\n<p>(c)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0Q3 +\u00a0 Q1 \u2212 2M<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Hence by using the above three measures one can check the skewness of the distribution. As we already discussed that in skewed distribution either the mean value is greater than median or mode or mode is greater than median and mean. Hence, these measure just give you an indication about the presence of skewness in the data expect when the value of these measure is zero in that case data is symmetric.<\/p>\n<p>&nbsp;<\/p>\n<p>However, these measures of skewness have limited utilization in practice due to these reasons. These are:<\/p>\n<p style=\"text-align: justify\">(i)\u00a0 The first and the most important thing is that these measures are based on the units of the observations. Hence the values that are derived from these measures cannot be used for comparison purposes.<\/p>\n<p style=\"text-align: justify\">(ii) If absolute measure of skewness values of two data sets are same. It does not mean that data set are always same as it may be possible that there may be variation between distributions in terms of mean and dispersion.<\/p>\n<p>&nbsp;<\/p>\n<p>Now to overcome these limitations, another measure that is independent from units is used is called relative measure of skewness or coefficient of skewness.<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Relative Measure of Skewness<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">In <\/span>these measure<span style=\"text-align: initial;font-size: 1em\">, the limitations of absolute measures have been removed by dividing the absolute measure by the suitable measure or quantity. The following are some coefficient of skewness which <\/span>are<span style=\"text-align: initial;font-size: 1em\"> commonly used.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">(a) Karl Peason Coefficient of Skewness<\/strong><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-156 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-73.png\" alt=\"\" width=\"767\" height=\"477\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-73.png 767w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-73-300x187.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-73-65x40.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-73-225x140.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-73-350x218.png 350w\" sizes=\"auto, (max-width: 767px) 100vw, 767px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>where\u00a0 u3 and\u00a0\u00a0 are the third order moment and standard deviation of the distribution.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Ques 4 <\/strong>Calculate the absolute measure of skewness and coefficient of skewness (a) Karl Pearson<\/p>\n<p>&nbsp;<\/p>\n<p>(b)\u00a0\u00a0 Bowley (c) Kelly (d) Based on moments of the marks of 15 students in statistics given below:<\/p>\n<p>54, 63, 78, 59, 69, 74, 85, 46, 63, 51, 58, 73, 86, 88, 93<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Ans <\/strong>Arrange the series in ascending order<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">46, 51, 54, 58, 59, 63, 63, 69, 73, 74, 78, 85, 86, 88, 93 <\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">Now compute arithmetic mean, median and mode from the series. <\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">A.M. is 69.33, Median is 69, Mode is 63<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Now 1 is 58 , 3 is 85 and 1 is 46+0.6 (51-46) = 49 and 9 is 88 + 0.4 (93-88) = 90<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Absolute measures values are<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Mean(\u03bc) \u2212 Mode (Md) = 69.33 \u2212 63 = 6.33<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Mean (\u03bc)\u2013 Median (M) = 69.33 \u2212 69 = 0.33<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">3 +\u00a0 1 \u2212 2\u00a0 = 85 + 58 \u2212 2 \u2217 69 = 5<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Hence all absolute measures show that the data is <\/span>positive<span style=\"text-align: initial;font-size: 1em\"> skewed.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Now calculate the relative measures of skewness. We need <\/span>standard<span style=\"text-align: initial;font-size: 1em\"> deviation of the data.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Let 1, 2, \u2026 , be the observations and n is the number of observations. S.D. is evaluated by using the formula<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-157 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-74.png\" alt=\"\" width=\"761\" height=\"447\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-74.png 761w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-74-300x176.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-74-65x38.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-74-225x132.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-74-350x206.png 350w\" sizes=\"auto, (max-width: 761px) 100vw, 761px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Hence from all the absolute and relative measure of skewness, we conclude that data is positively skewed. Although we also notice that different measures have different values for same dataset that is considered as the limitation of the measures of skewness.<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">4.<\/strong><span style=\"text-align: initial;font-size: 1em\">\u00a0\u00a0\u00a0 <\/span><strong style=\"text-align: initial;font-size: 1em\">Kurtosis<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Kurtosis word comes from the Greek language with a meaning curved arching. Kurtosis is basically used to measure the peakedness of the frequency distribution. It is possible that two data set have <\/span>same<span style=\"text-align: initial;font-size: 1em\"> arithmetic mean, standard deviation and coefficient of <\/span>skewnss<span style=\"text-align: initial;font-size: 1em\"> but still one has different concentration of values near the mode value. So the distribution can have more peakedness than the usual normal distribution, less peakedness than the usual normal curve and equal to the normal distribution curve. So basically kurtosis is a measure that <\/span>compare<span style=\"text-align: initial;font-size: 1em\"> the peakedness of the curve relative to the peakedness of a normal curve. So kurtosis is basically used to measure the extent how the distribution is more peaked or less peaked than the normal distribution curve.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Many authors give the definitions of the kurtosis as<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u201cA measure of kurtosis indicated the degree to which a curve of a frequency distribution is peaked or <\/span>flat topped<span style=\"text-align: initial;font-size: 1em\">\u201d by Croxton and Cowden<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">\u201c<\/strong><span style=\"text-align: initial;font-size: 1em\">Kurtosis is the degree of peakedness of <\/span>a distribution<span style=\"text-align: initial;font-size: 1em\">, usually taken relative to a normal distribution\u201d by Spiegel.