{"id":376,"date":"2018-10-31T11:42:32","date_gmt":"2018-10-31T11:42:32","guid":{"rendered":"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=376"},"modified":"2018-10-31T12:18:54","modified_gmt":"2018-10-31T12:18:54","slug":"correlation-coefficient-of-determination-testing-for-significance","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/chapter\/correlation-coefficient-of-determination-testing-for-significance\/","title":{"rendered":"Correlation: Coefficient of Determination: Testing for Significance"},"content":{"raw":"<div id=\"page_3\">\r\n\r\n&nbsp;\r\n<p class=\"p11 ft14\"><strong>1. Learning Outcome:<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li class=\"p12 ft10\">After completing this module the students will be able to:<\/li>\r\n \t<li class=\"p13 ft10\"><span class=\"ft16\">Understand the meaning and usefulness of correlation analysis<\/span><\/li>\r\n \t<li class=\"p14 ft10\"><span class=\"ft16\">Understand different techniques of finding out correlation<\/span><\/li>\r\n \t<li class=\"p14 ft10\"><span class=\"ft16\">Find out the significance of correlation coefficient<\/span><\/li>\r\n \t<li class=\"p15 ft10\"><span class=\"ft16\">Calculate coefficient of correlation and its importance<\/span><\/li>\r\n \t<li class=\"p14 ft10\"><span class=\"ft16\">Develop an understanding about various types of correlation<\/span><\/li>\r\n \t<li class=\"p15 ft10\"><span class=\"ft16\">Understand the significance and limitations of every technique of correlation<\/span><\/li>\r\n \t<li class=\"p14 ft10\"><span class=\"ft16\">Evaluate the relationship between the variables<\/span><\/li>\r\n<\/ul>\r\n&nbsp;\r\n<p class=\"p16 ft14\"><strong><span class=\"ft2\">2.<\/span><span class=\"ft17\">Introduction to the Concept of Correlation and Its Applications:<\/span><\/strong><\/p>\r\n&nbsp;\r\n<p class=\"p17 ft18\" style=\"text-align: justify\">Most of the times, we come across the problems that comprise of two or more variables. When two variables appear to move in the same direction i.e. both variables are either increasing or decreasing; or in opposite direction i.e. one is increasing and another decreasing; both variables are said to be associated\/ correlated to each other. When the variations in both variables take place in the same direction (both are either increasing or decreasing), they are assumed positively correlated. If variations in both variables travel in opposite director (one increases and another decreases), both are said to be negatively correlated. For example, in a class when homework grades of any student increase, his final grades also increase. This means that there is a positive relationship between homework grades and final grades. Similarly, when the price of a branded washing machine is decreased, its market demand will shoot up. This signifies a negative relationship between the price and demand of washing machine.<\/p>\r\n<img class=\"aligncenter size-full wp-image-380\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-189.png\" alt=\"\" width=\"396\" height=\"174\" \/>\r\n\r\n<\/div>\r\n<div id=\"page_4\">\r\n\r\n&nbsp;\r\n<p class=\"p18 ft20\" style=\"text-align: justify\">For instance, we assume two variables, X and Y. When we plot the values of X and Y on a graph and if \u2018Y\u2019 increases at a similar rate as \u2018X,\u2019 these two variables are said to be\u00a0<span class=\"ft19\">positively correlated<\/span>. The figure above on the left is a case of a positive correlation. Alternatively; if \u2018Y\u2019 decreases as \u2018X\u2019 increases then both the variables are said to be\u00a0<span class=\"ft19\">negatively correlated<\/span>. The figure above on the right provides an example of a negative correlation.<\/p>\r\n&nbsp;\r\n<p class=\"p19 ft10\">Some important definitions of correlation are mentioned as hereunder:<\/p>\r\n&nbsp;\r\n<p class=\"p20 ft10\"><span class=\"ft21\">Simpson and Kafka\u00a0<\/span>\u2013 correlation analysis deals with the association between two or more<\/p>\r\n<p class=\"p21 ft10\" style=\"text-align: justify\">variables.<\/p>\r\n&nbsp;\r\n<p class=\"p22 ft20\" style=\"text-align: justify\"><span class=\"ft22\">L.R. Conner<\/span>- If two or more quantities vary in sympathy so that movements in one tend to accompanied by corresponding movements in the other(s) the they tend are said to be correlated.<\/p>\r\n&nbsp;\r\n<p class=\"p23 ft10\" style=\"text-align: justify\"><span class=\"ft21\">Ya Lun Chou<\/span>- correlation analysis attempts to determine the \u2018degree of relationship\u2019 between<\/p>\r\n<p class=\"p21 ft10\">variables.<\/p>\r\n&nbsp;\r\n<p class=\"p24 ft24\" style=\"text-align: justify\"><span class=\"ft23\">A.M.Tuttle<\/span>- correlation is an analysis of the covariation between two or more variables. Thus, correlation may be considered as a technique that helps us in analyzing the covariation of two or more variables. To be more precise, it measures the extent of correspondence between the ordering of two random variables. It reflects the degree to which two variables share a common relationship.<\/p>\r\n&nbsp;\r\n<p class=\"p25 ft25\">The problem in analyzing the relationship\/association between the variables may be categorized in three stages as below:<\/p>\r\n\r\n<ul>\r\n \t<li class=\"p26 ft25\" style=\"text-align: justify\"><span class=\"ft26\">Firstly, we need to see whether the variables under study are related to each other or independent of each other;<\/span><\/li>\r\n \t<li class=\"p26 ft25\" style=\"text-align: justify\">Secondly, if any relationship is found than we move ahead to understand the nature and degree of this relationship;<\/li>\r\n \t<li class=\"p26 ft25\" style=\"text-align: justify\">Lastly, having calculated the degree of relationship, we may be interested in searching a<span style=\"text-align: initial;font-size: 1em\">cause-effect\u00a0(causal) relationship between the variables. That means variations in one variable cause variations in another variable.<\/span><\/li>\r\n \t<li class=\"p26 ft25\" style=\"text-align: justify\">It is important to mention that now a day\u2019s correlation is the most widely used technique in the problems pertaining to economics, business world, social science, biological problems, and psychology etc. It has become so important mainly because of the following reasons:<\/li>\r\n \t<li class=\"p26 ft25\" style=\"text-align: justify\">With the knowledge of degree of relationship between the variables under study, we can be very specific while appreciating this relationship. For example, we can easily appreciate the relationship between per capita income and per capita electricity consumption;<\/li>\r\n \t<li class=\"p26 ft25\" style=\"text-align: justify\">In business organizations, it is very significant concept as it enables the managers to estimate the costs, sales, price, and many other important variables with the help of other variables which are closely related to these variables.<\/li>\r\n \t<li class=\"p26 ft25\" style=\"text-align: justify\">If both variables show a specific and reliable relationship, we can predict the unknown value of one variable with the help of the given value of another variable. This is usually done with the help of simple regression analysis.Correlation and\u00a0Cause-effect\u00a0Relationship (Causal Relationship):If two variables are related to each other, it does not mean that there is essentially any cause- effect (causal) relationship. Causal relationship means that variations in one variable cause variations in another. In fact, two variables may be strongly correlated, but causal relationship is\u00a0non-existent.\u00a0Here, we need to understand that two variables may be correlated to each other because of following reasons:<\/li>\r\n \t<li class=\"p26 ft25\" style=\"text-align: justify\">In a small sample, two variables may show a strong relationship but in large population but no relationship is observed between both variables in large population. It means that both variables tend to be correlated only because of chance, actually they are not;<\/li>\r\n \t<li class=\"p26 ft25\" style=\"text-align: justify\">Two variables may be in relationship due to influence of one or more other variables. For example, in Madhya Pradesh, for last five years production of wheat and rice has been increasing. This shows a positive relationship between both variable. But actually, it is happening because of rainfall. Due to rainfall during that time period, production of rice and production of wheat is increasing. Thus, a third variable (rainfall) influences both the variables.<\/li>\r\n \t<li class=\"p26 ft25\" style=\"text-align: justify\">Sometimes, both the variables may be influencing each other so that it becomes difficult to say that which is the cause and which is the effect. For example, in first case, when a company increases its promotion expenditure, its sales also increase. Here promotion expenditure is cause and sale is effect. In second case, a decrease in company sales may force the company to cut its promotion expenditure, here sale is cause and promotion expenditure if effect. Therefore, it is very difficult to understand which variable is cause and which is effect as both influence each other.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"page_6\">\r\n\r\n&nbsp;\r\n<p class=\"p39 ft24\" style=\"text-align: justify\">From the above discussion, we may easily understand that existence of correlation does not indicate towards any\u00a0cause-effect\u00a0relation between the variables. But existence of a cause- effect relationship between two variables implies that both variables are necessarily correlated to each other.