{"id":34,"date":"2018-10-29T06:39:43","date_gmt":"2018-10-29T06:39:43","guid":{"rendered":"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=34"},"modified":"2018-10-29T09:35:51","modified_gmt":"2018-10-29T09:35:51","slug":"measures-of-central-tendency-mathematical-averages-am-gm-hm","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/chapter\/measures-of-central-tendency-mathematical-averages-am-gm-hm\/","title":{"rendered":"Measures of Central Tendency: Mathematical Averages (AM, GM, HM)."},"content":{"raw":"<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Objectives<\/strong>\r\n<ul>\r\n \t<li>After studying this module you would be able to understand the:<\/li>\r\n \t<li>Concept of measures of central tendency; Arithmetic Mean;<\/li>\r\n \t<li>Geometric Mean; Harmonic Mean;<\/li>\r\n \t<li>Methods of calculating AM, GM &amp; HM;<\/li>\r\n \t<li>Merits, demerits and uses of AM, GM &amp; HM; and Relation between AM, GM &amp; HM.<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n<strong>Introduction<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">\u201cThe International Monetary Fund (IMF) on Tuesday raised projections for India\u2019s economic growth by 0.2 percentage points to 7.6 percent for 2016-17 and 2017-18. The projections came in at a time when the Fund said global economic growth will be subdued this year, following a slowdown in the US and Britain\u2019s vote to exit the European Union. It, however, retained global economic growth at 3.1 percent for 2016 and 3.4 percent for 2017.\u201d<\/p>\r\n&nbsp;\r\n\r\n<em>Business Standard, New Delhi October 05, 2016.<\/em>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Statements like these which talk about the growth rates of nations\/states\/industries\/sectors\/areas\/etc. are quite common that we read daily in newspapers\/magazines\/journals\/etc. or hear it on TV channels or discussions among ourselves.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Similarly in our daily lives we often make statements like: the average income of Area \u201cA\u201d is Rs 15,000\/- per month; the commerce students study on an average 4 hrs daily after college; average wages of workers of Factory X are Rs 10,000\/- per month; etc.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">A careful analysis of these statements reveals that they are talking about some value or figure, not extreme but some central value, around which most of the observations cluster. This central value, around which most of the data points cluster, is used as a representative value for the data.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">Hence these central values which represent the data are known as Measures of Central Tendency. When people talk about an average value or the middle value or the most frequent value, they are talking informally about the some measure of central tendency.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Generally, it is very difficult rather impossible for a human mind to remember the huge and unwieldy set of numeric values which it comes across in life on daily basis; and even if it remembers them then also it is not possible to draw some valid conclusion from these tens\/hundreds\/thousands\/lakhs\/etc of figures. Measures of Central Tendency are the statistical tool which helps in condensing, simplifying and making the data more understandable. Hence Measures of Central Tendency occupy a place of pre eminence in all statistical analyses.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\nThe measures of central tendency which we discuss here in this module are:\r\n\r\n&nbsp;\r\n\r\nA.\u00a0\u00a0 Arithmetic mean\r\n\r\n&nbsp;\r\n\r\nB.\u00a0\u00a0 Geometric mean\r\n\r\n&nbsp;\r\n\r\nC.\u00a0\u00a0 Harmonic mean\r\n\r\n&nbsp;\r\n\r\n<strong>A. Arithmetic Mean<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The arithmetic mean or average as referred in common parlance is the most common measure of central tendency. It is obtained by adding all the observations and then dividing the sum by the number of observations. Depending on the type of data i.e. ungrouped (unclassified) data or grouped (classified) data, different methods for calculating the arithmetic mean are used.<\/p>\r\n&nbsp;\r\n\r\n<strong>Arithmetic Mean of Ungrouped Data: <\/strong>There are two methods for calculating arithmetic mean for ungrouped data.\r\n\r\n&nbsp;\r\n\r\ni) Direct method\r\n\r\n&nbsp;\r\n\r\nii) Indirect or short cut method\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\ni)\u00a0\u00a0\u00a0 <strong>Direct method:<\/strong>\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-37\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-1.png\" alt=\"\" width=\"654\" height=\"254\" \/>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Example1:<\/strong>- Find the arithmetic mean of marks obtained by 10 students in a test.\r\n\r\nThe marks are as follows:- 61, 81, 87, 78, 54, 56, 67, 65, 68, 69.\r\n\r\n&nbsp;\r\n\r\n<strong>Solution<\/strong>\r\n\r\nA.M. = (65+81+87+78+54+56+67+65+68+69)\/10\r\n\r\n=\u00a0 (690)\/10\r\n\r\n=\u00a0 69\r\n\r\nThe average marks are 69.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">ii)\u00a0<strong>Indirect or short cut method: <\/strong>In this method an arbitrary assumed mean is used. Deviations of individual observations from this assumed mean are taken for calculating arithmetic mean.<\/p>\r\n&nbsp;\r\n\r\nLet \u201cA\u201d be the arbitrary assumed mean and \u201cdi\u201d the new variable defined as follows:\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-38\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-2.png\" alt=\"\" width=\"191\" height=\"41\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>Example2:<\/strong>- Find the arithmetic mean of marks obtained by 10 students in a test.\r\n\r\n&nbsp;\r\n\r\nThe marks are as follows:- 63, 62, 67, 68, 64, 66, 67, 65, 68, 70.\r\n\r\n&nbsp;\r\n\r\n<strong>Solution<\/strong>\r\n<table class=\"aligncenter\" style=\"width: 60%\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td><strong>S. No.<\/strong><\/td>\r\n<td><strong>X<\/strong><\/td>\r\n<td><strong>d=x-A<\/strong>\r\n\r\n<strong>let \u201cA\u201d=60<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>63<\/td>\r\n<td>63-60=3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>62<\/td>\r\n<td>62-60=2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>67<\/td>\r\n<td>67-60=7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>68<\/td>\r\n<td>68-60=8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>64<\/td>\r\n<td>64-60=4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6<\/td>\r\n<td>66<\/td>\r\n<td>66-60=6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>7<\/td>\r\n<td>67<\/td>\r\n<td>67-60=7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>8<\/td>\r\n<td>65<\/td>\r\n<td>65-60=5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>9<\/td>\r\n<td>68<\/td>\r\n<td>68-60=8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>10<\/td>\r\n<td>70<\/td>\r\n<td>70-60=10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>\u2211d=60<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div><\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Arithmetic Mean of Grouped Data: <\/strong>There are two methods for calculating arithmetic mean of grouped data.