{"id":120,"date":"2019-03-11T08:35:29","date_gmt":"2019-03-11T08:35:29","guid":{"rendered":"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=120"},"modified":"2019-03-11T09:13:40","modified_gmt":"2019-03-11T09:13:40","slug":"measures-of-dispersion-i","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/chapter\/measures-of-dispersion-i\/","title":{"rendered":"Measures of Dispersion I"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/3rixaWi9R4A\" 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>Summary<\/strong><\/li>\r\n \t<li><strong>Suggested Readings<\/strong><\/li>\r\n<\/ul>\r\n<strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 1.\u00a0<\/strong><strong style=\"text-align: initial;font-size: 1em\">Learning Objectives<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In this module, a complete introduction to measures of dispersion and different measures of dispersion are discussed with examples. Properties of different measures are also discussed with their merits and demerits. Through this module, one can easily understand about which method to use under what type of conditions. The topic of measures of dispersion is covered in two modules. This module will cover the absolute measures of dispersions. Other topic of relative measures will be covered in the module \u201cMeasures of Dispersion- II with Skewness and Kurtosis\u201d. Questions with answers 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\">In this module, an important measure that is used to see how each observation varies from the mean value. When one is interested to know about the variation of observations from the mean value. The measures that are used to detect this variations in the observation are called Measures of Dispersions.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">In the module \u201cCentral Tendency Measures \u2013I\u201d and \u201cCentral Tendency Measures \u2013II\u201d we discussed about different measures of central tendency. We discussed about the mathematical averages and positional averages. The basic purpose of these measures is to find out a single value that represent the whole dataset. Also concentration of observations about the central part of the data were observed. But these measure will not take into account of the fact that whether two or more different dataset have same mean values but it does not mean that the observations are same.<\/p>\r\n&nbsp;\r\n\r\nFor example, we have two dataset given below that have same mean value but the observations are not same.\r\n\r\n&nbsp;\r\n\r\nDataset I: 8, 9, 10, 11, 13, 15\r\n\r\nDataset II: 3, 4, 6, 14, 16, 23\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Hence the total sum of both datasets are same i.e. 66 and the mean is also same i.e. 11. Now the question is how we can say that mean is the representation of the data? The answer is central tendency measure gives you just an idea about the concentration of the observation around a central value but it does not say anything about the variation of these observations from the central part. In dataset I, the values are very close to each other and also to the mean value so we can say that 11 is the correct representation of the dataset but in dataset II, observations are very scattered and also very far away from the mean value. Hence, one can see that 11as a mean value is not represent the whole data set in a correct manner.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Hence, one should not rely just on the central tendency measure to take any opinion about the observations but also one should also think about the dispersion or variation among the observation. In this module, <\/span>different<span style=\"text-align: initial;font-size: 1em\"> type of measure will be discussed that consider <\/span>variation<span style=\"text-align: initial;font-size: 1em\"> of observation from different values like <\/span>difference<span style=\"text-align: initial;font-size: 1em\"> between highest and smallest values, <\/span>absolute<span style=\"text-align: initial;font-size: 1em\"> difference of observations from a particular quantity etc. but mainly the <\/span>variation<span style=\"text-align: initial;font-size: 1em\"> of observations from its mean is considered the most in the literature.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Definitions<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Some authors have defined the measures of dispersion as:<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u201cDispersion is the measure of variation of the items\u201d by A.L. Bowley.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u201cDispersion or spread is the degree of the scatter or the variation of the variable about a central value\u201d by B.C. Brooks and W. F. L. Dicks.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In literature, the dispersion is also considered as <\/span>synonym<span style=\"text-align: initial;font-size: 1em\"> for heterogeneous in the data. As heterogeneous is basically used to understand the extent of variations among observations. <\/span>Dispersion<span style=\"text-align: initial;font-size: 1em\"> can only be zero if all the observations have <\/span>same<span style=\"text-align: initial;font-size: 1em\"> values. The dispersion is more when the difference between the observations is very large. So one can say that if the variation is small like in data set I then it is considered as insignificant but if the variation is large as shown from the data set II then it is considered as significant.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Also<span style=\"text-align: initial;font-size: 1em\"> dispersion is termed as second ordered means as central tendency measures <\/span>is<span style=\"text-align: initial;font-size: 1em\"> the first ordered means where one can see the tendency of the values around the middle of the data. As measures of dispersion are basically used to discuss <\/span>about<span style=\"text-align: initial;font-size: 1em\"> the variation, scatterness of the observations from the central tendency measure. The measures of central tendency and measures of dispersion both together discuss <\/span>about<span style=\"text-align: initial;font-size: 1em\"> the characteristics of the data set but they <\/span>do<span style=\"text-align: initial;font-size: 1em\"> 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. It also <\/span>give<span style=\"text-align: initial;font-size: 1em\"> us an idea of 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 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. Skewness and Kurtosis <\/span>is<span style=\"text-align: initial;font-size: 1em\"> discussed in the module \u201cMeasure of Dipersion \u2013II with Skewness and Kurtosis\u201d.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Importance of measures of dispersions<\/strong><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(a)\u00a0\u00a0 The main motive behind using these measures to check the authenticity of the central tendency measures. It is used to see whether the value of central tendency measure are reliable or not.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(b)\u00a0\u00a0 The second important thing about the measures of central tendency is that these are also used to compare the two datasets through relative values.