<\/p>\r\n&nbsp;\r\n<p class=\"p40 ft32\" style=\"text-align: justify\">Suppose, two variables are correlated to each other but have no causal relationship and one interprets the relationship as causal, such a correlation is described as spurious or\u00a0no-sensecorrelation. For example, salary of health sector employee and number of accidents in Delhi over a period 10 years tend to be positively correlated (r = 0.9) . But does that mean that these two variables are strongly correlated? Certainly not! Not even through the longest and most complex\u00a0cause-effect\u00a0chain. That is what spurious correlation is all about.\u00a0<span class=\"ft31\">Spurious correlation occurs between two variables that are supposed to be mutually independent<\/span>. Therefore, high correlation between the variables indicates only the mathematical result. One must arrive at the conclusion based on logical reasoning and intelligent investigation of significantly related variables.<\/p>\r\n&nbsp;\r\n<p class=\"p41 ft32\" style=\"text-align: justify\">It may be noted here that in simple correlation analysis we have only two variables and term \u2018dependent variable\u2019 and \u2018independent variable\u2019 refer to the mathematical or functional meaning of dependence; i.e. they do not imply that there is necessarily any cause and effect relationship between the variables. For instance, while estimating demand of a FMCG product from figures on sales promotion expenditures, demand is generally considered as the dependent variable. However, there may or may not be causal relationship between these two variables in the sense that changes in sales promotion cause changes in demand. In fact, in few cases, the\u00a0cause-effect\u00a0relationship may be just opposite what appears to be the obvious one.<\/p>\r\n&nbsp;\r\n<p class=\"p42 ft2\">Types of Correlation:<\/p>\r\n<p class=\"p43 ft10\">Some of the important types of correlation are as:<\/p>\r\n<p class=\"p44 ft25\" style=\"text-align: justify\"><span class=\"ft10\">(a)<\/span><span class=\"ft33\">Positive Correlation:\u00a0<\/span>when variations in two variables move in the same directions. That means both variables are either increasing or decreasing.<\/p>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div id=\"page_7\">\r\n<div id=\"p7dimg1\"><img class=\"aligncenter size-full wp-image-381\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-190.png\" alt=\"\" width=\"304\" height=\"188\" \/><\/div>\r\n&nbsp;\r\n<p class=\"p45 ft10\">In above graph, both variables are increasing that signifies a positive correlation.<\/p>\r\n&nbsp;\r\n<p class=\"p46 ft35\" style=\"text-align: justify\"><span class=\"ft10\">(b)<\/span><span class=\"ft34\">Negative Correlation:\u00a0<\/span>When variations in both variables travel in opposite direction. That means one is increasing and another is decreasing.<\/p>\r\n<img class=\"aligncenter size-full wp-image-382\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-191.png\" alt=\"\" width=\"309\" height=\"199\" \/>\r\n<p class=\"p47 ft10\">In the above graph, variable X increases and Y decreases. Thus, it is a negative correlation.<\/p>\r\n&nbsp;\r\n<p class=\"p30 ft20\" style=\"text-align: justify\"><span class=\"ft10\">(c)<\/span><span class=\"ft36\">Linear and\u00a0<\/span><span class=\"ft37\">Non-linear<\/span><span class=\"ft37\">\u00a0Correlation:\u00a0<\/span>When spread of change in one variable has a tendency to have a constant ratio with the spread of change in another variable, then it is said be a linear correlation.<\/p>\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-383\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-192.png\" alt=\"\" width=\"546\" height=\"164\" \/>\r\n<div id=\"page_8\">\r\n<div id=\"p8dimg1\"><\/div>\r\n<div id=\"id8_1\">\r\n<p class=\"p48 ft20\" style=\"text-align: justify\">In the above figure, it is easy to understand that changes in Y are proportional to the changes in X. Therefore, it is a perfect linear relationship. A linear correlation may be positive or negative.<\/p>\r\n<p class=\"p49 ft35\" style=\"text-align: justify\"><span class=\"ft10\">(d)<\/span><span class=\"ft34\">Non-linear<\/span><span class=\"ft38\">\u00a0Correlation:\u00a0<\/span>When the extent of variations in both the variables does not show a constant ratio, then it is said to be\u00a0non-linear\u00a0correlation.<\/p>\r\n&nbsp;\r\n\r\n<img id=\"p8img1\" class=\"aligncenter\" src=\"http:\/\/www.htmlpublish.com\/newTestDocStorage\/DocStorage\/c52abc008b984cababa1aaf7806e7c4a\/pdf-to-word_images\/pdf-to-word8x1.jpg\" \/>\r\n\r\n<\/div>\r\n<div id=\"id8_2\">\r\n<div id=\"id8_2_1\"><\/div>\r\n<div id=\"id8_2_2\">\r\n\r\n&nbsp;\r\n<p class=\"p53 ft40\"><span style=\"text-align: initial;font-size: 1em\">LLR Smoother<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"id8_3\">\r\n<p class=\"p18 ft20\" style=\"text-align: justify\">In the above graph, one can easily see that variations in study hours and variations in final score are not showing a constant ratio. Beyond a level, a little increase in study hours brings very high increase in final score.<\/p>\r\n<p class=\"p63 ft45\" style=\"text-align: justify\"><span class=\"ft10\">(e)<\/span><span class=\"ft44\">Simple, Multiple and Partial Correlation:\u00a0<\/span><span class=\"ft24\">In simple correlation with study only two variables simultaneously. Multiple correlations are used when we try to find out the relationship among more than two variables simultaneously. Partial correlation is used when we try to find out the relationship between two variables assuming the effect of other variables constant.<\/span><\/p>\r\n&nbsp;\r\n<p class=\"p64 ft2\"><span class=\"ft2\">3.<\/span><span class=\"ft46\">Methods of Correlation:<\/span><\/p>\r\n&nbsp;\r\n<p class=\"p65 ft24\" style=\"text-align: justify\">There are different methods of determining the association\/ relationship between variables, but none of them can inform us with certainty that a correlation is pinpointing of causal relationship. Therefore, we have to answer two types of questions in bivariate (only two variables) population viz.<\/p>\r\n&nbsp;\r\n<p class=\"p65 ft24\" style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">The first question is answered by the use of correlation technique and the second question by the technique of regression. In case of bivariate population (two variables), correlation can be studied through<\/span><\/p>\r\n&nbsp;\r\n<p class=\"p65 ft24\" style=\"text-align: justify\"><span class=\"ft10\" style=\"text-align: initial;font-size: 1em\">(a)<\/span><span class=\"ft47\" style=\"text-align: initial;font-size: 1em\">Scatter Diagramme;<\/span><\/p>\r\n<p class=\"p65 ft24\" style=\"text-align: justify\"><span class=\"ft10\" style=\"text-align: initial;font-size: 1em\">(b)<\/span><span class=\"ft48\" style=\"text-align: initial;font-size: 1em\">Karl Pearson\u2019s coefficient of correlation;<\/span><\/p>\r\n<p class=\"p65 ft24\" style=\"text-align: justify\"><span class=\"ft10\" style=\"text-align: initial;font-size: 1em\">(c)<\/span><span class=\"ft47\" style=\"text-align: initial;font-size: 1em\">Charles Spearman\u2019s coefficient of correlation;<\/span><\/p>\r\n&nbsp;\r\n<p class=\"p65 ft24\"><span style=\"text-align: initial;font-size: 1em\">Whereas cause and effect relationship can be studied through simple regression equations.\u00a0<\/span><span class=\"ft49\" style=\"text-align: initial;font-size: 1em\">Finding a correlation among more than two variables is beyond the scope of this module.<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"page_9\">\r\n<div id=\"id9_1\">\r\n\r\n&nbsp;\r\n<p class=\"p69 ft50\"><strong>Scatter Diagramme Method:<\/strong><\/p>\r\n&nbsp;\r\n<p class=\"p65 ft35\" style=\"text-align: justify\">Scatter Diagram is a graph of observed plotted points where each points represents the values of X &amp; Y as its coordinate. It gives us a superficial idea about the relationship between two variables graphically.<\/p>\r\n&nbsp;\r\n<p class=\"p72 ft52\"><img class=\"aligncenter size-full wp-image-384\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-193.png\" alt=\"\" width=\"542\" height=\"332\" \/><\/p>\r\n\r\n<\/div>\r\n<div id=\"id9_2\">\r\n\r\n&nbsp;\r\n<p class=\"p48 ft20\" style=\"text-align: justify\">Above figure depicts different scatter diagrammes. One can easily take an idea about the direction and strength of relationship. However, exact degree of relationship cannot be ascertained from the above diagrammes.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"page_10\">\r\n<div id=\"p10dimg1\"><\/div>\r\n&nbsp;\r\n<p class=\"p0 ft54\"><strong>Advantages<\/strong><span class=\"ft10\">:<\/span><\/p>\r\n\r\n<ul>\r\n \t<li class=\"p73 ft10\"><span class=\"ft55\">First step in investigating the relationship between two variables<\/span><\/li>\r\n \t<li class=\"p67 ft10\"><span class=\"ft56\">Simple &amp; non mathematical method<\/span><\/li>\r\n \t<li class=\"p73 ft10\"><span class=\"ft56\">Not influenced by the size of extreme item<\/span><\/li>\r\n<\/ul>\r\n&nbsp;\r\n<p class=\"p74 ft54\"><strong>Disadvantage:<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li class=\"p74 ft10\">This cannot measure the exact degree of relationship.<\/li>\r\n<\/ul>\r\n&nbsp;\r\n<p class=\"p42 ft2\"><span class=\"ft50\">Karl Pearson\u2019s Method<\/span>:<\/p>\r\n&nbsp;\r\n<p class=\"p34 ft20\" style=\"text-align: justify\">It is most commonly used method for finding out the relationship between the variables. It measures the direction as well as degree of relationship. The degree of correlation between two variables (measured on interval and ratio scale) can be measured through PERASON\u2019S CORRELATION COEFFICIENT (r).