\r\n\r\n&nbsp;\r\n\r\ni) Direct method\r\n\r\n&nbsp;\r\n\r\nii) Indirect or step-deviation method\r\n\r\n&nbsp;\r\n\r\n<strong>i)\u00a0<\/strong><strong>Direct method: <\/strong>Suppose we have data in form of X1, X2\u2026\u2026.\u2026\u2026.Xn observations with corresponding frequencies f1, f2\u2026\u2026\u2026\u2026\u2026\u2026\u2026fn. The arithmetic mean will be\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-39\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-3.png\" alt=\"\" width=\"248\" height=\"37\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>Example 3:<\/strong>- Calculate the average number of children per family from the following data.\r\n\r\n&nbsp;\r\n<table class=\"aligncenter\" style=\"width: 60%\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td>\u00a0No. of children<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<td>5<\/td>\r\n<td>6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0No. of families<\/td>\r\n<td>30<\/td>\r\n<td>52<\/td>\r\n<td>60<\/td>\r\n<td>65<\/td>\r\n<td>18<\/td>\r\n<td>10<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<table class=\"aligncenter\" style=\"width: 60%\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td>No. of\u00a0children (x)<\/td>\r\n<td>No. of\u00a0family (f)<\/td>\r\n<td>f.x<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>30<\/td>\r\n<td>0x30=0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>52<\/td>\r\n<td>1x52=52<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>60<\/td>\r\n<td>2x60=120<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>65<\/td>\r\n<td>3x65=195<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>18<\/td>\r\n<td>4x18=72<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5<\/td>\r\n<td>10<\/td>\r\n<td>5x10=50<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>6<\/td>\r\n<td>5<\/td>\r\n<td>6x5=30<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>\u2211f= 240<\/td>\r\n<td>\u2211f.x= 519<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">A.M. = \u2211\u00a0 =1 \u00a0\u00a0\u00a0\u00a0\/=\u00a0 519\/240<\/p>\r\n<p style=\"text-align: center\">=\u00a0 2.1625<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>ii)\u00a0\u00a0\u00a0 <\/strong><strong>Indirect or step-deviation method: <\/strong>Steps we follow in this method are as follows-\r\n\r\n<strong>\u00a0<\/strong>\r\n\r\n<strong>a)\u00a0\u00a0\u00a0 <\/strong>First find out the mid points of different classes (X)\r\n\r\n<strong>\u00a0<\/strong>\r\n\r\n<strong>b)\u00a0\u00a0\u00a0 <\/strong>Then decide about the value of assumed mean. Let it be \u201cA\u201d\r\n\r\n<strong>\u00a0<\/strong>\r\n\r\n<strong>c)\u00a0\u00a0\u00a0\u00a0 <\/strong>Calculate the value of dx. If class interval is denoted by \u2018h\u2019 and \u2018A\u2019 is assumed mean then <strong>dx = (X-A)\/h<\/strong>.\r\n\r\n<strong>\u00a0<\/strong>\r\n\r\n<strong>d)\u00a0\u00a0\u00a0 <\/strong>Multiply these deviations with corresponding frequency and calculate the value of \u2211 fdx.\r\n\r\n<strong>\u00a0<\/strong>\r\n\r\n<strong>e)\u00a0\u00a0\u00a0\u00a0 <\/strong>Apply the formula-\r\n\r\n<strong>A.M. = A+ ( <\/strong>\u2211\u00a0 \/\u00a0 <strong>\u00a0)h<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Example:-4 <\/strong>The following table shows the daily income distribution of 500 workers. Find the average income.\r\n\r\n&nbsp;\r\n<table class=\"aligncenter\" style=\"width: 60%\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td>Income<\/td>\r\n<td>0-50<\/td>\r\n<td>50-100<\/td>\r\n<td>100-150<\/td>\r\n<td>150-200<\/td>\r\n<td>200-250<\/td>\r\n<td>250-300<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>No. of Workers<\/td>\r\n<td>90<\/td>\r\n<td>150<\/td>\r\n<td>100<\/td>\r\n<td>80<\/td>\r\n<td>70<\/td>\r\n<td>10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>Solution<\/strong>\r\n<table class=\"aligncenter\" style=\"width: 60%\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 17.1717%\">Income[footnote]null[\/footnote]<\/td>\r\n<td style=\"width: 18.1818%\">Workers\u00a0(f)<\/td>\r\n<td style=\"width: 19.1919%\">Mid Value(x)<\/td>\r\n<td style=\"width: 25.6566%\">dx=(X-125)\/50<\/td>\r\n<td style=\"width: 19.596%\">fdx<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 17.1717%\">0-50<\/td>\r\n<td style=\"width: 18.1818%\">90<\/td>\r\n<td style=\"width: 19.1919%\">25<\/td>\r\n<td style=\"width: 25.6566%\">-2<\/td>\r\n<td style=\"width: 19.596%\">-180<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 17.1717%\">50-100<\/td>\r\n<td style=\"width: 18.1818%\">150<\/td>\r\n<td style=\"width: 19.1919%\">75<\/td>\r\n<td style=\"width: 25.6566%\">-1<\/td>\r\n<td style=\"width: 19.596%\">-150<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 17.1717%\">100-150<\/td>\r\n<td style=\"width: 18.1818%\">100<\/td>\r\n<td style=\"width: 19.1919%\">125<\/td>\r\n<td style=\"width: 25.6566%\">0<\/td>\r\n<td style=\"width: 19.596%\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 17.1717%\">150-200<\/td>\r\n<td style=\"width: 18.1818%\">80<\/td>\r\n<td style=\"width: 19.1919%\">175<\/td>\r\n<td style=\"width: 25.6566%\">1<\/td>\r\n<td style=\"width: 19.596%\">80<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 17.1717%\">200-250<\/td>\r\n<td style=\"width: 18.1818%\">70<\/td>\r\n<td style=\"width: 19.1919%\">225<\/td>\r\n<td style=\"width: 25.6566%\">2<\/td>\r\n<td style=\"width: 19.596%\">140<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 17.1717%\">250-300<\/td>\r\n<td style=\"width: 18.1818%\">10<\/td>\r\n<td style=\"width: 19.1919%\">275<\/td>\r\n<td style=\"width: 25.6566%\">3<\/td>\r\n<td style=\"width: 19.596%\">30<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 17.1717%\"><\/td>\r\n<td style=\"width: 18.1818%\">\u2211f=500<\/td>\r\n<td style=\"width: 19.1919%\"><\/td>\r\n<td style=\"width: 25.6566%\"><\/td>\r\n<td style=\"width: 19.596%\">\u2211 fdx=-80<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<strong>A.M. = A+ ( <\/strong>\u2211\u00a0 \u00a0\/\u00a0 <strong>\u00a0)h<\/strong>\r\n\r\n&nbsp;\r\n\r\n=\u00a0 125 + (-80) x 50\r\n\r\n&nbsp;\r\n\r\n500\r\n\r\n&nbsp;\r\n\r\n=\u00a0 117\r\n\r\n&nbsp;\r\n\r\nThus, average income is Rs. 117.