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(c)\u00a0\u00a0\u00a0 This measure is also useful in identifying the reasons behind the variations in order to control them but these are not helpful in <\/span>give<span style=\"text-align: initial;font-size: 1em\"> the exact reason behind variations in the observations. For\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">example, in <\/span>electronic<span style=\"text-align: initial;font-size: 1em\"> industry quality of an <\/span>item<span style=\"text-align: initial;font-size: 1em\"> cannot be judged only by through whether an item is produced under defined limits but also through the variations between the characteristics of observations.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(d)\u00a0\u00a0 Measures of dispersions are also used to help for further analysis of the data like correlation, regression, testing of hypothesis and ANOVA etc<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">As there are many measures of dispersion, the ideal measure prevails the following characteristics:<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(a)\u00a0\u00a0 The values of these measures should be rigid.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(b)\u00a0\u00a0 It should be calculated on all the observations.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(c)\u00a0\u00a0 The method should be easily calculated and understood by non-mathematical background person.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(d)\u00a0\u00a0 The measure should be used for further algebraic treatment.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(e)\u00a0\u00a0 The measure should not be affected by extreme values and fluctuation of the observations.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The measures of dispersion are further categorized into two types. These are shown in the following flowchart<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-125\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-50.png\" alt=\"\" width=\"740\" height=\"362\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">From <\/span>the Figure<span style=\"text-align: initial;font-size: 1em\"> 1, the measure of dispersion is categorized into two measures: Absolute Measures and Relative Measures. As dispersion measure is used to detect the deviation of the observations from the central tendency. If the measures of dispersion express the dispersion of the observations in the original units then the measures are called absolute measures of dispersion. One can only compare the variations between two series if both series are in <\/span>same<span style=\"text-align: initial;font-size: 1em\"> units otherwise <\/span>comparison<span style=\"text-align: initial;font-size: 1em\"> is not meaningless. To\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">overcome this problem, those measures that give the dispersion values in terms of ratio and percentage are called relative measures.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">A relative measure of dispersion is the ratio of <\/span>absolute<span style=\"text-align: initial;font-size: 1em\"> measure of dispersion with its appropriate average. These <\/span>measure<span style=\"text-align: initial;font-size: 1em\"> are independent of units and these are termed as <\/span>coefficient<span style=\"text-align: initial;font-size: 1em\"> of dispersion. One important thing, while the calculation of relative measures, is that the units of absolute measure and the appropriate average must be <\/span>same<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In this module, absolute measures will be discussed with examples. Relative measures will be discussed in the module \u201cMeasures of Dispersion II with Skewness and Kurtosis\u201d.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Range<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Range<span style=\"text-align: initial;font-size: 1em\"> is considered as the simplest absolute measure of dispersion. It is evaluated by just taking the difference between the maximum value and minimum value in the data set.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Range = Maximum value- Minimum value<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Ques 1<\/strong><span style=\"text-align: initial;font-size: 1em\">: Calculate the range of the following dataset:<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">23, 45, 56, 52, 64, 35, 42, 51, 76, 65.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Ans<\/strong><span style=\"text-align: initial;font-size: 1em\">: Maximum value is 76 and minimum value is 23. Hence<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Range = 76-23 = 53.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Merits of Range<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">These are the merits of the range measure<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(i)\u00a0\u00a0<\/span>Range<span style=\"text-align: initial;font-size: 1em\"> is the simplest among all the absolute measures of dispersion.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(ii)\u00a0<\/span>Range<span style=\"text-align: initial;font-size: 1em\"> is defined in a rigid manner and easily calculated. Due to this <\/span>property<span style=\"text-align: initial;font-size: 1em\"> it is widely used in the industry as a quality control tool.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iii)\u00a0<\/span>Range<span style=\"text-align: initial;font-size: 1em\"> is computed within second hence one can get a complete picture of variability in the dataset in a short time.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Demerits of Range<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There are few demerits of range also. These are:<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(i) As <\/span>range<span style=\"text-align: initial;font-size: 1em\"> is evaluated just from the difference of maximum and <\/span>minimum<span style=\"text-align: initial;font-size: 1em\"> value of the dataset. Hence it does not depend on all the observations. So it is possible that range will be <\/span>same<span style=\"text-align: initial;font-size: 1em\"> for two <\/span>dataset<span style=\"text-align: initial;font-size: 1em\"> that <\/span>have same<span style=\"text-align: initial;font-size: 1em\"> maximum and minimum value but different intermediate values that make no sense.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(ii) As we already discuss that range depend on just two values and both of these values are extreme values. Hence range is affected a lot by extreme values.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iii)\u00a0<\/span>Also range<span style=\"text-align: initial;font-size: 1em\"> will not take into account the shape of the distribution. It may be possible that range will be <\/span>same<span style=\"text-align: initial;font-size: 1em\"> for the dataset whether it is symmetric or <\/span>skew symmetric<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iv)\u00a0<\/span>Range<span style=\"text-align: initial;font-size: 1em\"> is affected by the change in the observations.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(v)\u00a0<\/span>Range<span style=\"text-align: initial;font-size: 1em\"> cannot be evaluated from the grouped frequency distribution with open end class.