<\/p>\r\n&nbsp;\r\n<p class=\"p75 ft2\"><span class=\"ft57\">When deviation taken from actual mean:<\/span><\/p>\r\n<p class=\"p76 ft10\">r = \u03a3dxdy \/\u221a \u03a3dx\u00b2 \u03a3dy\u00b2<\/p>\r\n<p class=\"p77 ft54\">(also known as covariance method<span class=\"ft21\">)<\/span><\/p>\r\n<p class=\"p75 ft2\"><span class=\"ft57\">When deviation taken from an assumed mean:<\/span><\/p>\r\n<p class=\"p77 ft10\">r\u00a0<span class=\"ft2\">=\u00a0<\/span>N \u03a3dxdy \u2013 \u03a3dx \u03a3dy<\/p>\r\n<p class=\"p78 ft10\"><span class=\"ft10\">\u221a<\/span><span class=\"ft13\">[N \u03a3d\u00b2x - (\u03a3dx)\u00b2] [N \u03a3d\u00b2y - (\u03a3dy)\u00b2]<\/span><\/p>\r\n<p class=\"p75 ft2\"><span class=\"ft57\">Value of correlation coefficient for bivariate frequency data<\/span><\/p>\r\n<p class=\"p79 ft10\">r\u00a0<span class=\"ft2\">=\u00a0<\/span>N \u03a3fdxdy \u2013 \u03a3fdx \u03a3fdy<\/p>\r\n<p class=\"p80 ft10\"><span class=\"ft10\">\u221a<\/span><span class=\"ft13\">[N \u03a3fd\u00b2x - (\u03a3fdx)\u00b2] [N \u03a3fd\u00b2y - (\u03a3fdy)\u00b2]<\/span><\/p>\r\n<p class=\"p81 ft20\" style=\"text-align: justify\">The value of\u00a0<span class=\"ft19\">r\u00a0<\/span>always lies between\u00a0\u20131\u00a0and 1, i.e., \u22121 \u2264\u00a0<span class=\"ft19\">r\u00a0<\/span>\u22641. If\u00a0<span class=\"ft19\">both variables\u00a0<\/span>increase or decreases (same direction), we say that there is\u00a0<span class=\"ft19\">positive\u00a0<\/span>correlation between them. However, if\u00a0<span class=\"ft19\">one\u00a0<\/span>decreases when\u00a0<span class=\"ft19\">another\u00a0<\/span>increases (or\u00a0<span class=\"ft19\">vice versa<\/span>), then we say that both are\u00a0<span class=\"ft19\">negatively\u00a0<\/span>or\u00a0<span class=\"ft19\">inversely\u00a0<\/span>correlated.<\/p>\r\n<p class=\"p82 ft2\">Procedure for Calculating Pearson\u2019s coefficient of correlation:<\/p>\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div id=\"page_11\">\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p class=\"p83 ft10\"><span class=\"ft58\">\u2022\u00a0<\/span><span class=\"ft59\">Calculate the mean of the two series \u2018X\u2019 &amp;\u2019Y\u2019<\/span><\/p>\r\n<p class=\"p67 ft10\"><span class=\"ft58\">\u2022\u00a0<\/span><span class=\"ft59\">Calculate the deviations \u2018dx\u2019 &amp; \u2018dy\u2019 in two series from their respective mean.<\/span><\/p>\r\n<p class=\"p84 ft25\"><span class=\"ft58\">\u2022\u00a0<\/span><span class=\"ft60\">Square each deviation of \u2018dx\u2019 &amp; \u2019dy\u2019 then obtain the sum of the squared deviation i.e.\u2211dx<\/span><span class=\"ft61\">2\u00a0<\/span>&amp; \u2211dy<span class=\"ft61\">2<\/span><\/p>\r\n<p class=\"p85 ft10\"><span class=\"ft58\">\u2022\u00a0<\/span><span class=\"ft59\">Multiply each deviation under x with each deviation under y &amp; obtain the product of<\/span><\/p>\r\n<p class=\"p86 ft10\">\u2018dxdy\u2019. Then obtain the sum of the product of dx, dy i.e. \u2211dxdy<\/p>\r\n<p class=\"p67 ft10\"><span class=\"ft58\">\u2022\u00a0<\/span><span class=\"ft59\">Substitute the value in the formula.<\/span><\/p>\r\n<p class=\"p42 ft2\">Interpretation of Correlation Coefficient (r):<\/p>\r\n<p class=\"p87 ft20\" style=\"text-align: justify\">The extreme values of\u00a0<span class=\"ft19\">r<\/span>, i.e., when\u00a0<span class=\"ft19\">r\u00a0<\/span>=\u00a0<span class=\"ft62\">\u00b11<\/span>, shows that there is\u00a0<span class=\"ft19\">perfect\u00a0<\/span>(positive or negative) correlation between\u00a0<span class=\"ft19\">X\u00a0<\/span>and\u00a0<span class=\"ft19\">Y.\u00a0<\/span>The remaining values of r that lie in subintervals of\u00a0{\u20131,\u00a01}, explain the association\/ relationship in terms of its strength. One may use the following figure as a guideline as to what\u00a0<span class=\"ft19\">adjective\u00a0<\/span>must be used for the values of\u00a0<span class=\"ft19\">r\u00a0<\/span>obtained after calculation to describe the relationship.<\/p>\r\n<p class=\"p19 ft21\">Note<\/p>\r\n<p class=\"p88 ft20\" style=\"text-align: justify\">When\u00a0<span class=\"ft19\">r\u00a0<\/span>= 0, we cannot say that there is no correlation\u00a0<span class=\"ft19\">at all\u00a0<\/span>between\u00a0<span class=\"ft19\">X\u00a0<\/span>and\u00a0<span class=\"ft19\">Y<\/span>. Pearson\u2019s correlation coefficient is meant to measure\u00a0<span class=\"ft19\">linear\u00a0<\/span>relationship only. It should\u00a0<span class=\"ft19\">not\u00a0<\/span>be used in the case of\u00a0<span class=\"ft19\">non-linear<\/span>relationships since it will obviously lead to a wrong interpretation.<\/p>\r\n<p class=\"p101 ft10\"><\/p>\r\n\r\n<\/div>\r\n<div id=\"page_12\">\r\n<div id=\"p12dimg1\"><img id=\"p12img1\" src=\"http:\/\/www.htmlpublish.com\/newTestDocStorage\/DocStorage\/c52abc008b984cababa1aaf7806e7c4a\/pdf-to-word_images\/pdf-to-word12x1.jpg\" \/><img class=\"aligncenter size-full wp-image-385\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-194.png\" alt=\"\" width=\"422\" height=\"392\" \/><\/div>\r\n<div><img class=\"aligncenter size-full wp-image-386\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-195.png\" alt=\"\" width=\"406\" height=\"237\" \/><\/div>\r\n&nbsp;\r\n<p class=\"p108 ft2\">Advantages of Correlation Coefficient:<\/p>\r\n\r\n<ul>\r\n \t<li class=\"p109 ft10\"><span class=\"ft55\">It summarizes in one value, the degree of correlation &amp; direction of correlation also;<\/span><\/li>\r\n \t<li class=\"p109 ft10\">Since its value is unit less therefore, it is independent of the change of origin &amp; scale.<span style=\"text-align: initial;font-size: 1em\">Limitations of Correlation Coefficient:<\/span><\/li>\r\n \t<li class=\"p109 ft10\">Always assume linear relationship between variables<\/li>\r\n \t<li class=\"p109 ft10\">Interpreting the value of r is difficult. There are chances of wrong interpretation.<\/li>\r\n \t<li class=\"p109 ft10\">Value of Correlation Coefficient is affected by the extreme values<\/li>\r\n \t<li class=\"p109 ft10\">Time consuming methods<\/li>\r\n \t<li class=\"p109 ft10\">Usually, Coefficient of Determination (r<span class=\"ft61\" style=\"text-align: initial;font-size: 1em\">2<\/span><span style=\"text-align: initial;font-size: 1em\">) is used to interpret the value of coefficient of correlation (r). Coefficient of determination (r<\/span><span class=\"ft61\" style=\"text-align: initial;font-size: 1em\">2<\/span><span style=\"text-align: initial;font-size: 1em\">) measures common variance.<\/span><\/li>\r\n \t<li class=\"p109 ft10\">Charles Spearman\u2019s Method<\/li>\r\n<\/ul>\r\n&nbsp;\r\n<p class=\"p113 ft25\">This method is used when we are given with rank data (ordinal data). This method is based on the order (ranks) of the given observations.<\/p>\r\n&nbsp;\r\n<p class=\"p114 ft10\">Spearman\u2019s coefficient of correlation is represented by R and it is calculated as:<\/p>\r\n<p class=\"p115 ft2\">R = 1- (6 \u2211D<span class=\"ft71\">2<\/span>) \/ N (N<span class=\"ft71\">2\u00a0<\/span>\u2013 1)<\/p>\r\n<p class=\"p116 ft10\">Where<\/p>\r\n<p class=\"p117 ft10\">R = Rank correlation coefficient<\/p>\r\n<p class=\"p118 ft10\">D = Difference of rank between paired item in two series.<\/p>\r\n<p class=\"p118 ft10\">N = Total number of observation.<\/p>\r\n\r\n<\/div>\r\n<div id=\"page_13\">\r\n\r\n&nbsp;\r\n<p class=\"p119 ft2\">Interpretation of Rank Correlation Coefficient (R)<\/p>\r\n<p class=\"p120 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">The value of rank correlation coefficient, R ranges from\u00a0<\/span>-1\u00a0to +1<\/p>\r\n<p class=\"p121 ft25\"><span class=\"ft72\">\u2022<\/span><span class=\"ft74\">If R = +1, then there is complete agreement in the order of the ranks and the ranks are in the same direction<\/span><\/p>\r\n<p class=\"p122 ft25\"><span class=\"ft72\">\u2022<\/span><span class=\"ft74\">If R =\u00a0<\/span>-1,\u00a0then there is complete agreement in the order of the ranks and the ranks are in the opposite direction<\/p>\r\n<p class=\"p123 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">If R = 0, then there is no correlation<\/span><\/p>\r\n<p class=\"p124 ft2\">Problems in Rank Correlation Coefficient:<\/p>\r\n<p class=\"p125 ft20\"><span class=\"ft15\">\uf0d8<\/span><span class=\"ft75\">Problems where Ranks are not given:\u00a0<\/span>If the ranks are not given, then we need to assign ranks to the data series. The ranking can be done in ascending or descending order. We need to follow the same scheme of ranking for the other series.<\/p>\r\n<p class=\"p126 ft10\"><span class=\"ft15\">\uf0d8<\/span><span class=\"ft76\">Equal Ranks or tie in Ranks:\u00a0<\/span>In such cases average ranks should be assigned to each individual.<\/p>\r\n<p class=\"p127 ft2\">R = 1- (6 \u2211D<span class=\"ft71\">2<\/span>) + AF \/ N (N<span class=\"ft71\">2\u00a0<\/span>\u2013 1)<\/p>\r\n<p class=\"p128 ft10\">Where<\/p>\r\n<p class=\"p129 ft10\">AF = 1\/12(m<span class=\"ft77\">1<\/span><span class=\"ft78\">3\u00a0<\/span>\u2013 m<span class=\"ft77\">1<\/span>) + 1\/12(m<span class=\"ft77\">2<\/span><span class=\"ft78\">3\u00a0<\/span>\u2013 m<span class=\"ft77\">2<\/span>) +\u2026. 1\/12(m<span class=\"ft77\">2<\/span><span class=\"ft78\">3\u00a0<\/span>\u2013 m<span class=\"ft77\">2<\/span>)<\/p>\r\n<p class=\"p130 ft10\">m = Number of time an item is repeated<\/p>\r\n<p class=\"p131 ft2\">Merits\/ Demerits of Spearman\u2019s Rank Correlation:<\/p>\r\n&nbsp;\r\n<p class=\"p124 ft2\">Merits:<\/p>\r\n<p class=\"p132 ft35\"><span class=\"ft72\">\u2022<\/span><span class=\"ft79\">This method is simpler to understand and easier to apply compared to karl Pearson\u2019s correlation method.<\/span><\/p>\r\n<p class=\"p133 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">This method is useful where we can give the ranks and not the actual data. (qualitative term)<\/span><\/p>\r\n<p class=\"p115 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">This method is to use where the initial data in the form of ranks.