\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Merits of Arithmetic Mean:<\/strong>\r\n\r\n&nbsp;\r\n\r\ni) It is easy to understand and calculate\r\n\r\n&nbsp;\r\n\r\nii) It is based on all observations\r\n\r\n&nbsp;\r\n\r\niii) It is rigidly defined\r\n\r\n&nbsp;\r\n\r\niv) It is capable of further mathematical treatment\r\n\r\n&nbsp;\r\n\r\nv)\u00a0 It is least affected by sampling fluctuation.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>Demerits of Arithmetic Mean:<\/strong>\r\n\r\n&nbsp;\r\n\r\ni) It is unduly affected by extreme values.\r\n\r\n&nbsp;\r\n\r\nii) In case of open ended classes it cannot be calculated.\r\n\r\n&nbsp;\r\n\r\n<strong>B. Geometric Mean:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">When we are interested in measuring average rate of change over time then we use geometric mean. Geometric mean is defined as the nth root of the product of n items (or) values.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Calculation of Geometric Mean (G.M.) - Individual series<\/strong>:\r\n\r\nIf<em>x<\/em>1,<em>x<\/em>2,<em>x<\/em>3,.......,<em>x<\/em><em>n\u00a0<\/em><span style=\"font-size: 1em;text-align: initial\">be n observations studied on a variable X, then the G.M of the observations is defined as<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-41\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-4.png\" alt=\"\" width=\"194\" height=\"60\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Applying log both sides<\/span><img class=\"aligncenter size-full wp-image-42\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-5.png\" alt=\"\" width=\"282\" height=\"132\" \/><img class=\"aligncenter size-full wp-image-43\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-6.png\" alt=\"\" width=\"222\" height=\"176\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Calculation of G.M. - Discrete series<\/strong>:\r\n\r\n&nbsp;\r\n\r\nIf\u00a0 <em>x<\/em>1 , <em>x<\/em>2 , <em>x<\/em>3 ,.......,<em>x<\/em><em>n<\/em> be n observations of a\u00a0variable X with frequencies\u00a0 <em>f<\/em>1 , <em>f<\/em> 2 , <em>f<\/em> 3 ,.......,<em>f<\/em> <em>n\u00a0<\/em>respectively then\r\n\r\nthe G.M is defined as\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-44\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-7.png\" alt=\"\" width=\"262\" height=\"67\" \/>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<em>n<\/em>\r\n\r\nWhere N =\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <em>f <\/em><em>i<\/em><em>\u00a0 <\/em>i.e. total frequency\r\n\r\n<em>i <\/em>1\r\n\r\nApplying log both sides in (i) we get\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-45\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-8.png\" alt=\"\" width=\"244\" height=\"72\" \/><strong>\r\nCalculation\u00a0 of\u00a0 G.M.\u00a0 -Continuous\u00a0 Series<\/strong>:<strong>\u00a0 <\/strong>In\u00a0 continuous\u00a0 series\u00a0 the\u00a0 G.M.\u00a0 is\u00a0<span style=\"font-size: 1em\">calculated by replacing the value of xi by the mid points of the class\u2019s i.e. mi<\/span>\r\n\r\n&nbsp;\r\n\r\nWhere <em>m<\/em><em>i<\/em>\u00a0 is the mid value of the ith class interval.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>Merits of Geometric Mean<\/strong>:\r\n\r\n<\/div>\r\n<div>\r\n<ol>\r\n \t<li style=\"text-align: justify\">It is rigidly defined.<\/li>\r\n \t<li style=\"text-align: justify\">\u00a0It is based on all the observations.<\/li>\r\n \t<li style=\"text-align: justify\">If G1 and G2 are geometric means of two groups having n1 and n2 observations, respectively, then the geometric mean G of the combined group of (n1+n2) values is given by log G = (n1log G1 + n2 log G2) \/ (n1 + n2)<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Uses of Geometric Mean: <\/strong>Geometrical Mean is especially useful in the following cases.\r\n\r\n&nbsp;\r\n<ol>\r\n \t<li style=\"text-align: justify\">The G.M is used to find the average percentage increase in sales, production, or other economic or business series.<\/li>\r\n<\/ol>\r\n<p style=\"text-align: justify\">For example, from 1992 to 1994 prices increased by 5%,10%,and 18% respectively, then the average annual income is not 11% which is calculated by A.M but it is 10.9 which is calculated by G.M.<\/p>\r\n&nbsp;\r\n\r\n2) G.M is theoretically considered to be best average in the construction of Index numbers.\r\n\r\n&nbsp;\r\n\r\nC. <strong>Harmonic Mean<\/strong>:\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The Harmonic Mean (H.M.) is defined as the reciprocal of the arithmetic mean of the reciprocals of the individual observations.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Calculation of H.M -Individual series<\/strong><span style=\"text-align: initial;font-size: 1em\">: If <\/span><em style=\"text-align: initial;font-size: 1em\">x<\/em><span style=\"text-align: initial;font-size: 1em\">1 variable X then harmonic mean is defined as\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">be \u2018n\u2019 observations of a variable X then harmonic means defined as\u00a0<\/span><\/p>\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-46\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-9.png\" alt=\"\" width=\"263\" height=\"103\" \/>\r\n<div><\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-47\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-10.png\" alt=\"\" width=\"147\" height=\"97\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>Calculation\u00a0 of\u00a0 H.M.\u00a0 -Discrete\u00a0\u00a0 series<\/strong>:\u00a0\u00a0\u00a0\u00a0 If X<sub>1,<\/sub>X<sub>2,<\/sub>X<sub>3---------------<\/sub>X<sub>n\u00a0<\/sub>be an observations occuring with frequencies f<sub>1,<\/sub>f<sub>2,-------------<\/sub>f<sub>n <\/sub>respectively then H.M defined as\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-48\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-11.png\" alt=\"\" width=\"245\" height=\"213\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<strong>Calculation of H.M \u2013 Continuous series: <\/strong>In case of continuous series H.M can be\u00a0calculated by taking mid values ( mi ) in place of xi ' s . Hence H.M is given by\u00a0<img class=\"aligncenter size-full wp-image-49\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-12.png\" alt=\"\" width=\"532\" height=\"113\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>Example:- 5 <\/strong>A cyclist pedals from his house to his college at a speed of 12 km.p.h and back from the college to his house at 15 km.p.h Find the average speed.