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Quartile Deviation<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This measure of dispersion is related with range and it removes the drawbacks of range upto some extent. It is defined as the difference between third quartile and first quartile divided by 2. In other words,<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-126\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-51.png\" alt=\"\" width=\"753\" height=\"404\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>Merits of Quartile Deviation<\/strong>\r\n\r\n&nbsp;\r\n\r\nThere are some merits of quartile deviation. These are\r\n<p style=\"text-align: justify\">(i)\u00a0 \u00a0Quartile deviation is easy to understand like range.<\/p>\r\n<p style=\"text-align: justify\">(ii)\u00a0 As quartile deviation is based on first and third quarter hence it removes the limitations of the range.<\/p>\r\n<p style=\"text-align: justify\">(iii) As quartile deviation consider only first quarter and third quarter value. These values are derived by neglecting first and last 25% of the observations respectively. Hence it is not affected by the extreme values.<\/p>\r\n<p style=\"text-align: justify\">(iv) Quartile deviation is more useful to consider in case of skewed symmetrical data.<\/p>\r\n<p style=\"text-align: justify\">(v)\u00a0 Quartile deviation can be evaluated for open end class interval as it is not depend on the extreme values.<\/p>\r\n&nbsp;\r\n\r\n<strong>Demerits of Quartile deviation<\/strong>\r\n\r\n&nbsp;\r\n\r\nThere are some demerits of quartile deviation. These are\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(i) Quartile deviation is evaluated on just half of the observations as it neglect first and last 25% of the observations. Hence it is not considered as a fully appropriate measure for detecting variation among observations.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(ii) Quartile measure is based on quartiles which are positional averages. It represents the deviation of observations among quartiles that is just a distance on a scale.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iii) Quartile deviation is very much influenced by the change in the observations. A change in single observations <\/span>lead<span style=\"text-align: initial;font-size: 1em\"> to <\/span>change<span style=\"text-align: initial;font-size: 1em\"> in the value of quartile deviation.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iv) Quartile deviation just <\/span>give<span style=\"text-align: initial;font-size: 1em\"> an idea about the deviation like range. So it is not the best measure for measuring dispersion.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Mean deviation<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">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.<\/span><\/p>\r\n<img class=\"size-full wp-image-127 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-52.png\" alt=\"\" width=\"772\" height=\"563\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\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\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>Ques 3<\/strong>. Calculate the mean deviation about mean for the following observations.\r\n\r\n30, 40, 50, 55, 60, 65, 70, 80, 90, 100\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>Ans <\/strong>First find out the A.M. of the observations. It is 64. Now take absolute deviation of each observation from 64. The values are 34, 24, 14, 9, 4, 1, 6, 16, 26. Take the sum of these observation and divide it by 10 i.e. number of observations.<\/p>\r\n&nbsp;\r\n\r\nThe answer is 134\/10=13.4.\r\n\r\n&nbsp;\r\n\r\n<strong>Ques 4. <\/strong>Calculate the mean deviation about mean for the following dataset.\r\n\r\n<img class=\"aligncenter size-full wp-image-128\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-53.png\" alt=\"\" width=\"763\" height=\"355\" \/>\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-129\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-54.png\" alt=\"\" width=\"751\" height=\"305\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-130\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-55.png\" alt=\"\" width=\"774\" height=\"458\" \/>\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-131\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-56.png\" alt=\"\" width=\"733\" height=\"318\" \/>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Merits and Demerits of Mean deviation<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Merits<\/strong>\r\n\r\n&nbsp;\r\n\r\nThere are some merits of mean deviation measure. These are\r\n\r\n(i)\u00a0 \u00a0Mean deviation is defined in a rigid manner.\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">(ii)\u00a0 Mean deviation is an easy method for a non-mathematical background person.<\/span>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iii) Mean deviation is based on all the observations so it is better than range and quartile deviation measures.<\/span><\/p>\r\n<span style=\"text-align: initial;font-size: 1em\">(iv) Mean deviation is less influenced by the extreme values than <\/span>range<span style=\"text-align: initial;font-size: 1em\">, standard deviation.<\/span>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(v)\u00a0 Mean deviation is appropriate for comparison purpose because it is derived from the deviation of observations from central values i.e. mean, median or mode.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(vi) Mean deviation is basically an average of absolute deviation from central value. Hence it <\/span>remove biasness<span style=\"text-align: initial;font-size: 1em\"> to some extent that <\/span>occur<span style=\"text-align: initial;font-size: 1em\"> due to observations while calculating and provide an accurate value for dispersion.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Demerits<\/strong>\r\n\r\n&nbsp;\r\n\r\nThere are few demerits of mean deviation measure. These are\r\n<p style=\"text-align: justify\">(i)\u00a0 The major drawback of mean deviation measure is that while computing it we ignore the signs of deviation and take absolute of it. So one cannot get any information about the concentration of the observations above the mean or below the mean. So this measure is not useful for further mathematical treatment.<\/p>\r\n(ii)\u00a0\u00a0Mean deviation cannot be computed for open end classes distributions.\r\n\r\n(iii) Mean deviation is not an accurate measure of dispersion for highly skewed data.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Although mean deviation has some limitations. Due to its simplicity and easy to understand features it has great importance in accountancy, economics, forecasting and business sectors.<\/p>\r\n&nbsp;\r\n\r\n<strong>Standard Deviation<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Karl Pearson in 1823 first introduced this measure and after that it is the most widely used measure of dispersion till date. Standard deviation is the measure that prevails all the features that other measures lack of. So basically it is considered as an ideal measure of deviation. It is also know as square root of variance, root mean square deviation etc and denoted by the Greek letter (sigma). Standard deviation value is high or low when there is more or less variation among observations respectively.