<\/span><\/p>\r\n<p class=\"p134 ft2\">Demerits:<\/p>\r\n<p class=\"p130 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">Cannot be used for finding out correlation in a grouped frequency distribution.<\/span><\/p>\r\n<p class=\"p130 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">This method should be applied where N exceeds 30.<\/span><\/p>\r\n\r\n<\/div>\r\n<div id=\"page_14\">\r\n\r\n&nbsp;\r\n<p class=\"p0 ft2\"><strong><span class=\"ft2\">4.<\/span><span class=\"ft46\">Finding out Significance of Pearson\u2019s Correlation Coefficient (r):<\/span><\/strong><\/p>\r\n&nbsp;\r\n<p class=\"p135 ft20\" style=\"text-align: justify\">As we know that correlation coefficient (r) is usually calculated to have an idea about the extent of relationship between two variables. Most of time, we consider a sample from a large population to calculate correlation coefficient (r) that is further used as a point estimate of population correlation coefficient (\u01bf). It means \u2018\u01bf\u2019 is estimated by \u2018r\u2019.<\/p>\r\n&nbsp;\r\n<p class=\"p136 ft10\">Therefore, we can say that \u2018r\u2019 is used as a population parameter (\u01bf).<\/p>\r\n&nbsp;\r\n<p class=\"p137 ft25\" style=\"text-align: justify\">Here, one can take note that \u2018r\u2019 may be considered as an estimate of \u2018\u01bf\u2019 if the assumption of normal distribution of both the variables holds true.<\/p>\r\n&nbsp;\r\n<p class=\"p138 ft25\" style=\"text-align: justify\">The most widely used test to investigate whether both variables X and Y are correlated to each other or not, is the\u00a0t-test.\u00a0For using\u00a0t-test,\u00a0we take our hypothesis as follows:<\/p>\r\n&nbsp;\r\n<p class=\"p139 ft10\">H<span class=\"ft77\">0<\/span>: \u01bf = 0<\/p>\r\n<p class=\"p140 ft10\">H<span class=\"ft77\">a<\/span>: \u01bf \u2260 0<\/p>\r\n<p class=\"p141 ft20\">Null hypothesis (H<span class=\"ft80\">0<\/span>) assumes that both the variables X and Y are not correlated in the entire population. Alternative hypothesis (H<span class=\"ft80\">a<\/span>) is just opposite to null hypothesis that assumes both variables are strongly correlated in population.<\/p>\r\n<p class=\"p142 ft10\">Now we can calculate the\u00a0t-value\u00a0using the following formula:<\/p>\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-387\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-196.png\" alt=\"\" width=\"168\" height=\"57\" \/>\r\n<div id=\"page_14\">\r\n\r\n&nbsp;\r\n<p class=\"p11 ft10\">Where<\/p>\r\n<p class=\"p145 ft10\">r = Pearson\u2019s coefficient of correlation<\/p>\r\n<p class=\"p145 ft10\"><span class=\"ft10\">\u01bf<\/span><span class=\"ft13\">= Population correlation<\/span><\/p>\r\n<p class=\"p146 ft25\">n = number of observations in sample\u00a0(n-2)\u00a0= degree of freedom<\/p>\r\n&nbsp;\r\n<p class=\"p147 ft25\">Here, we have assumed population correlation (\u01bf) equals to zero. We can get the value of t after putting the values of r and n. This is said to be calculated value of t.<\/p>\r\n&nbsp;\r\n<p class=\"p148 ft25\">Now, we see the table value of t for level of significance (\u03b1) = 0.05 and degree of freedom =\u00a0(n-2).<\/p>\r\n<p class=\"p149 ft20\"><span class=\"ft10\">(a)\u00a0<\/span><span class=\"ft87\">If\u00a0<\/span><span class=\"ft88\">t<\/span><span class=\"ft80\">cal\u00a0<\/span>&gt;\u00a0<span class=\"ft88\">t<\/span><span class=\"ft80\">table value\u00a0<\/span>; we will reject H<span class=\"ft80\">0<\/span>; it means r is significantly different from zero and both variables are strongly correlated in population.<\/p>\r\n<p class=\"p149 ft20\"><span class=\"ft10\" style=\"text-align: initial;font-size: 1em\">(b)\u00a0<\/span><span class=\"ft89\" style=\"text-align: initial;font-size: 1em\">If\u00a0<\/span><span class=\"ft88\" style=\"text-align: initial;font-size: 1em\">t<\/span><span class=\"ft80\" style=\"text-align: initial;font-size: 1em\">cal\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">&lt;\u00a0<\/span><span class=\"ft88\" style=\"text-align: initial;font-size: 1em\">t<\/span><span class=\"ft80\" style=\"text-align: initial;font-size: 1em\">table value\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">; we will accept H<\/span><span class=\"ft80\" style=\"text-align: initial;font-size: 1em\">0<\/span><span style=\"text-align: initial;font-size: 1em\">; it means r is almost equal to zero and both variables are not correlated in population.<\/span><\/p>\r\n&nbsp;\r\n<p class=\"p149 ft20\"><span style=\"text-align: initial;font-size: 1em\">Standard Error of Correlation Coefficient (r):<\/span><\/p>\r\n&nbsp;\r\n<p class=\"p149 ft20\"><span style=\"text-align: initial;font-size: 1em\">We can also calculate the standard error of r using following formula:<\/span><\/p>\r\n&nbsp;\r\n<p class=\"p149 ft20\"><span style=\"text-align: initial;font-size: 1em\">(1\u2212\ufffd\u00b2)<\/span><\/p>\r\n<p class=\"p149 ft20\"><span style=\"text-align: initial;font-size: 1em\">S.E.=\u00a0<\/span><span class=\"ft91\" style=\"text-align: initial;font-size: 1em\">\u221a\ufffd<\/span><\/p>\r\n\r\n<\/div>\r\n<div id=\"page_15\">\r\n<p class=\"p154 ft10\">Where<\/p>\r\n<p class=\"p155 ft10\">r = Correlation coefficient<\/p>\r\n<p class=\"p156 ft10\">n = Number of observations in sample<\/p>\r\n<p class=\"p157 ft2\">Probable Error of Correlation Coefficient (r):<\/p>\r\n<p class=\"p158 ft25\" style=\"text-align: justify\">We can also calculate the range of degree of correlation for the entire population with the help of \u2018probable error\u2019.<\/p>\r\n<p class=\"p161 ft99\"><span class=\"ft96\">\u01bf<\/span><span class=\"ft97\">\u00b1\u00a0<\/span>P.E.<span class=\"ft98\">r<\/span><\/p>\r\n<p class=\"p162 ft10\">Lower limit of population correlation coefficient =\u00a0<span class=\"ft100\">\u01bf -\u00a0<\/span><span class=\"ft99\">P.E.<\/span><span class=\"ft98\">r<\/span><\/p>\r\n<p class=\"p163 ft10\">Upper limit of population correlation coefficient =\u00a0<span class=\"ft100\">\u01bf +\u00a0<\/span><span class=\"ft99\">P.E.<\/span><span class=\"ft98\">r<\/span><\/p>\r\n<p class=\"p164 ft2\">5. Coefficient of Determination (r\u00b2):<\/p>\r\n<p class=\"p34 ft20\" style=\"text-align: justify\">The most widely used and convenient way to interpret the value of correlation coefficient between two variables is the coefficient of determination. The square of r is called as Coefficient of Determination (r<span class=\"ft101\">2<\/span>) that measures the common variations in both the variables. This is very useful measure of linear covariation of two variables.<\/p>\r\n<p class=\"p33 ft103\">Coefficient of Determination (r<span class=\"ft77\">2<\/span>) =\u00a0<span class=\"ft102\">\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd<\/span><\/p>\r\n<p class=\"p33 ft103\">\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd<\/p>\r\n\r\n<\/div>\r\n<div id=\"page_16\">\r\n\r\n&nbsp;\r\n<p class=\"p166 ft35\">For example, if value of r = 0.50, then r<span class=\"ft105\">2\u00a0<\/span>= 0.25 i.e r<span class=\"ft105\">2\u00a0<\/span>= 25\/ 100<\/p>\r\n&nbsp;\r\n<p class=\"p167 ft20\" style=\"text-align: justify\">It may be interpreted as out of total variations (100 percent), 25 percent variations in dependent variable may be explained by variations in independent variable. The value of r<span class=\"ft106\">2\u00a0<\/span>is ranging from 0 to 1.<\/p>\r\n&nbsp;\r\n<p class=\"p168 ft20\" style=\"text-align: justify\">Coefficient of determination is widely used in regression analysis to assess the goodness of fit of the regression model. The greater the value of r<span class=\"ft106\">2\u00a0<\/span>the better is the fit and the more useful the regression equation as a predictive instrument.<\/p>\r\n&nbsp;\r\n<p class=\"p33 ft2\"><strong>6. Limitations of Correlation analysis:<\/strong><\/p>\r\n&nbsp;\r\n<p class=\"p34 ft20\" style=\"text-align: justify\">From the discussion made so far, one can easily understand that correlation analysis is a statistical tool that is used to find out the association\/ relationship between the variables. It must be used very carefully to avoid misleading conclusions\/interpretations. The most common mistakes that are usually made by us are as hereunder:<\/p>\r\n\r\n<ul>\r\n \t<li class=\"p169 ft20\" style=\"text-align: justify\"><span class=\"ft27\">Correlation coefficient (r) gives us an idea about the\u00a0<\/span><span class=\"ft19\">linear relationship\u00a0<\/span>between the variables. As the value of r increase from 0 to 1, it means linear association between the variables also increases. A value of r = 0 does not show the absence of relationship. In this case, both the variables are not linearly related to one another but they may be associated in any other manner.<\/li>\r\n \t<li class=\"p169 ft20\" style=\"text-align: justify\">For the value of r = 0.8 and r = 0.4; we cannot say that in first case degree of relationship between the variables is just double\/two times as compared to second case. It means correlation coefficient does not follow the principle of proportionality.<\/li>\r\n \t<li class=\"p169 ft20\" style=\"text-align: justify\">A value of r = 0.6 does not mean that correlation explains 60% variations of total variations. Rather, r = 0.6 means that correlation explains only 36% variations in both the variables (as r\u00b2 = 0.36).<\/li>\r\n \t<li class=\"p169 ft20\" style=\"text-align: justify\">One may consider presence of strong correlation between the variables as\u00a0<span style=\"text-align: initial;font-size: 1em\">cause-effectrelationship (causation). Actually, correlation speaks nothing about the causation. In case of two variables, causation may be established only with the help of simple regression analysis.