<\/p>\r\n&nbsp;\r\n\r\n<strong>Solution<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Let the distance from the house to the college be <em>x<\/em> kms. So the total distance travelled by cyclist in going to college and then coming back to house is 2x kms. Since the speed of cyclist in going from house to college is 12 km.p.h. therefore the time taken to cover this distance is x\/12 hours. Similarly the time taken to reach house from college is x \/ 15 hours. Thus a total distance of 2x kms is covered in (12 +15 <strong>)<\/strong>hours<strong>.<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">Speed = Distance\/Time<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-50\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-13.png\" alt=\"\" width=\"420\" height=\"125\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>Merits of Harmonic Mean<\/strong>:\r\n\r\n&nbsp;\r\n\r\n1)\u00a0 Its value is based on all the observations of the data.\r\n\r\n&nbsp;\r\n\r\n2) It is less affected by the extreme values.\r\n\r\n&nbsp;\r\n\r\n3)\u00a0 It is strictly defined.\r\n\r\n&nbsp;\r\n\r\n<strong>Demerits of Harmonic Mean<\/strong>:\r\n\r\n&nbsp;\r\n\r\n1)\u00a0 It is not simple to calculate and easy to understand.\r\n\r\n&nbsp;\r\n\r\n2)\u00a0 It cannot be calculated if one of the observations is zero.\r\n\r\n&nbsp;\r\n\r\n3)\u00a0 The H.M is always less than A.M and G.M.\r\n\r\n&nbsp;\r\n\r\n<strong>Uses of Harmonic Mean:<\/strong>\r\n\r\n&nbsp;\r\n\r\nThe H.M is used to calculate the averages where two units are involved like rates, speed, etc.\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Relation between A.M., G.M. and H.M.<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">The relation between A.M, G.M, and H.M is given by<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-51\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-14.png\" alt=\"\" width=\"247\" height=\"43\" \/>\r\n\r\n<strong>Note<\/strong>: The equality condition holds true only if all the items are equal in the distribution.\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Prove that if a and b are two positive numbers then<\/strong>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>Solution:<\/strong>\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-52\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-15.png\" alt=\"\" width=\"438\" height=\"440\" \/><img class=\"aligncenter size-full wp-image-53\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-16.png\" alt=\"\" width=\"552\" height=\"130\" \/><img class=\"aligncenter size-full wp-image-54\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-17.png\" alt=\"\" width=\"577\" height=\"334\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>Summary<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The measures of central tendency give us an idea about the central value around which the data values cluster. That\u2019s why these values are considered to be representative values i.e. the values which represent the data. Arithmetic mean is the most common measure of central tendency which is obtained by adding all the observations and then dividing the sum by the number of observations. Geometric mean is used for measuring the average rate of change over time. It is defined as the nth root of the product of n items (or) values. Harmonic Mean (H.M.) is defined as the reciprocal of the arithmetic mean of the reciprocals of the individual observations.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center\"><strong>Learn More:<\/strong><\/p>\r\n\r\n<ol>\r\n \t<li><em>Business Research Methods\u2019 Authored by Naval Bajpai, Published by Pearson\u2019s India PHI<\/em><\/li>\r\n \t<li><em>Business Statistics Authored by Dr. K.L. Gupta, Published by Nirupam Publications.<\/em><\/li>\r\n \t<li><em>Business Statistics Authored by G.C. Beri, Published by TMH Publications.<\/em><\/li>\r\n \t<li><em style=\"text-align: initial;font-size: 1em\">.Statistics For Managers using Microsoft Excel by David M. Levine David F. Stephan Timothy C. Krehbiel Mark L. Berenson, Published by PEARSON<\/em><\/li>\r\n \t<li><em>Business Statistics by Ken Black, Published by John Wiley &amp; Sons, Inc.<\/em><\/li>\r\n<\/ol>","rendered":"<div>\n<p>&nbsp;<\/p>\n<p><strong>Objectives<\/strong><\/p>\n<ul>\n<li>After studying this module you would be able to understand the:<\/li>\n<li>Concept of measures of central tendency; Arithmetic Mean;<\/li>\n<li>Geometric Mean; Harmonic Mean;<\/li>\n<li>Methods of calculating AM, GM &amp; HM;<\/li>\n<li>Merits, demerits and uses of AM, GM &amp; HM; and Relation between AM, GM &amp; HM.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><strong>Introduction<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">\u201cThe International Monetary Fund (IMF) on Tuesday raised projections for India\u2019s economic growth by 0.2 percentage points to 7.6 percent for 2016-17 and 2017-18. The projections came in at a time when the Fund said global economic growth will be subdued this year, following a slowdown in the US and Britain\u2019s vote to exit the European Union. It, however, retained global economic growth at 3.1 percent for 2016 and 3.4 percent for 2017.\u201d<\/p>\n<p>&nbsp;<\/p>\n<p><em>Business Standard, New Delhi October 05, 2016.<\/em><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Statements like these which talk about the growth rates of nations\/states\/industries\/sectors\/areas\/etc. are quite common that we read daily in newspapers\/magazines\/journals\/etc. or hear it on TV channels or discussions among ourselves.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Similarly in our daily lives we often make statements like: the average income of Area \u201cA\u201d is Rs 15,000\/- per month; the commerce students study on an average 4 hrs daily after college; average wages of workers of Factory X are Rs 10,000\/- per month; etc.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">A careful analysis of these statements reveals that they are talking about some value or figure, not extreme but some central value, around which most of the observations cluster. This central value, around which most of the data points cluster, is used as a representative value for the data.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">Hence these central values which represent the data are known as Measures of Central Tendency. When people talk about an average value or the middle value or the most frequent value, they are talking informally about the some measure of central tendency.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Generally, it is very difficult rather impossible for a human mind to remember the huge and unwieldy set of numeric values which it comes across in life on daily basis; and even if it remembers them then also it is not possible to draw some valid conclusion from these tens\/hundreds\/thousands\/lakhs\/etc of figures. Measures of Central Tendency are the statistical tool which helps in condensing, simplifying and making the data more understandable. Hence Measures of Central Tendency occupy a place of pre eminence in all statistical analyses.