<\/p>\r\n<img class=\"size-full wp-image-132 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-57.png\" alt=\"\" width=\"767\" height=\"265\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div><img class=\"size-full wp-image-133 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-58.png\" alt=\"\" width=\"770\" height=\"423\" \/><\/div>\r\n<div><\/div>\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\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<img class=\"size-full wp-image-134 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-59.png\" alt=\"\" width=\"766\" height=\"395\" \/>\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<\/div>\r\n<img class=\"aligncenter size-full wp-image-135\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-60.png\" alt=\"\" width=\"724\" height=\"396\" \/>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Ques 8. <\/strong>Calculate the standard deviation for the following dataset.\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-136\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-61.png\" alt=\"\" width=\"763\" height=\"321\" \/>\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-137\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-62.png\" alt=\"\" width=\"721\" height=\"390\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Merits and Demerits of Standard deviation<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Merits<\/strong>\r\n<p style=\"text-align: justify\">As<span style=\"text-align: initial;font-size: 1em\"> standard deviation is considered as the best absolute measure of dispersion. These are the merits of standard deviation measure<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(i)\u00a0 \u00a0Standard deviation is rigidly defined measure.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(ii)\u00a0 Standard deviation is based on all the observations.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iii) Standard deviation is evaluated by squaring of deviation of observations from the mean. This makes the standard deviation for further mathematical treatment.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iv) Standard deviation has less influenced by the change in the sample observations.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(v)\u00a0 It is also possible to compute the combined standard deviation of two or more dataset from individual standard deviation.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(vi)\u00a0 Standard deviation is used for further statistical analysis like skewness and correlation.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Demerits<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Although standard deviation is considered as the best absolute measure of dispersion but it is not free from demerits. These are<\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(i) Standard deviation is very difficult to understand for a non-mathematical background person.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">(ii) Standard deviation gives more weight to extreme values and less weight to values close to mean. As while squaring the deviation this bias will be increased.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Although, standard deviation has some demerits or limitations <\/span>but<span style=\"text-align: initial;font-size: 1em\"> this is the widely used method for the absolute measure of dispersion.<\/span><\/p>\r\n\r\n<\/div>\r\n<ol start=\"3\">\r\n \t<li><strong>Summary<\/strong><\/li>\r\n<\/ol>\r\n<p style=\"text-align: justify\">\u00a0 \u00a0 In this module<strong>,<\/strong> one category of measures of dispersion i.e. absolute measures of dispersion are discussed in detail. These are range, quartile deviation, mean deviation and standard deviation. We first give an introduction about these measures. Questions and answers are also included for better understanding of the topic. Merits and demerits of each measure are discussed for better understanding and comparison purpose. Other branch of dispersion measure that is relative measure of dispersion will be discussed in the module \u201cMeasures of Dispersion- II with Skewness and Kurtosis\u201d.<\/p>\r\n\r\n<ol start=\"4\">\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 I<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/3rixaWi9R4A\" 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\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>\r\n\r\n&nbsp;","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/3rixaWi9R4A\" 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>Summary<\/strong><\/li>\n<li><strong>Suggested Readings<\/strong><\/li>\n<\/ul>\n<p><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 1.\u00a0<\/strong><strong style=\"text-align: initial;font-size: 1em\">Learning Objectives<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In this module, a complete introduction to measures of dispersion and different measures of dispersion are discussed with examples. Properties of different measures are also discussed with their merits and demerits. Through this module, one can easily understand about which method to use under what type of conditions. The topic of measures of dispersion is covered in two modules. This module will cover the absolute measures of dispersions. Other topic of relative measures will be covered in the module \u201cMeasures of Dispersion- II with Skewness and Kurtosis\u201d. Questions with answers 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\">In this module, an important measure that is used to see how each observation varies from the mean value. When one is interested to know about the variation of observations from the mean value. The measures that are used to detect this variations in the observation are called Measures of Dispersions.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In the module \u201cCentral Tendency Measures \u2013I\u201d and \u201cCentral Tendency Measures \u2013II\u201d we discussed about different measures of central tendency. We discussed about the mathematical averages and positional averages. The basic purpose of these measures is to find out a single value that represent the whole dataset. Also concentration of observations about the central part of the data were observed. But these measure will not take into account of the fact that whether two or more different dataset have same mean values but it does not mean that the observations are same.<\/p>\n<p>&nbsp;<\/p>\n<p>For example, we have two dataset given below that have same mean value but the observations are not same.<\/p>\n<p>&nbsp;<\/p>\n<p>Dataset I: 8, 9, 10, 11, 13, 15<\/p>\n<p>Dataset II: 3, 4, 6, 14, 16, 23<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Hence the total sum of both datasets are same i.e. 66 and the mean is also same i.e. 11. Now the question is how we can say that mean is the representation of the data? The answer is central tendency measure gives you just an idea about the concentration of the observation around a central value but it does not say anything about the variation of these observations from the central part. In dataset I, the values are very close to each other and also to the mean value so we can say that 11 is the correct representation of the dataset but in dataset II, observations are very scattered and also very far away from the mean value. Hence, one can see that 11as a mean value is not represent the whole data set in a correct manner.