<\/span><\/li>\r\n \t<li class=\"p169 ft20\">A very common mistake in interpretation of correlation coefficient takes place when we conclude a strong relationship between the variables but actually they are not related. For example, sales of Hero bikes in New Delhi and number of accidents in Mumbai. Both the variables seem to be correlated as they show similar movements but actually, it is not possible to link them.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"page_17\">\r\n\r\n&nbsp;\r\n<p class=\"p172 ft2\"><strong><span class=\"ft2\">7.<\/span><span class=\"ft108\">Summary:<\/span><\/strong><\/p>\r\n<p class=\"p173 ft18\" style=\"text-align: justify\">Correlation analysis is a very good technique to find out the association or relationship between the two or more variables. In simple correlation we consider only two variables. In multiple correlations we try to find out the correlation among more than two variables. In partial correlation, we try to find out the correlation between two variables by partial out the effect of other variables. Pearson\u2019s coefficient of correlation is most widely used technique for finding out the correlation between the variables (recorded using interval\/ ratio scale). This assumes a linear relationship among the variables and normal distribution of variables. The calculation of correlation coefficient is little complex and is affected by the extreme values. For the rank data, Spearman\u2019s correlation coefficient is used. It is distribution free coefficient. Significance of Pearson\u2019s coefficient of correlation can be found out using\u00a0t-test.\u00a0Coefficient of determination (r\u00b2) is widely used in determining the common variations between the related variables. Existence of correlation between the variable does not signify any\u00a0cause-effect\u00a0relationship. Causation may be established by simple regression analysis.<\/p>\r\n\r\n<\/div>","rendered":"<div id=\"page_3\">\n<p>&nbsp;<\/p>\n<p class=\"p11 ft14\"><strong>1. Learning Outcome:<\/strong><\/p>\n<ul>\n<li class=\"p12 ft10\">After completing this module the students will be able to:<\/li>\n<li class=\"p13 ft10\"><span class=\"ft16\">Understand the meaning and usefulness of correlation analysis<\/span><\/li>\n<li class=\"p14 ft10\"><span class=\"ft16\">Understand different techniques of finding out correlation<\/span><\/li>\n<li class=\"p14 ft10\"><span class=\"ft16\">Find out the significance of correlation coefficient<\/span><\/li>\n<li class=\"p15 ft10\"><span class=\"ft16\">Calculate coefficient of correlation and its importance<\/span><\/li>\n<li class=\"p14 ft10\"><span class=\"ft16\">Develop an understanding about various types of correlation<\/span><\/li>\n<li class=\"p15 ft10\"><span class=\"ft16\">Understand the significance and limitations of every technique of correlation<\/span><\/li>\n<li class=\"p14 ft10\"><span class=\"ft16\">Evaluate the relationship between the variables<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p class=\"p16 ft14\"><strong><span class=\"ft2\">2.<\/span><span class=\"ft17\">Introduction to the Concept of Correlation and Its Applications:<\/span><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p class=\"p17 ft18\" style=\"text-align: justify\">Most of the times, we come across the problems that comprise of two or more variables. When two variables appear to move in the same direction i.e. both variables are either increasing or decreasing; or in opposite direction i.e. one is increasing and another decreasing; both variables are said to be associated\/ correlated to each other. When the variations in both variables take place in the same direction (both are either increasing or decreasing), they are assumed positively correlated. If variations in both variables travel in opposite director (one increases and another decreases), both are said to be negatively correlated. For example, in a class when homework grades of any student increase, his final grades also increase. This means that there is a positive relationship between homework grades and final grades. Similarly, when the price of a branded washing machine is decreased, its market demand will shoot up. This signifies a negative relationship between the price and demand of washing machine.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-380\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-189.png\" alt=\"\" width=\"396\" height=\"174\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-189.png 396w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-189-300x132.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-189-65x29.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-189-225x99.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-189-350x154.png 350w\" sizes=\"auto, (max-width: 396px) 100vw, 396px\" \/><\/p>\n<\/div>\n<div id=\"page_4\">\n<p>&nbsp;<\/p>\n<p class=\"p18 ft20\" style=\"text-align: justify\">For instance, we assume two variables, X and Y. When we plot the values of X and Y on a graph and if \u2018Y\u2019 increases at a similar rate as \u2018X,\u2019 these two variables are said to be\u00a0<span class=\"ft19\">positively correlated<\/span>. The figure above on the left is a case of a positive correlation. Alternatively; if \u2018Y\u2019 decreases as \u2018X\u2019 increases then both the variables are said to be\u00a0<span class=\"ft19\">negatively correlated<\/span>. The figure above on the right provides an example of a negative correlation.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p19 ft10\">Some important definitions of correlation are mentioned as hereunder:<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p20 ft10\"><span class=\"ft21\">Simpson and Kafka\u00a0<\/span>\u2013 correlation analysis deals with the association between two or more<\/p>\n<p class=\"p21 ft10\" style=\"text-align: justify\">variables.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p22 ft20\" style=\"text-align: justify\"><span class=\"ft22\">L.R. Conner<\/span>&#8211; If two or more quantities vary in sympathy so that movements in one tend to accompanied by corresponding movements in the other(s) the they tend are said to be correlated.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p23 ft10\" style=\"text-align: justify\"><span class=\"ft21\">Ya Lun Chou<\/span>&#8211; correlation analysis attempts to determine the \u2018degree of relationship\u2019 between<\/p>\n<p class=\"p21 ft10\">variables.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p24 ft24\" style=\"text-align: justify\"><span class=\"ft23\">A.M.Tuttle<\/span>&#8211; correlation is an analysis of the covariation between two or more variables. Thus, correlation may be considered as a technique that helps us in analyzing the covariation of two or more variables. To be more precise, it measures the extent of correspondence between the ordering of two random variables. It reflects the degree to which two variables share a common relationship.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p25 ft25\">The problem in analyzing the relationship\/association between the variables may be categorized in three stages as below:<\/p>\n<ul>\n<li class=\"p26 ft25\" style=\"text-align: justify\"><span class=\"ft26\">Firstly, we need to see whether the variables under study are related to each other or independent of each other;<\/span><\/li>\n<li class=\"p26 ft25\" style=\"text-align: justify\">Secondly, if any relationship is found than we move ahead to understand the nature and degree of this relationship;<\/li>\n<li class=\"p26 ft25\" style=\"text-align: justify\">Lastly, having calculated the degree of relationship, we may be interested in searching a<span style=\"text-align: initial;font-size: 1em\">cause-effect\u00a0(causal) relationship between the variables. That means variations in one variable cause variations in another variable.<\/span><\/li>\n<li class=\"p26 ft25\" style=\"text-align: justify\">It is important to mention that now a day\u2019s correlation is the most widely used technique in the problems pertaining to economics, business world, social science, biological problems, and psychology etc. It has become so important mainly because of the following reasons:<\/li>\n<li class=\"p26 ft25\" style=\"text-align: justify\">With the knowledge of degree of relationship between the variables under study, we can be very specific while appreciating this relationship. For example, we can easily appreciate the relationship between per capita income and per capita electricity consumption;<\/li>\n<li class=\"p26 ft25\" style=\"text-align: justify\">In business organizations, it is very significant concept as it enables the managers to estimate the costs, sales, price, and many other important variables with the help of other variables which are closely related to these variables.<\/li>\n<li class=\"p26 ft25\" style=\"text-align: justify\">If both variables show a specific and reliable relationship, we can predict the unknown value of one variable with the help of the given value of another variable. This is usually done with the help of simple regression analysis.Correlation and\u00a0Cause-effect\u00a0Relationship (Causal Relationship):If two variables are related to each other, it does not mean that there is essentially any cause- effect (causal) relationship. Causal relationship means that variations in one variable cause variations in another. In fact, two variables may be strongly correlated, but causal relationship is\u00a0non-existent.