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>The measures of central tendency which we discuss here in this module are:<\/p>\n<p>&nbsp;<\/p>\n<p>A.\u00a0\u00a0 Arithmetic mean<\/p>\n<p>&nbsp;<\/p>\n<p>B.\u00a0\u00a0 Geometric mean<\/p>\n<p>&nbsp;<\/p>\n<p>C.\u00a0\u00a0 Harmonic mean<\/p>\n<p>&nbsp;<\/p>\n<p><strong>A. Arithmetic Mean<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The arithmetic mean or average as referred in common parlance is the most common measure of central tendency. It is obtained by adding all the observations and then dividing the sum by the number of observations. Depending on the type of data i.e. ungrouped (unclassified) data or grouped (classified) data, different methods for calculating the arithmetic mean are used.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Arithmetic Mean of Ungrouped Data: <\/strong>There are two methods for calculating arithmetic mean for ungrouped data.<\/p>\n<p>&nbsp;<\/p>\n<p>i) Direct method<\/p>\n<p>&nbsp;<\/p>\n<p>ii) Indirect or short cut method<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>i)\u00a0\u00a0\u00a0 <strong>Direct method:<\/strong><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-37\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-1.png\" alt=\"\" width=\"654\" height=\"254\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-1.png 654w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-1-300x117.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-1-65x25.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-1-225x87.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-1-350x136.png 350w\" sizes=\"auto, (max-width: 654px) 100vw, 654px\" \/><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Example1:<\/strong>&#8211; Find the arithmetic mean of marks obtained by 10 students in a test.<\/p>\n<p>The marks are as follows:- 61, 81, 87, 78, 54, 56, 67, 65, 68, 69.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>A.M. = (65+81+87+78+54+56+67+65+68+69)\/10<\/p>\n<p>=\u00a0 (690)\/10<\/p>\n<p>=\u00a0 69<\/p>\n<p>The average marks are 69.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">ii)\u00a0<strong>Indirect or short cut method: <\/strong>In this method an arbitrary assumed mean is used. Deviations of individual observations from this assumed mean are taken for calculating arithmetic mean.<\/p>\n<p>&nbsp;<\/p>\n<p>Let \u201cA\u201d be the arbitrary assumed mean and \u201cdi\u201d the new variable defined as follows:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-38\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-2.png\" alt=\"\" width=\"191\" height=\"41\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-2.png 191w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-2-65x14.png 65w\" sizes=\"auto, (max-width: 191px) 100vw, 191px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example2:<\/strong>&#8211; Find the arithmetic mean of marks obtained by 10 students in a test.<\/p>\n<p>&nbsp;<\/p>\n<p>The marks are as follows:- 63, 62, 67, 68, 64, 66, 67, 65, 68, 70.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Solution<\/strong><\/p>\n<table class=\"aligncenter\" style=\"width: 60%\">\n<tbody>\n<tr>\n<td><strong>S. No.<\/strong><\/td>\n<td><strong>X<\/strong><\/td>\n<td><strong>d=x-A<\/strong><\/p>\n<p><strong>let \u201cA\u201d=60<\/strong><\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>63<\/td>\n<td>63-60=3<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>62<\/td>\n<td>62-60=2<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>67<\/td>\n<td>67-60=7<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>68<\/td>\n<td>68-60=8<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>64<\/td>\n<td>64-60=4<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>66<\/td>\n<td>66-60=6<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>67<\/td>\n<td>67-60=7<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>65<\/td>\n<td>65-60=5<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>68<\/td>\n<td>68-60=8<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>70<\/td>\n<td>70-60=10<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td>\u2211d=60<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div><\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Arithmetic Mean of Grouped Data: <\/strong>There are two methods for calculating arithmetic mean of grouped data.<\/p>\n<p>&nbsp;<\/p>\n<p>i) Direct method<\/p>\n<p>&nbsp;<\/p>\n<p>ii) Indirect or step-deviation method<\/p>\n<p>&nbsp;<\/p>\n<p><strong>i)\u00a0<\/strong><strong>Direct method: <\/strong>Suppose we have data in form of X1, X2\u2026\u2026.\u2026\u2026.Xn observations with corresponding frequencies f1, f2\u2026\u2026\u2026\u2026\u2026\u2026\u2026fn. The arithmetic mean will be<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-39\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-3.png\" alt=\"\" width=\"248\" height=\"37\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-3.png 248w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-3-65x10.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-3-225x34.png 225w\" sizes=\"auto, (max-width: 248px) 100vw, 248px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 3:<\/strong>&#8211; Calculate the average number of children per family from the following data.<\/p>\n<p>&nbsp;<\/p>\n<table class=\"aligncenter\" style=\"width: 60%\">\n<tbody>\n<tr>\n<td>\u00a0No. of children<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>\u00a0No. of families<\/td>\n<td>30<\/td>\n<td>52<\/td>\n<td>60<\/td>\n<td>65<\/td>\n<td>18<\/td>\n<td>10<\/td>\n<td>5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table class=\"aligncenter\" style=\"width: 60%\">\n<tbody>\n<tr>\n<td>No. of\u00a0children (x)<\/td>\n<td>No. of\u00a0family (f)<\/td>\n<td>f.x<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>30<\/td>\n<td>0x30=0<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>52<\/td>\n<td>1&#215;52=52<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>60<\/td>\n<td>2&#215;60=120<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>65<\/td>\n<td>3&#215;65=195<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>18<\/td>\n<td>4&#215;18=72<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>10<\/td>\n<td>5&#215;10=50<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>5<\/td>\n<td>6&#215;5=30<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\u2211f= 240<\/td>\n<td>\u2211f.x= 519<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">A.M. = \u2211\u00a0 =1 \u00a0\u00a0\u00a0\u00a0\/=\u00a0 519\/240<\/p>\n<p style=\"text-align: center\">=\u00a0 2.1625<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>ii)\u00a0\u00a0\u00a0 <\/strong><strong>Indirect or step-deviation method: <\/strong>Steps we follow in this method are as follows-<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p><strong>a)\u00a0\u00a0\u00a0 <\/strong>First find out the mid points of different classes (X)<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p><strong>b)\u00a0\u00a0\u00a0 <\/strong>Then decide about the value of assumed mean. Let it be \u201cA\u201d<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p><strong>c)\u00a0\u00a0\u00a0\u00a0 <\/strong>Calculate the value of dx. If class interval is denoted by \u2018h\u2019 and \u2018A\u2019 is assumed mean then <strong>dx = (X-A)\/h<\/strong>.