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Hence, one should not rely just on the central tendency measure to take any opinion about the observations but also one should also think about the dispersion or variation among the observation. In this module, <\/span>different<span style=\"text-align: initial;font-size: 1em\"> type of measure will be discussed that consider <\/span>variation<span style=\"text-align: initial;font-size: 1em\"> of observation from different values like <\/span>difference<span style=\"text-align: initial;font-size: 1em\"> between highest and smallest values, <\/span>absolute<span style=\"text-align: initial;font-size: 1em\"> difference of observations from a particular quantity etc. but mainly the <\/span>variation<span style=\"text-align: initial;font-size: 1em\"> of observations from its mean is considered the most in the literature.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Definitions<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Some authors have defined the measures of dispersion as:<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u201cDispersion is the measure of variation of the items\u201d by A.L. Bowley.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u201cDispersion or spread is the degree of the scatter or the variation of the variable about a central value\u201d by B.C. Brooks and W. F. L. Dicks.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In literature, the dispersion is also considered as <\/span>synonym<span style=\"text-align: initial;font-size: 1em\"> for heterogeneous in the data. As heterogeneous is basically used to understand the extent of variations among observations. <\/span>Dispersion<span style=\"text-align: initial;font-size: 1em\"> can only be zero if all the observations have <\/span>same<span style=\"text-align: initial;font-size: 1em\"> values. The dispersion is more when the difference between the observations is very large. So one can say that if the variation is small like in data set I then it is considered as insignificant but if the variation is large as shown from the data set II then it is considered as significant.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Also<span style=\"text-align: initial;font-size: 1em\"> dispersion is termed as second ordered means as central tendency measures <\/span>is<span style=\"text-align: initial;font-size: 1em\"> the first ordered means where one can see the tendency of the values around the middle of the data. As measures of dispersion are basically used to discuss <\/span>about<span style=\"text-align: initial;font-size: 1em\"> the variation, scatterness of the observations from the central tendency measure. The measures of central tendency and measures of dispersion both together discuss <\/span>about<span style=\"text-align: initial;font-size: 1em\"> the characteristics of the data set but they <\/span>do<span style=\"text-align: initial;font-size: 1em\"> 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. It also <\/span>give<span style=\"text-align: initial;font-size: 1em\"> us an idea of 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 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. Skewness and Kurtosis <\/span>is<span style=\"text-align: initial;font-size: 1em\"> discussed in the module \u201cMeasure of Dipersion \u2013II with Skewness and Kurtosis\u201d.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Importance of measures of dispersions<\/strong><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(a)\u00a0\u00a0 The main motive behind using these measures to check the authenticity of the central tendency measures. It is used to see whether the value of central tendency measure are reliable or not.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(b)\u00a0\u00a0 The second important thing about the measures of central tendency is that these are also used to compare the two datasets through relative values.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(c)\u00a0\u00a0\u00a0 This measure is also useful in identifying the reasons behind the variations in order to control them but these are not helpful in <\/span>give<span style=\"text-align: initial;font-size: 1em\"> the exact reason behind variations in the observations. For\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">example, in <\/span>electronic<span style=\"text-align: initial;font-size: 1em\"> industry quality of an <\/span>item<span style=\"text-align: initial;font-size: 1em\"> cannot be judged only by through whether an item is produced under defined limits but also through the variations between the characteristics of observations.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(d)\u00a0\u00a0 Measures of dispersions are also used to help for further analysis of the data like correlation, regression, testing of hypothesis and ANOVA etc<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">As there are many measures of dispersion, the ideal measure prevails the following characteristics:<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(a)\u00a0\u00a0 The values of these measures should be rigid.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(b)\u00a0\u00a0 It should be calculated on all the observations.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(c)\u00a0\u00a0 The method should be easily calculated and understood by non-mathematical background person.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(d)\u00a0\u00a0 The measure should be used for further algebraic treatment.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(e)\u00a0\u00a0 The measure should not be affected by extreme values and fluctuation of the observations.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The measures of dispersion are further categorized into two types. These are shown in the following flowchart<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-125\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-50.png\" alt=\"\" width=\"740\" height=\"362\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-50.png 740w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-50-300x147.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-50-65x32.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-50-225x110.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-50-350x171.png 350w\" sizes=\"auto, (max-width: 740px) 100vw, 740px\" \/><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">From <\/span>the Figure<span style=\"text-align: initial;font-size: 1em\"> 1, the measure of dispersion is categorized into two measures: Absolute Measures and Relative Measures. As dispersion measure is used to detect the deviation of the observations from the central tendency. If the measures of dispersion express the dispersion of the observations in the original units then the measures are called absolute measures of dispersion. One can only compare the variations between two series if both series are in <\/span>same<span style=\"text-align: initial;font-size: 1em\"> units otherwise <\/span>comparison<span style=\"text-align: initial;font-size: 1em\"> is not meaningless. To\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">overcome this problem, those measures that give the dispersion values in terms of ratio and percentage are called relative measures.