\u00a0Here, we need to understand that two variables may be correlated to each other because of following reasons:<\/li>\n<li class=\"p26 ft25\" style=\"text-align: justify\">In a small sample, two variables may show a strong relationship but in large population but no relationship is observed between both variables in large population. It means that both variables tend to be correlated only because of chance, actually they are not;<\/li>\n<li class=\"p26 ft25\" style=\"text-align: justify\">Two variables may be in relationship due to influence of one or more other variables. For example, in Madhya Pradesh, for last five years production of wheat and rice has been increasing. This shows a positive relationship between both variable. But actually, it is happening because of rainfall. Due to rainfall during that time period, production of rice and production of wheat is increasing. Thus, a third variable (rainfall) influences both the variables.<\/li>\n<li class=\"p26 ft25\" style=\"text-align: justify\">Sometimes, both the variables may be influencing each other so that it becomes difficult to say that which is the cause and which is the effect. For example, in first case, when a company increases its promotion expenditure, its sales also increase. Here promotion expenditure is cause and sale is effect. In second case, a decrease in company sales may force the company to cut its promotion expenditure, here sale is cause and promotion expenditure if effect. Therefore, it is very difficult to understand which variable is cause and which is effect as both influence each other.<\/li>\n<\/ul>\n<\/div>\n<div id=\"page_6\">\n<p>&nbsp;<\/p>\n<p class=\"p39 ft24\" style=\"text-align: justify\">From the above discussion, we may easily understand that existence of correlation does not indicate towards any\u00a0cause-effect\u00a0relation between the variables. But existence of a cause- effect relationship between two variables implies that both variables are necessarily correlated to each other.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p40 ft32\" style=\"text-align: justify\">Suppose, two variables are correlated to each other but have no causal relationship and one interprets the relationship as causal, such a correlation is described as spurious or\u00a0no-sensecorrelation. For example, salary of health sector employee and number of accidents in Delhi over a period 10 years tend to be positively correlated (r = 0.9) . But does that mean that these two variables are strongly correlated? Certainly not! Not even through the longest and most complex\u00a0cause-effect\u00a0chain. That is what spurious correlation is all about.\u00a0<span class=\"ft31\">Spurious correlation occurs between two variables that are supposed to be mutually independent<\/span>. Therefore, high correlation between the variables indicates only the mathematical result. One must arrive at the conclusion based on logical reasoning and intelligent investigation of significantly related variables.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p41 ft32\" style=\"text-align: justify\">It may be noted here that in simple correlation analysis we have only two variables and term \u2018dependent variable\u2019 and \u2018independent variable\u2019 refer to the mathematical or functional meaning of dependence; i.e. they do not imply that there is necessarily any cause and effect relationship between the variables. For instance, while estimating demand of a FMCG product from figures on sales promotion expenditures, demand is generally considered as the dependent variable. However, there may or may not be causal relationship between these two variables in the sense that changes in sales promotion cause changes in demand. In fact, in few cases, the\u00a0cause-effect\u00a0relationship may be just opposite what appears to be the obvious one.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p42 ft2\">Types of Correlation:<\/p>\n<p class=\"p43 ft10\">Some of the important types of correlation are as:<\/p>\n<p class=\"p44 ft25\" style=\"text-align: justify\"><span class=\"ft10\">(a)<\/span><span class=\"ft33\">Positive Correlation:\u00a0<\/span>when variations in two variables move in the same directions. That means both variables are either increasing or decreasing.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"page_7\">\n<div id=\"p7dimg1\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-381\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-190.png\" alt=\"\" width=\"304\" height=\"188\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-190.png 304w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-190-300x186.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-190-65x40.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-190-225x139.png 225w\" sizes=\"auto, (max-width: 304px) 100vw, 304px\" \/><\/div>\n<p>&nbsp;<\/p>\n<p class=\"p45 ft10\">In above graph, both variables are increasing that signifies a positive correlation.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p46 ft35\" style=\"text-align: justify\"><span class=\"ft10\">(b)<\/span><span class=\"ft34\">Negative Correlation:\u00a0<\/span>When variations in both variables travel in opposite direction. That means one is increasing and another is decreasing.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-382\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-191.png\" alt=\"\" width=\"309\" height=\"199\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-191.png 309w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-191-300x193.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-191-65x42.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-191-225x145.png 225w\" sizes=\"auto, (max-width: 309px) 100vw, 309px\" \/><\/p>\n<p class=\"p47 ft10\">In the above graph, variable X increases and Y decreases. Thus, it is a negative correlation.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p30 ft20\" style=\"text-align: justify\"><span class=\"ft10\">(c)<\/span><span class=\"ft36\">Linear and\u00a0<\/span><span class=\"ft37\">Non-linear<\/span><span class=\"ft37\">\u00a0Correlation:\u00a0<\/span>When spread of change in one variable has a tendency to have a constant ratio with the spread of change in another variable, then it is said be a linear correlation.<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-383\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-192.png\" alt=\"\" width=\"546\" height=\"164\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-192.png 546w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-192-300x90.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-192-65x20.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-192-225x68.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-192-350x105.png 350w\" sizes=\"auto, (max-width: 546px) 100vw, 546px\" \/><\/p>\n<div id=\"page_8\">\n<div id=\"p8dimg1\"><\/div>\n<div id=\"id8_1\">\n<p class=\"p48 ft20\" style=\"text-align: justify\">In the above figure, it is easy to understand that changes in Y are proportional to the changes in X. Therefore, it is a perfect linear relationship. A linear correlation may be positive or negative.<\/p>\n<p class=\"p49 ft35\" style=\"text-align: justify\"><span class=\"ft10\">(d)<\/span><span class=\"ft34\">Non-linear<\/span><span class=\"ft38\">\u00a0Correlation:\u00a0<\/span>When the extent of variations in both the variables does not show a constant ratio, then it is said to be\u00a0non-linear\u00a0correlation.<\/p>\n<p>&nbsp;<\/p>\n<p><img decoding=\"async\" id=\"p8img1\" class=\"aligncenter\" src=\"http:\/\/www.htmlpublish.com\/newTestDocStorage\/DocStorage\/c52abc008b984cababa1aaf7806e7c4a\/pdf-to-word_images\/pdf-to-word8x1.jpg\" alt=\"image\" \/><\/p>\n<\/div>\n<div id=\"id8_2\">\n<div id=\"id8_2_1\"><\/div>\n<div id=\"id8_2_2\">\n<p>&nbsp;<\/p>\n<p class=\"p53 ft40\"><span style=\"text-align: initial;font-size: 1em\">LLR Smoother<\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"id8_3\">\n<p class=\"p18 ft20\" style=\"text-align: justify\">In the above graph, one can easily see that variations in study hours and variations in final score are not showing a constant ratio. Beyond a level, a little increase in study hours brings very high increase in final score.<\/p>\n<p class=\"p63 ft45\" style=\"text-align: justify\"><span class=\"ft10\">(e)<\/span><span class=\"ft44\">Simple, Multiple and Partial Correlation:\u00a0<\/span><span class=\"ft24\">In simple correlation with study only two variables simultaneously. Multiple correlations are used when we try to find out the relationship among more than two variables simultaneously. Partial correlation is used when we try to find out the relationship between two variables assuming the effect of other variables constant.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"p64 ft2\"><span class=\"ft2\">3.<\/span><span class=\"ft46\">Methods of Correlation:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"p65 ft24\" style=\"text-align: justify\">There are different methods of determining the association\/ relationship between variables, but none of them can inform us with certainty that a correlation is pinpointing of causal relationship. Therefore, we have to answer two types of questions in bivariate (only two variables) population viz.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p65 ft24\" style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">The first question is answered by the use of correlation technique and the second question by the technique of regression. In case of bivariate population (two variables), correlation can be studied through<\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"p65 ft24\" style=\"text-align: justify\"><span class=\"ft10\" style=\"text-align: initial;font-size: 1em\">(a)<\/span><span class=\"ft47\" style=\"text-align: initial;font-size: 1em\">Scatter Diagramme;<\/span><\/p>\n<p class=\"p65 ft24\" style=\"text-align: justify\"><span class=\"ft10\" style=\"text-align: initial;font-size: 1em\">(b)<\/span><span class=\"ft48\" style=\"text-align: initial;font-size: 1em\">Karl Pearson\u2019s coefficient of correlation;<\/span><\/p>\n<p class=\"p65 ft24\" style=\"text-align: justify\"><span class=\"ft10\" style=\"text-align: initial;font-size: 1em\">(c)<\/span><span class=\"ft47\" style=\"text-align: initial;font-size: 1em\">Charles Spearman\u2019s coefficient of correlation;<\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"p65 ft24\"><span style=\"text-align: initial;font-size: 1em\">Whereas cause and effect relationship can be studied through simple regression equations.