<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-158\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-75.png\" alt=\"\" width=\"580\" height=\"444\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-75.png 580w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-75-300x230.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-75-65x50.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-75-225x172.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-75-350x268.png 350w\" sizes=\"auto, (max-width: 580px) 100vw, 580px\" \/><\/p>\n<p style=\"text-align: center\"><strong>Figure 2<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Courtesy for image is<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>whatilearned.wikia.com<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">So if the distribution curve (blue curve) is more peaked than the normal distribution as shown with blue curve in Figure 2, then the distribution is called <strong>Leptokurtic<\/strong>. If the distribution curve (red curve) is more flat than the normal distribution curve then the distribution is called <strong>Platykurtic.<\/strong> Hence<strong>,<\/strong> the black curve represent the normal curve is also known as <strong>Mesokurtic<\/strong>.<\/p>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0 Measure of Kurtosis<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Kurtosis is defined as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-159 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-76.png\" alt=\"\" width=\"637\" height=\"520\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-76.png 637w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-76-300x245.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-76-65x53.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-76-225x184.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-76-350x286.png 350w\" sizes=\"auto, (max-width: 637px) 100vw, 637px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-160 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-77.png\" alt=\"\" width=\"387\" height=\"143\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-77.png 387w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-77-300x111.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-77-65x24.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-77-225x83.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-77-350x129.png 350w\" sizes=\"auto, (max-width: 387px) 100vw, 387px\" \/><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<ol start=\"5\">\n<li><strong>Summary<\/strong><\/li>\n<\/ol>\n<p style=\"text-align: justify\">\u00a0 \u00a0 In this module, first we introduced the relative measures of dispersion. There are different relative measures like coefficient of range, coefficient of mean deviation, coefficient of variation etc. Properties of different measures are also discussed with merits and demerits. Difference between dispersion and skewness are also discussed. Through this module, based on the merits and demerits one can easily understand about which method to use under what type of conditions. In this module, another important measure like Skewness and Kurtosis are discussed.<\/p>\n<ol start=\"6\">\n<li><strong> Suggested Readings<\/strong><\/li>\n<\/ol>\n<p>Agresti, A. and B. Finlay, Statistical Methods for the Social Science, 3rd Edition, Prentice Hall, 1997.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Daniel, W. W. and C. L. Cross, C. L., Biostatistics<em>:<\/em> A Foundation for Analysis in the Health Sciences<em>,<\/em> 10th Edition<em>,<\/em> John Wiley &amp; Sons, 2013.<\/p>\n<p>&nbsp;<\/p>\n<p>Hogg, R. V., J. Mckean and A. Craig, Introduction to Mathematical Statistics, Macmillan Pub. Co. Inc., 1978.<\/p>\n<p>&nbsp;<\/p>\n<p>Meyer, P. L., Introductory Probability and Statistical Applications, Oxford &amp; IBH Pub, 1975.<\/p>\n<p>&nbsp;<\/p>\n<p>Triola, M. F., Elementary Statistics, 13th\u00a0 Edition, Pearson, 2017.<\/p>\n<p>&nbsp;<\/p>\n<p>Weiss, N. A., Introductory Statistics, 10th Edition, Pearson, 2017.<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Measures of Dispersion II with Skewness and Kurtosis<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/sPXVfyVuxLA\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>One can refer to the following links for further understanding of the statistics terms.<\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"http:\/\/biostat.mc.vanderbilt.edu\/wiki\/pub\/Main\/ClinStat\/glossary.pdf\">http:\/\/biostat.mc.vanderbilt.edu\/wiki\/pub\/Main\/ClinStat\/glossary.pdf<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"http:\/\/www.stats.gla.ac.uk\/steps\/glossary\/alphabet.html\">http:\/\/www.stats.gla.ac.uk\/steps\/glossary\/alphabet.html<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"http:\/\/www.reading.ac.uk\/ssc\/resources\/Docs\/Statistical_Glossary.pdf\">http:\/\/www.reading.ac.uk\/ssc\/resources\/Docs\/Statistical_Glossary.pdf<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/stats.oecd.org\/glossary\/\">https:\/\/stats.oecd.org\/glossary\/<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"http:\/\/www.statsoft.com\/Textbook\/Statistics-Glossary\">http:\/\/www.statsoft.com\/Textbook\/Statistics-Glossary<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/www.stat.berkeley.edu\/~stark\/SticiGui\/Text\/gloss.htm\">https:\/\/www.stat.berkeley.edu\/~stark\/SticiGui\/Text\/gloss.htm<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/stats.oecd.org\/glossary\/alpha.asp?Let=A\">https:\/\/stats.oecd.org\/glossary\/alpha.asp?Let=A<\/a><\/p>\n","protected":false},"author":3,"menu_order":9,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["dr-harmanpreet-singh-kapoor"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-140","chapter","type-chapter","status-publish","hentry","contributor-dr-harmanpreet-singh-kapoor"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapters\/140","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":6,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapters\/140\/revisions"}],"predecessor-version":[{"id":161,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapters\/140\/revisions\/161"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapters\/140\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/media?parent=140"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapter-type?post=140"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/contributor?post=140"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/license?post=140"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}