<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p><strong>d)\u00a0\u00a0\u00a0 <\/strong>Multiply these deviations with corresponding frequency and calculate the value of \u2211 fdx.<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p><strong>e)\u00a0\u00a0\u00a0\u00a0 <\/strong>Apply the formula-<\/p>\n<p><strong>A.M. = A+ ( <\/strong>\u2211\u00a0 \/\u00a0 <strong>\u00a0)h<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example:-4 <\/strong>The following table shows the daily income distribution of 500 workers. Find the average income.<\/p>\n<p>&nbsp;<\/p>\n<table class=\"aligncenter\" style=\"width: 60%\">\n<tbody>\n<tr>\n<td>Income<\/td>\n<td>0-50<\/td>\n<td>50-100<\/td>\n<td>100-150<\/td>\n<td>150-200<\/td>\n<td>200-250<\/td>\n<td>250-300<\/td>\n<\/tr>\n<tr>\n<td>No. of Workers<\/td>\n<td>90<\/td>\n<td>150<\/td>\n<td>100<\/td>\n<td>80<\/td>\n<td>70<\/td>\n<td>10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Solution<\/strong><\/p>\n<table class=\"aligncenter\" style=\"width: 60%\">\n<tbody>\n<tr>\n<td style=\"width: 17.1717%\">Income<a class=\"footnote\" title=\"null\" id=\"return-footnote-34-1\" href=\"#footnote-34-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/td>\n<td style=\"width: 18.1818%\">Workers\u00a0(f)<\/td>\n<td style=\"width: 19.1919%\">Mid Value(x)<\/td>\n<td style=\"width: 25.6566%\">dx=(X-125)\/50<\/td>\n<td style=\"width: 19.596%\">fdx<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 17.1717%\">0-50<\/td>\n<td style=\"width: 18.1818%\">90<\/td>\n<td style=\"width: 19.1919%\">25<\/td>\n<td style=\"width: 25.6566%\">-2<\/td>\n<td style=\"width: 19.596%\">-180<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 17.1717%\">50-100<\/td>\n<td style=\"width: 18.1818%\">150<\/td>\n<td style=\"width: 19.1919%\">75<\/td>\n<td style=\"width: 25.6566%\">-1<\/td>\n<td style=\"width: 19.596%\">-150<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 17.1717%\">100-150<\/td>\n<td style=\"width: 18.1818%\">100<\/td>\n<td style=\"width: 19.1919%\">125<\/td>\n<td style=\"width: 25.6566%\">0<\/td>\n<td style=\"width: 19.596%\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 17.1717%\">150-200<\/td>\n<td style=\"width: 18.1818%\">80<\/td>\n<td style=\"width: 19.1919%\">175<\/td>\n<td style=\"width: 25.6566%\">1<\/td>\n<td style=\"width: 19.596%\">80<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 17.1717%\">200-250<\/td>\n<td style=\"width: 18.1818%\">70<\/td>\n<td style=\"width: 19.1919%\">225<\/td>\n<td style=\"width: 25.6566%\">2<\/td>\n<td style=\"width: 19.596%\">140<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 17.1717%\">250-300<\/td>\n<td style=\"width: 18.1818%\">10<\/td>\n<td style=\"width: 19.1919%\">275<\/td>\n<td style=\"width: 25.6566%\">3<\/td>\n<td style=\"width: 19.596%\">30<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 17.1717%\"><\/td>\n<td style=\"width: 18.1818%\">\u2211f=500<\/td>\n<td style=\"width: 19.1919%\"><\/td>\n<td style=\"width: 25.6566%\"><\/td>\n<td style=\"width: 19.596%\">\u2211 fdx=-80<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p><strong>A.M. = A+ ( <\/strong>\u2211\u00a0 \u00a0\/\u00a0 <strong>\u00a0)h<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>=\u00a0 125 + (-80) x 50<\/p>\n<p>&nbsp;<\/p>\n<p>500<\/p>\n<p>&nbsp;<\/p>\n<p>=\u00a0 117<\/p>\n<p>&nbsp;<\/p>\n<p>Thus, average income is Rs. 117.<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Merits of Arithmetic Mean:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>i) It is easy to understand and calculate<\/p>\n<p>&nbsp;<\/p>\n<p>ii) It is based on all observations<\/p>\n<p>&nbsp;<\/p>\n<p>iii) It is rigidly defined<\/p>\n<p>&nbsp;<\/p>\n<p>iv) It is capable of further mathematical treatment<\/p>\n<p>&nbsp;<\/p>\n<p>v)\u00a0 It is least affected by sampling fluctuation.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Demerits of Arithmetic Mean:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>i) It is unduly affected by extreme values.<\/p>\n<p>&nbsp;<\/p>\n<p>ii) In case of open ended classes it cannot be calculated.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>B. Geometric Mean:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">When we are interested in measuring average rate of change over time then we use geometric mean. Geometric mean is defined as the nth root of the product of n items (or) values.<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Calculation of Geometric Mean (G.M.) &#8211; Individual series<\/strong>:<\/p>\n<p>If<em>x<\/em>1,<em>x<\/em>2,<em>x<\/em>3,&#8230;&#8230;.,<em>x<\/em><em>n\u00a0<\/em><span style=\"font-size: 1em;text-align: initial\">be n observations studied on a variable X, then the G.M of the observations is defined as<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-41\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-4.png\" alt=\"\" width=\"194\" height=\"60\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-4.png 194w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-4-65x20.png 65w\" sizes=\"auto, (max-width: 194px) 100vw, 194px\" \/><\/p>\n<\/div>\n<div>\n<p><span style=\"text-align: initial;font-size: 1em\">Applying log both sides<\/span><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-42\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-5.png\" alt=\"\" width=\"282\" height=\"132\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-5.png 282w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-5-65x30.