\u00a0<\/span><span class=\"ft49\" style=\"text-align: initial;font-size: 1em\">Finding a correlation among more than two variables is beyond the scope of this module.<\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"page_9\">\n<div id=\"id9_1\">\n<p>&nbsp;<\/p>\n<p class=\"p69 ft50\"><strong>Scatter Diagramme Method:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p class=\"p65 ft35\" style=\"text-align: justify\">Scatter Diagram is a graph of observed plotted points where each points represents the values of X &amp; Y as its coordinate. It gives us a superficial idea about the relationship between two variables graphically.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p72 ft52\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-384\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-193.png\" alt=\"\" width=\"542\" height=\"332\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-193.png 542w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-193-300x184.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-193-65x40.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-193-225x138.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-193-350x214.png 350w\" sizes=\"auto, (max-width: 542px) 100vw, 542px\" \/><\/p>\n<\/div>\n<div id=\"id9_2\">\n<p>&nbsp;<\/p>\n<p class=\"p48 ft20\" style=\"text-align: justify\">Above figure depicts different scatter diagrammes. One can easily take an idea about the direction and strength of relationship. However, exact degree of relationship cannot be ascertained from the above diagrammes.<\/p>\n<\/div>\n<\/div>\n<div id=\"page_10\">\n<div id=\"p10dimg1\"><\/div>\n<p>&nbsp;<\/p>\n<p class=\"p0 ft54\"><strong>Advantages<\/strong><span class=\"ft10\">:<\/span><\/p>\n<ul>\n<li class=\"p73 ft10\"><span class=\"ft55\">First step in investigating the relationship between two variables<\/span><\/li>\n<li class=\"p67 ft10\"><span class=\"ft56\">Simple &amp; non mathematical method<\/span><\/li>\n<li class=\"p73 ft10\"><span class=\"ft56\">Not influenced by the size of extreme item<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p class=\"p74 ft54\"><strong>Disadvantage:<\/strong><\/p>\n<ul>\n<li class=\"p74 ft10\">This cannot measure the exact degree of relationship.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p class=\"p42 ft2\"><span class=\"ft50\">Karl Pearson\u2019s Method<\/span>:<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p34 ft20\" style=\"text-align: justify\">It is most commonly used method for finding out the relationship between the variables. It measures the direction as well as degree of relationship. The degree of correlation between two variables (measured on interval and ratio scale) can be measured through PERASON\u2019S CORRELATION COEFFICIENT (r).<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p75 ft2\"><span class=\"ft57\">When deviation taken from actual mean:<\/span><\/p>\n<p class=\"p76 ft10\">r = \u03a3dxdy \/\u221a \u03a3dx\u00b2 \u03a3dy\u00b2<\/p>\n<p class=\"p77 ft54\">(also known as covariance method<span class=\"ft21\">)<\/span><\/p>\n<p class=\"p75 ft2\"><span class=\"ft57\">When deviation taken from an assumed mean:<\/span><\/p>\n<p class=\"p77 ft10\">r\u00a0<span class=\"ft2\">=\u00a0<\/span>N \u03a3dxdy \u2013 \u03a3dx \u03a3dy<\/p>\n<p class=\"p78 ft10\"><span class=\"ft10\">\u221a<\/span><span class=\"ft13\">[N \u03a3d\u00b2x &#8211; (\u03a3dx)\u00b2] [N \u03a3d\u00b2y &#8211; (\u03a3dy)\u00b2]<\/span><\/p>\n<p class=\"p75 ft2\"><span class=\"ft57\">Value of correlation coefficient for bivariate frequency data<\/span><\/p>\n<p class=\"p79 ft10\">r\u00a0<span class=\"ft2\">=\u00a0<\/span>N \u03a3fdxdy \u2013 \u03a3fdx \u03a3fdy<\/p>\n<p class=\"p80 ft10\"><span class=\"ft10\">\u221a<\/span><span class=\"ft13\">[N \u03a3fd\u00b2x &#8211; (\u03a3fdx)\u00b2] [N \u03a3fd\u00b2y &#8211; (\u03a3fdy)\u00b2]<\/span><\/p>\n<p class=\"p81 ft20\" style=\"text-align: justify\">The value of\u00a0<span class=\"ft19\">r\u00a0<\/span>always lies between\u00a0\u20131\u00a0and 1, i.e., \u22121 \u2264\u00a0<span class=\"ft19\">r\u00a0<\/span>\u22641. If\u00a0<span class=\"ft19\">both variables\u00a0<\/span>increase or decreases (same direction), we say that there is\u00a0<span class=\"ft19\">positive\u00a0<\/span>correlation between them. However, if\u00a0<span class=\"ft19\">one\u00a0<\/span>decreases when\u00a0<span class=\"ft19\">another\u00a0<\/span>increases (or\u00a0<span class=\"ft19\">vice versa<\/span>), then we say that both are\u00a0<span class=\"ft19\">negatively\u00a0<\/span>or\u00a0<span class=\"ft19\">inversely\u00a0<\/span>correlated.<\/p>\n<p class=\"p82 ft2\">Procedure for Calculating Pearson\u2019s coefficient of correlation:<\/p>\n<\/div>\n<div><\/div>\n<div><\/div>\n<div id=\"page_11\">\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p83 ft10\"><span class=\"ft58\">\u2022\u00a0<\/span><span class=\"ft59\">Calculate the mean of the two series \u2018X\u2019 &amp;\u2019Y\u2019<\/span><\/p>\n<p class=\"p67 ft10\"><span class=\"ft58\">\u2022\u00a0<\/span><span class=\"ft59\">Calculate the deviations \u2018dx\u2019 &amp; \u2018dy\u2019 in two series from their respective mean.<\/span><\/p>\n<p class=\"p84 ft25\"><span class=\"ft58\">\u2022\u00a0<\/span><span class=\"ft60\">Square each deviation of \u2018dx\u2019 &amp; \u2019dy\u2019 then obtain the sum of the squared deviation i.e.\u2211dx<\/span><span class=\"ft61\">2\u00a0<\/span>&amp; \u2211dy<span class=\"ft61\">2<\/span><\/p>\n<p class=\"p85 ft10\"><span class=\"ft58\">\u2022\u00a0<\/span><span class=\"ft59\">Multiply each deviation under x with each deviation under y &amp; obtain the product of<\/span><\/p>\n<p class=\"p86 ft10\">\u2018dxdy\u2019. Then obtain the sum of the product of dx, dy i.e. \u2211dxdy<\/p>\n<p class=\"p67 ft10\"><span class=\"ft58\">\u2022\u00a0<\/span><span class=\"ft59\">Substitute the value in the formula.<\/span><\/p>\n<p class=\"p42 ft2\">Interpretation of Correlation Coefficient (r):<\/p>\n<p class=\"p87 ft20\" style=\"text-align: justify\">The extreme values of\u00a0<span class=\"ft19\">r<\/span>, i.e., when\u00a0<span class=\"ft19\">r\u00a0<\/span>=\u00a0<span class=\"ft62\">\u00b11<\/span>, shows that there is\u00a0<span class=\"ft19\">perfect\u00a0<\/span>(positive or negative) correlation between\u00a0<span class=\"ft19\">X\u00a0<\/span>and\u00a0<span class=\"ft19\">Y.\u00a0<\/span>The remaining values of r that lie in subintervals of\u00a0{\u20131,\u00a01}, explain the association\/ relationship in terms of its strength. One may use the following figure as a guideline as to what\u00a0<span class=\"ft19\">adjective\u00a0<\/span>must be used for the values of\u00a0<span class=\"ft19\">r\u00a0<\/span>obtained after calculation to describe the relationship.<\/p>\n<p class=\"p19 ft21\">Note<\/p>\n<p class=\"p88 ft20\" style=\"text-align: justify\">When\u00a0<span class=\"ft19\">r\u00a0<\/span>= 0, we cannot say that there is no correlation\u00a0<span class=\"ft19\">at all\u00a0<\/span>between\u00a0<span class=\"ft19\">X\u00a0<\/span>and\u00a0<span class=\"ft19\">Y<\/span>. Pearson\u2019s correlation coefficient is meant to measure\u00a0<span class=\"ft19\">linear\u00a0<\/span>relationship only. It should\u00a0<span class=\"ft19\">not\u00a0<\/span>be used in the case of\u00a0<span class=\"ft19\">non-linear<\/span>relationships since it will obviously lead to a wrong interpretation.<\/p>\n<p class=\"p101 ft10\">\n<\/div>\n<div id=\"page_12\">\n<div id=\"p12dimg1\"><img decoding=\"async\" id=\"p12img1\" src=\"http:\/\/www.htmlpublish.com\/newTestDocStorage\/DocStorage\/c52abc008b984cababa1aaf7806e7c4a\/pdf-to-word_images\/pdf-to-word12x1.jpg\" alt=\"image\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-385\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-194.png\" alt=\"\" width=\"422\" height=\"392\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-194.png 422w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-194-300x279.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-194-65x60.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-194-225x209.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-194-350x325.png 350w\" sizes=\"auto, (max-width: 422px) 100vw, 422px\" \/><\/div>\n<div><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-386\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-195.png\" alt=\"\" width=\"406\" height=\"237\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-195.png 406w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-195-300x175.png 300w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-195-65x38.png 65w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-195-225x131.png 225w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-195-350x204.png 350w\" sizes=\"auto, (max-width: 406px) 100vw, 406px\" \/><\/div>\n<p>&nbsp;<\/p>\n<p class=\"p108 ft2\">Advantages of Correlation Coefficient:<\/p>\n<ul>\n<li class=\"p109 ft10\"><span class=\"ft55\">It summarizes in one value, the degree of correlation &amp; direction of correlation also;<\/span><\/li>\n<li class=\"p109 ft10\">Since its value is unit less therefore, it is independent of the change of origin &amp; scale.<span style=\"text-align: initial;font-size: 1em\">Limitations of Correlation Coefficient:<\/span><\/li>\n<li class=\"p109 ft10\">Always assume linear relationship between variables<\/li>\n<li class=\"p109 ft10\">Interpreting the value of r is difficult. There are chances of wrong interpretation.