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">A relative measure of dispersion is the ratio of <\/span>absolute<span style=\"text-align: initial;font-size: 1em\"> measure of dispersion with its appropriate average. These <\/span>measure<span style=\"text-align: initial;font-size: 1em\"> are independent of units and these are termed as <\/span>coefficient<span style=\"text-align: initial;font-size: 1em\"> of dispersion. One important thing, while the calculation of relative measures, is that the units of absolute measure and the appropriate average must be <\/span>same<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In this module, absolute measures will be discussed with examples. Relative measures will be discussed in the module \u201cMeasures of Dispersion II with Skewness and Kurtosis\u201d.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Range<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Range<span style=\"text-align: initial;font-size: 1em\"> is considered as the simplest absolute measure of dispersion. It is evaluated by just taking the difference between the maximum value and minimum value in the data set.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Range = Maximum value- Minimum value<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Ques 1<\/strong><span style=\"text-align: initial;font-size: 1em\">: Calculate the range of the following dataset:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">23, 45, 56, 52, 64, 35, 42, 51, 76, 65.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Ans<\/strong><span style=\"text-align: initial;font-size: 1em\">: Maximum value is 76 and minimum value is 23. Hence<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Range = 76-23 = 53.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Merits of Range<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">These are the merits of the range measure<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(i)\u00a0\u00a0<\/span>Range<span style=\"text-align: initial;font-size: 1em\"> is the simplest among all the absolute measures of dispersion.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(ii)\u00a0<\/span>Range<span style=\"text-align: initial;font-size: 1em\"> is defined in a rigid manner and easily calculated. Due to this <\/span>property<span style=\"text-align: initial;font-size: 1em\"> it is widely used in the industry as a quality control tool.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iii)\u00a0<\/span>Range<span style=\"text-align: initial;font-size: 1em\"> is computed within second hence one can get a complete picture of variability in the dataset in a short time.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Demerits of Range<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There are few demerits of range also. These are:<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(i) As <\/span>range<span style=\"text-align: initial;font-size: 1em\"> is evaluated just from the difference of maximum and <\/span>minimum<span style=\"text-align: initial;font-size: 1em\"> value of the dataset. Hence it does not depend on all the observations. So it is possible that range will be <\/span>same<span style=\"text-align: initial;font-size: 1em\"> for two <\/span>dataset<span style=\"text-align: initial;font-size: 1em\"> that <\/span>have same<span style=\"text-align: initial;font-size: 1em\"> maximum and minimum value but different intermediate values that make no sense.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(ii) As we already discuss that range depend on just two values and both of these values are extreme values. Hence range is affected a lot by extreme values.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iii)\u00a0<\/span>Also range<span style=\"text-align: initial;font-size: 1em\"> will not take into account the shape of the distribution. It may be possible that range will be <\/span>same<span style=\"text-align: initial;font-size: 1em\"> for the dataset whether it is symmetric or <\/span>skew symmetric<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iv)\u00a0<\/span>Range<span style=\"text-align: initial;font-size: 1em\"> is affected by the change in the observations.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(v)\u00a0<\/span>Range<span style=\"text-align: initial;font-size: 1em\"> cannot be evaluated from the grouped frequency distribution with open end class.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Quartile Deviation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This measure of dispersion is related with range and it removes the drawbacks of range upto some extent. It is defined as the difference between third quartile and first quartile divided by 2. In other words,<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-126\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-51.png\" alt=\"\" width=\"753\" height=\"404\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-51.png 753w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-51-300x161.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-51-65x35.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-51-225x121.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-51-350x188.png 350w\" sizes=\"auto, (max-width: 753px) 100vw, 753px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Merits of Quartile Deviation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>There are some merits of quartile deviation. These are<\/p>\n<p style=\"text-align: justify\">(i)\u00a0 \u00a0Quartile deviation is easy to understand like range.<\/p>\n<p style=\"text-align: justify\">(ii)\u00a0 As quartile deviation is based on first and third quarter hence it removes the limitations of the range.<\/p>\n<p style=\"text-align: justify\">(iii) As quartile deviation consider only first quarter and third quarter value. These values are derived by neglecting first and last 25% of the observations respectively. Hence it is not affected by the extreme values.<\/p>\n<p style=\"text-align: justify\">(iv) Quartile deviation is more useful to consider in case of skewed symmetrical data.<\/p>\n<p style=\"text-align: justify\">(v)\u00a0 Quartile deviation can be evaluated for open end class interval as it is not depend on the extreme values.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Demerits of Quartile deviation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>There are some demerits of quartile deviation. These are<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(i) Quartile deviation is evaluated on just half of the observations as it neglect first and last 25% of the observations. Hence it is not considered as a fully appropriate measure for detecting variation among observations.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(ii) Quartile measure is based on quartiles which are positional averages. It represents the deviation of observations among quartiles that is just a distance on a scale.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iii) Quartile deviation is very much influenced by the change in the observations. A change in single observations <\/span>lead<span style=\"text-align: initial;font-size: 1em\"> to <\/span>change<span style=\"text-align: initial;font-size: 1em\"> in the value of quartile deviation.