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-5-225x105.png 225w\" sizes=\"auto, (max-width: 282px) 100vw, 282px\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-43\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-6.png\" alt=\"\" width=\"222\" height=\"176\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-6.png 222w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-6-65x52.png 65w\" sizes=\"auto, (max-width: 222px) 100vw, 222px\" \/><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Calculation of G.M. &#8211; Discrete series<\/strong>:<\/p>\n<p>&nbsp;<\/p>\n<p>If\u00a0 <em>x<\/em>1 , <em>x<\/em>2 , <em>x<\/em>3 ,&#8230;&#8230;.,<em>x<\/em><em>n<\/em> be n observations of a\u00a0variable X with frequencies\u00a0 <em>f<\/em>1 , <em>f<\/em> 2 , <em>f<\/em> 3 ,&#8230;&#8230;.,<em>f<\/em> <em>n\u00a0<\/em>respectively then<\/p>\n<p>the G.M is defined as<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-44\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-7.png\" alt=\"\" width=\"262\" height=\"67\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-7.png 262w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-7-65x17.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-7-225x58.png 225w\" sizes=\"auto, (max-width: 262px) 100vw, 262px\" \/><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p><em>n<\/em><\/p>\n<p>Where N =\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <em>f <\/em><em>i<\/em><em>\u00a0 <\/em>i.e. total frequency<\/p>\n<p><em>i <\/em>1<\/p>\n<p>Applying log both sides in (i) we get<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-45\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-8.png\" alt=\"\" width=\"244\" height=\"72\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-8.png 244w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-8-65x19.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-8-225x66.png 225w\" sizes=\"auto, (max-width: 244px) 100vw, 244px\" \/><strong><br \/>\nCalculation\u00a0 of\u00a0 G.M.\u00a0 -Continuous\u00a0 Series<\/strong>:<strong>\u00a0 <\/strong>In\u00a0 continuous\u00a0 series\u00a0 the\u00a0 G.M.\u00a0 is\u00a0<span style=\"font-size: 1em\">calculated by replacing the value of xi by the mid points of the class\u2019s i.e. mi<\/span><\/p>\n<p>&nbsp;<\/p>\n<p>Where <em>m<\/em><em>i<\/em>\u00a0 is the mid value of the ith class interval.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Merits of Geometric Mean<\/strong>:<\/p>\n<\/div>\n<div>\n<ol>\n<li style=\"text-align: justify\">It is rigidly defined.<\/li>\n<li style=\"text-align: justify\">\u00a0It is based on all the observations.<\/li>\n<li style=\"text-align: justify\">If G1 and G2 are geometric means of two groups having n1 and n2 observations, respectively, then the geometric mean G of the combined group of (n1+n2) values is given by log G = (n1log G1 + n2 log G2) \/ (n1 + n2)<\/li>\n<\/ol>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Uses of Geometric Mean: <\/strong>Geometrical Mean is especially useful in the following cases.<\/p>\n<p>&nbsp;<\/p>\n<ol>\n<li style=\"text-align: justify\">The G.M is used to find the average percentage increase in sales, production, or other economic or business series.<\/li>\n<\/ol>\n<p style=\"text-align: justify\">For example, from 1992 to 1994 prices increased by 5%,10%,and 18% respectively, then the average annual income is not 11% which is calculated by A.M but it is 10.9 which is calculated by G.M.<\/p>\n<p>&nbsp;<\/p>\n<p>2) G.M is theoretically considered to be best average in the construction of Index numbers.<\/p>\n<p>&nbsp;<\/p>\n<p>C. <strong>Harmonic Mean<\/strong>:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The Harmonic Mean (H.M.) is defined as the reciprocal of the arithmetic mean of the reciprocals of the individual observations.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Calculation of H.M -Individual series<\/strong><span style=\"text-align: initial;font-size: 1em\">: If <\/span><em style=\"text-align: initial;font-size: 1em\">x<\/em><span style=\"text-align: initial;font-size: 1em\">1 variable X then harmonic mean is defined as\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">be \u2018n\u2019 observations of a variable X then harmonic means defined as\u00a0<\/span><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-46\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-9.png\" alt=\"\" width=\"263\" height=\"103\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-9.png 263w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-9-65x25.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-9-225x88.png 225w\" sizes=\"auto, (max-width: 263px) 100vw, 263px\" \/><\/p>\n<div><\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-47\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-10.png\" alt=\"\" width=\"147\" height=\"97\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-10.png 147w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-10-65x43.png 65w\" sizes=\"auto, (max-width: 147px) 100vw, 147px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Calculation\u00a0 of\u00a0 H.M.\u00a0 -Discrete\u00a0\u00a0 series<\/strong>:\u00a0\u00a0\u00a0\u00a0 If X<sub>1,<\/sub>X<sub>2,<\/sub>X<sub>3&#8212;&#8212;&#8212;&#8212;&#8212;<\/sub>X<sub>n\u00a0<\/sub>be an observations occuring with frequencies f<sub>1,<\/sub>f<sub>2,&#8212;&#8212;&#8212;&#8212;-<\/sub>f<sub>n <\/sub>respectively then H.M defined as<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-48\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-11.png\" alt=\"\" width=\"245\" height=\"213\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-11.png 245w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-11-65x57.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-11-225x196.png 225w\" sizes=\"auto, (max-width: 245px) 100vw, 245px\" \/><\/p>\n<\/div>\n<div>\n<p><strong>Calculation of H.M \u2013 Continuous series: <\/strong>In case of continuous series H.M can be\u00a0calculated by taking mid values ( mi ) in place of xi &#8216; s . Hence H.M is given by\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-49\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-12.png\" alt=\"\" width=\"532\" height=\"113\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-12.png 532w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-12-300x64.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-12-65x14.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-12-225x48.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-12-350x74.png 350w\" sizes=\"auto, (max-width: 532px) 100vw, 532px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>Example:- 5 <\/strong>A cyclist pedals from his house to his college at a speed of 12 km.p.h and back from the college to his house at 15 km.p.h Find the average speed.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Let the distance from the house to the college be <em>x<\/em> kms. So the total distance travelled by cyclist in going to college and then coming back to house is 2x kms. Since the speed of cyclist in going from house to college is 12 km.p.h. therefore the time taken to cover this distance is x\/12 hours. Similarly the time taken to reach house from college is x \/ 15 hours. Thus a total distance of 2x kms is covered in (12 +15 <strong>)<\/strong>hours<strong>.