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iv) Quartile deviation just <\/span>give<span style=\"text-align: initial;font-size: 1em\"> an idea about the deviation like range. So it is not the best measure for measuring dispersion.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Mean deviation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">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.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-127 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-52.png\" alt=\"\" width=\"772\" height=\"563\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-52.png 772w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-52-300x219.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-52-768x560.png 768w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-52-65x47.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-52-225x164.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-52-350x255.png 350w\" sizes=\"auto, (max-width: 772px) 100vw, 772px\" \/><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\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<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>Ques 3<\/strong>. Calculate the mean deviation about mean for the following observations.<\/p>\n<p>30, 40, 50, 55, 60, 65, 70, 80, 90, 100<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>Ans <\/strong>First find out the A.M. of the observations. It is 64. Now take absolute deviation of each observation from 64. The values are 34, 24, 14, 9, 4, 1, 6, 16, 26. Take the sum of these observation and divide it by 10 i.e. number of observations.<\/p>\n<p>&nbsp;<\/p>\n<p>The answer is 134\/10=13.4.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Ques 4. <\/strong>Calculate the mean deviation about mean for the following dataset.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-128\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-53.png\" alt=\"\" width=\"763\" height=\"355\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-53.png 763w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-53-300x140.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-53-65x30.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-53-225x105.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-53-350x163.png 350w\" sizes=\"auto, (max-width: 763px) 100vw, 763px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-129\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-54.png\" alt=\"\" width=\"751\" height=\"305\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-54.png 751w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-54-300x122.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-54-65x26.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-54-225x91.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-54-350x142.png 350w\" sizes=\"auto, (max-width: 751px) 100vw, 751px\" \/><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-130\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-55.png\" alt=\"\" width=\"774\" height=\"458\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-55.png 774w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-55-300x178.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-55-768x454.png 768w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-55-65x38.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-55-225x133.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-55-350x207.png 350w\" sizes=\"auto, (max-width: 774px) 100vw, 774px\" \/><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-131\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-56.png\" alt=\"\" width=\"733\" height=\"318\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-56.png 733w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-56-300x130.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-56-65x28.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-56-225x98.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-56-350x152.png 350w\" sizes=\"auto, (max-width: 733px) 100vw, 733px\" \/><\/p>\n<div>\n<p><strong>\u00a0 \u00a0 Merits and Demerits of Mean deviation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Merits<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>There are some merits of mean deviation measure. These are<\/p>\n<p>(i)\u00a0 \u00a0Mean deviation is defined in a rigid manner.<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">(ii)\u00a0 Mean deviation is an easy method for a non-mathematical background person.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iii) Mean deviation is based on all the observations so it is better than range and quartile deviation measures.<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">(iv) Mean deviation is less influenced by the extreme values than <\/span>range<span style=\"text-align: initial;font-size: 1em\">, standard deviation.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(v)\u00a0 Mean deviation is appropriate for comparison purpose because it is derived from the deviation of observations from central values i.e. mean, median or mode.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(vi) Mean deviation is basically an average of absolute deviation from central value. Hence it <\/span>remove biasness<span style=\"text-align: initial;font-size: 1em\"> to some extent that <\/span>occur<span style=\"text-align: initial;font-size: 1em\"> due to observations while calculating and provide an accurate value for dispersion.<\/span><\/p>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0 Demerits<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>There are few demerits of mean deviation measure. These are<\/p>\n<p style=\"text-align: justify\">(i)\u00a0 The major drawback of mean deviation measure is that while computing it we ignore the signs of deviation and take absolute of it. So one cannot get any information about the concentration of the observations above the mean or below the mean. So this measure is not useful for further mathematical treatment.<\/p>\n<p>(ii)\u00a0\u00a0Mean deviation cannot be computed for open end classes distributions.<\/p>\n<p>(iii) Mean deviation is not an accurate measure of dispersion for highly skewed data.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Although mean deviation has some limitations. Due to its simplicity and easy to understand features it has great importance in accountancy, economics, forecasting and business sectors.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Standard Deviation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Karl Pearson in 1823 first introduced this measure and after that it is the most widely used measure of dispersion till date. Standard deviation is the measure that prevails all the features that other measures lack of. So basically it is considered as an ideal measure of deviation. It is also know as square root of variance, root mean square deviation etc and denoted by the Greek letter (sigma). Standard deviation value is high or low when there is more or less variation among observations respectively.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-132 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-57.png\" alt=\"\" width=\"767\" height=\"265\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-57.png 767w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-57-300x104.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-57-65x22.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-57-225x78.