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">Speed = Distance\/Time<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-50\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-13.png\" alt=\"\" width=\"420\" height=\"125\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-13.png 420w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-13-300x89.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-13-65x19.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-13-225x67.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-13-350x104.png 350w\" sizes=\"auto, (max-width: 420px) 100vw, 420px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Merits of Harmonic Mean<\/strong>:<\/p>\n<p>&nbsp;<\/p>\n<p>1)\u00a0 Its value is based on all the observations of the data.<\/p>\n<p>&nbsp;<\/p>\n<p>2) It is less affected by the extreme values.<\/p>\n<p>&nbsp;<\/p>\n<p>3)\u00a0 It is strictly defined.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Demerits of Harmonic Mean<\/strong>:<\/p>\n<p>&nbsp;<\/p>\n<p>1)\u00a0 It is not simple to calculate and easy to understand.<\/p>\n<p>&nbsp;<\/p>\n<p>2)\u00a0 It cannot be calculated if one of the observations is zero.<\/p>\n<p>&nbsp;<\/p>\n<p>3)\u00a0 The H.M is always less than A.M and G.M.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Uses of Harmonic Mean:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>The H.M is used to calculate the averages where two units are involved like rates, speed, etc.<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Relation between A.M., G.M. and H.M.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">The relation between A.M, G.M, and H.M is given by<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-51\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-14.png\" alt=\"\" width=\"247\" height=\"43\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-14.png 247w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-14-65x11.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-14-225x39.png 225w\" sizes=\"auto, (max-width: 247px) 100vw, 247px\" \/><\/p>\n<p><strong>Note<\/strong>: The equality condition holds true only if all the items are equal in the distribution.<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Prove that if a and b are two positive numbers then<\/strong><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>Solution:<\/strong><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-52\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-15.png\" alt=\"\" width=\"438\" height=\"440\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-15.png 438w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-15-150x150.png 150w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-15-300x300.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-15-65x65.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-15-225x226.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-15-350x352.png 350w\" sizes=\"auto, (max-width: 438px) 100vw, 438px\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-53\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-16.png\" alt=\"\" width=\"552\" height=\"130\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-16.png 552w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-16-300x71.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-16-65x15.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-16-225x53.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-16-350x82.png 350w\" sizes=\"auto, (max-width: 552px) 100vw, 552px\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-54\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-17.png\" alt=\"\" width=\"577\" height=\"334\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-17.png 577w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-17-300x174.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-17-65x38.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-17-225x130.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-17-350x203.png 350w\" sizes=\"auto, (max-width: 577px) 100vw, 577px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Summary<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The measures of central tendency give us an idea about the central value around which the data values cluster. That\u2019s why these values are considered to be representative values i.e. the values which represent the data. Arithmetic mean is the most common measure of central tendency which is obtained by adding all the observations and then dividing the sum by the number of observations. Geometric mean is used for measuring the average rate of change over time. It is defined as the nth root of the product of n items (or) values. Harmonic Mean (H.M.) is defined as the reciprocal of the arithmetic mean of the reciprocals of the individual observations.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\"><strong>Learn More:<\/strong><\/p>\n<ol>\n<li><em>Business Research Methods\u2019 Authored by Naval Bajpai, Published by Pearson\u2019s India PHI<\/em><\/li>\n<li><em>Business Statistics Authored by Dr. K.L. Gupta, Published by Nirupam Publications.<\/em><\/li>\n<li><em>Business Statistics Authored by G.C. Beri, Published by TMH Publications.<\/em><\/li>\n<li><em style=\"text-align: initial;font-size: 1em\">.Statistics For Managers using Microsoft Excel by David M. Levine David F. Stephan Timothy C. Krehbiel Mark L. Berenson, Published by PEARSON<\/em><\/li>\n<li><em>Business Statistics by Ken Black, Published by John Wiley &amp; Sons, Inc.<\/em><\/li>\n<\/ol>\n<hr class=\"before-footnotes\" \/><div class=\"footnotes\"><ol><li id=\"footnote-34-1\">null <a href=\"#return-footnote-34-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":3,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["sanjay-mishra"],"pb_section_license":""},"chapter-type":[],"contributor":[60],"license":[],"class_list":["post-34","chapter","type-chapter","status-publish","hentry","contributor-sanjay-mishra"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/34","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/34\/revisions"}],"predecessor-version":[{"id":56,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/34\/revisions\/56"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/34\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/media?parent=34"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapter-type?post=34"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/contributor?post=34"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/license?post=34"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}