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-57-350x121.png 350w\" sizes=\"auto, (max-width: 767px) 100vw, 767px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div><\/div>\n<div><\/div>\n<div><\/div>\n<div><\/div>\n<div><\/div>\n<div><\/div>\n<div><\/div>\n<div><\/div>\n<div><\/div>\n<div><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-133 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-58.png\" alt=\"\" width=\"770\" height=\"423\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-58.png 770w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-58-300x165.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-58-768x422.png 768w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-58-65x36.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-58-225x124.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-58-350x192.png 350w\" sizes=\"auto, (max-width: 770px) 100vw, 770px\" \/><\/div>\n<div><\/div>\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>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-134 alignleft\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-59.png\" alt=\"\" width=\"766\" height=\"395\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-59.png 766w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-59-300x155.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-59-65x34.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-59-225x116.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-59-350x180.png 350w\" sizes=\"auto, (max-width: 766px) 100vw, 766px\" \/><\/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<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-135\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-60.png\" alt=\"\" width=\"724\" height=\"396\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-60.png 724w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-60-300x164.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-60-65x36.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-60-225x123.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-60-350x191.png 350w\" sizes=\"auto, (max-width: 724px) 100vw, 724px\" \/><\/p>\n<div>\n<p><strong>\u00a0 \u00a0 Ques 8. <\/strong>Calculate the standard deviation for the following dataset.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-136\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-61.png\" alt=\"\" width=\"763\" height=\"321\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-61.png 763w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-61-300x126.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-61-65x27.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-61-225x95.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-61-350x147.png 350w\" sizes=\"auto, (max-width: 763px) 100vw, 763px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-137\" src=\"http:\/\/esp14.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-62.png\" alt=\"\" width=\"721\" height=\"390\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-62.png 721w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-62-300x162.png 300w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-62-65x35.png 65w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-62-225x122.png 225w, https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-content\/uploads\/sites\/176\/2019\/03\/Untitled-62-350x189.png 350w\" sizes=\"auto, (max-width: 721px) 100vw, 721px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Merits and Demerits of Standard deviation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Merits<\/strong><\/p>\n<p style=\"text-align: justify\">As<span style=\"text-align: initial;font-size: 1em\"> standard deviation is considered as the best absolute measure of dispersion. These are the merits of standard deviation measure<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(i)\u00a0 \u00a0Standard deviation is rigidly defined measure.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(ii)\u00a0 Standard deviation is based on all the observations.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iii) Standard deviation is evaluated by squaring of deviation of observations from the mean. This makes the standard deviation for further mathematical treatment.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(iv) Standard deviation has less influenced by the change in the sample observations.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(v)\u00a0 It is also possible to compute the combined standard deviation of two or more dataset from individual standard deviation.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(vi)\u00a0 Standard deviation is used for further statistical analysis like skewness and correlation.<\/span><\/p>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0 Demerits<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Although standard deviation is considered as the best absolute measure of dispersion but it is not free from demerits. These are<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">(i) Standard deviation is very difficult to understand for a non-mathematical background person.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">(ii) Standard deviation gives more weight to extreme values and less weight to values close to mean. As while squaring the deviation this bias will be increased.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Although, standard deviation has some demerits or limitations <\/span>but<span style=\"text-align: initial;font-size: 1em\"> this is the widely used method for the absolute measure of dispersion.<\/span><\/p>\n<\/div>\n<ol start=\"3\">\n<li><strong>Summary<\/strong><\/li>\n<\/ol>\n<p style=\"text-align: justify\">\u00a0 \u00a0 In this module<strong>,<\/strong> one category of measures of dispersion i.e. absolute measures of dispersion are discussed in detail. These are range, quartile deviation, mean deviation and standard deviation. We first give an introduction about these measures. Questions and answers are also included for better understanding of the topic. Merits and demerits of each measure are discussed for better understanding and comparison purpose. Other branch of dispersion measure that is relative measure of dispersion will be discussed in the module \u201cMeasures of Dispersion- II with Skewness and Kurtosis\u201d.<\/p>\n<ol start=\"4\">\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 I<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/3rixaWi9R4A\" 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>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<p>&nbsp;<\/p>\n","protected":false},"author":3,"menu_order":8,"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-120","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\/120","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\/120\/revisions"}],"predecessor-version":[{"id":139,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapters\/120\/revisions\/139"}],"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\/120\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/media?parent=120"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/pressbooks\/v2\/chapter-type?post=120"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/contributor?post=120"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/esp14\/wp-json\/wp\/v2\/license?post=120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}