<\/li>\n<li class=\"p109 ft10\">Value of Correlation Coefficient is affected by the extreme values<\/li>\n<li class=\"p109 ft10\">Time consuming methods<\/li>\n<li class=\"p109 ft10\">Usually, Coefficient of Determination (r<span class=\"ft61\" style=\"text-align: initial;font-size: 1em\">2<\/span><span style=\"text-align: initial;font-size: 1em\">) is used to interpret the value of coefficient of correlation (r). Coefficient of determination (r<\/span><span class=\"ft61\" style=\"text-align: initial;font-size: 1em\">2<\/span><span style=\"text-align: initial;font-size: 1em\">) measures common variance.<\/span><\/li>\n<li class=\"p109 ft10\">Charles Spearman\u2019s Method<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p class=\"p113 ft25\">This method is used when we are given with rank data (ordinal data). This method is based on the order (ranks) of the given observations.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p114 ft10\">Spearman\u2019s coefficient of correlation is represented by R and it is calculated as:<\/p>\n<p class=\"p115 ft2\">R = 1- (6 \u2211D<span class=\"ft71\">2<\/span>) \/ N (N<span class=\"ft71\">2\u00a0<\/span>\u2013 1)<\/p>\n<p class=\"p116 ft10\">Where<\/p>\n<p class=\"p117 ft10\">R = Rank correlation coefficient<\/p>\n<p class=\"p118 ft10\">D = Difference of rank between paired item in two series.<\/p>\n<p class=\"p118 ft10\">N = Total number of observation.<\/p>\n<\/div>\n<div id=\"page_13\">\n<p>&nbsp;<\/p>\n<p class=\"p119 ft2\">Interpretation of Rank Correlation Coefficient (R)<\/p>\n<p class=\"p120 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">The value of rank correlation coefficient, R ranges from\u00a0<\/span>-1\u00a0to +1<\/p>\n<p class=\"p121 ft25\"><span class=\"ft72\">\u2022<\/span><span class=\"ft74\">If R = +1, then there is complete agreement in the order of the ranks and the ranks are in the same direction<\/span><\/p>\n<p class=\"p122 ft25\"><span class=\"ft72\">\u2022<\/span><span class=\"ft74\">If R =\u00a0<\/span>-1,\u00a0then there is complete agreement in the order of the ranks and the ranks are in the opposite direction<\/p>\n<p class=\"p123 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">If R = 0, then there is no correlation<\/span><\/p>\n<p class=\"p124 ft2\">Problems in Rank Correlation Coefficient:<\/p>\n<p class=\"p125 ft20\"><span class=\"ft15\">\uf0d8<\/span><span class=\"ft75\">Problems where Ranks are not given:\u00a0<\/span>If the ranks are not given, then we need to assign ranks to the data series. The ranking can be done in ascending or descending order. We need to follow the same scheme of ranking for the other series.<\/p>\n<p class=\"p126 ft10\"><span class=\"ft15\">\uf0d8<\/span><span class=\"ft76\">Equal Ranks or tie in Ranks:\u00a0<\/span>In such cases average ranks should be assigned to each individual.<\/p>\n<p class=\"p127 ft2\">R = 1- (6 \u2211D<span class=\"ft71\">2<\/span>) + AF \/ N (N<span class=\"ft71\">2\u00a0<\/span>\u2013 1)<\/p>\n<p class=\"p128 ft10\">Where<\/p>\n<p class=\"p129 ft10\">AF = 1\/12(m<span class=\"ft77\">1<\/span><span class=\"ft78\">3\u00a0<\/span>\u2013 m<span class=\"ft77\">1<\/span>) + 1\/12(m<span class=\"ft77\">2<\/span><span class=\"ft78\">3\u00a0<\/span>\u2013 m<span class=\"ft77\">2<\/span>) +\u2026. 1\/12(m<span class=\"ft77\">2<\/span><span class=\"ft78\">3\u00a0<\/span>\u2013 m<span class=\"ft77\">2<\/span>)<\/p>\n<p class=\"p130 ft10\">m = Number of time an item is repeated<\/p>\n<p class=\"p131 ft2\">Merits\/ Demerits of Spearman\u2019s Rank Correlation:<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p124 ft2\">Merits:<\/p>\n<p class=\"p132 ft35\"><span class=\"ft72\">\u2022<\/span><span class=\"ft79\">This method is simpler to understand and easier to apply compared to karl Pearson\u2019s correlation method.<\/span><\/p>\n<p class=\"p133 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">This method is useful where we can give the ranks and not the actual data. (qualitative term)<\/span><\/p>\n<p class=\"p115 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">This method is to use where the initial data in the form of ranks.<\/span><\/p>\n<p class=\"p134 ft2\">Demerits:<\/p>\n<p class=\"p130 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">Cannot be used for finding out correlation in a grouped frequency distribution.<\/span><\/p>\n<p class=\"p130 ft10\"><span class=\"ft72\">\u2022<\/span><span class=\"ft73\">This method should be applied where N exceeds 30.<\/span><\/p>\n<\/div>\n<div id=\"page_14\">\n<p>&nbsp;<\/p>\n<p class=\"p0 ft2\"><strong><span class=\"ft2\">4.<\/span><span class=\"ft46\">Finding out Significance of Pearson\u2019s Correlation Coefficient (r):<\/span><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p class=\"p135 ft20\" style=\"text-align: justify\">As we know that correlation coefficient (r) is usually calculated to have an idea about the extent of relationship between two variables. Most of time, we consider a sample from a large population to calculate correlation coefficient (r) that is further used as a point estimate of population correlation coefficient (\u01bf). It means \u2018\u01bf\u2019 is estimated by \u2018r\u2019.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p136 ft10\">Therefore, we can say that \u2018r\u2019 is used as a population parameter (\u01bf).<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p137 ft25\" style=\"text-align: justify\">Here, one can take note that \u2018r\u2019 may be considered as an estimate of \u2018\u01bf\u2019 if the assumption of normal distribution of both the variables holds true.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p138 ft25\" style=\"text-align: justify\">The most widely used test to investigate whether both variables X and Y are correlated to each other or not, is the\u00a0t-test.\u00a0For using\u00a0t-test,\u00a0we take our hypothesis as follows:<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p139 ft10\">H<span class=\"ft77\">0<\/span>: \u01bf = 0<\/p>\n<p class=\"p140 ft10\">H<span class=\"ft77\">a<\/span>: \u01bf \u2260 0<\/p>\n<p class=\"p141 ft20\">Null hypothesis (H<span class=\"ft80\">0<\/span>) assumes that both the variables X and Y are not correlated in the entire population. Alternative hypothesis (H<span class=\"ft80\">a<\/span>) is just opposite to null hypothesis that assumes both variables are strongly correlated in population.<\/p>\n<p class=\"p142 ft10\">Now we can calculate the\u00a0t-value\u00a0using the following formula:<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-387\" src=\"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/81\/2018\/10\/2-196.png\" alt=\"\" width=\"168\" height=\"57\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-196.png 168w, https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-content\/uploads\/sites\/81\/2018\/10\/2-196-65x22.png 65w\" sizes=\"auto, (max-width: 168px) 100vw, 168px\" \/><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p class=\"p11 ft10\">Where<\/p>\n<p class=\"p145 ft10\">r = Pearson\u2019s coefficient of correlation<\/p>\n<p class=\"p145 ft10\"><span class=\"ft10\">\u01bf<\/span><span class=\"ft13\">= Population correlation<\/span><\/p>\n<p class=\"p146 ft25\">n = number of observations in sample\u00a0(n-2)\u00a0= degree of freedom<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p147 ft25\">Here, we have assumed population correlation (\u01bf) equals to zero. We can get the value of t after putting the values of r and n. This is said to be calculated value of t.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p148 ft25\">Now, we see the table value of t for level of significance (\u03b1) = 0.05 and degree of freedom =\u00a0(n-2).<\/p>\n<p class=\"p149 ft20\"><span class=\"ft10\">(a)\u00a0<\/span><span class=\"ft87\">If\u00a0<\/span><span class=\"ft88\">t<\/span><span class=\"ft80\">cal\u00a0<\/span>&gt;\u00a0<span class=\"ft88\">t<\/span><span class=\"ft80\">table value\u00a0<\/span>; we will reject H<span class=\"ft80\">0<\/span>; it means r is significantly different from zero and both variables are strongly correlated in population.<\/p>\n<p class=\"p149 ft20\"><span class=\"ft10\" style=\"text-align: initial;font-size: 1em\">(b)\u00a0<\/span><span class=\"ft89\" style=\"text-align: initial;font-size: 1em\">If\u00a0<\/span><span class=\"ft88\" style=\"text-align: initial;font-size: 1em\">t<\/span><span class=\"ft80\" style=\"text-align: initial;font-size: 1em\">cal\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">&lt;\u00a0<\/span><span class=\"ft88\" style=\"text-align: initial;font-size: 1em\">t<\/span><span class=\"ft80\" style=\"text-align: initial;font-size: 1em\">table value\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">; we will accept H<\/span><span class=\"ft80\" style=\"text-align: initial;font-size: 1em\">0<\/span><span style=\"text-align: initial;font-size: 1em\">; it means r is almost equal to zero and both variables are not correlated in population.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"p149 ft20\"><span style=\"text-align: initial;font-size: 1em\">Standard Error of Correlation Coefficient (r):<\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"p149 ft20\"><span style=\"text-align: initial;font-size: 1em\">We can also calculate the standard error of r using following formula:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"p149 ft20\"><span style=\"text-align: initial;font-size: 1em\">(1\u2212\ufffd\u00b2)<\/span><\/p>\n<p class=\"p149 ft20\"><span style=\"text-align: initial;font-size: 1em\">S.E.=\u00a0<\/span><span class=\"ft91\" style=\"text-align: initial;font-size: 1em\">\u221a\ufffd<\/span><\/p>\n<\/div>\n<div id=\"page_15\">\n<p class=\"p154 ft10\">Where<\/p>\n<p class=\"p155 ft10\">r = Correlation coefficient<\/p>\n<p class=\"p156 ft10\">n = Number of observations in sample<\/p>\n<p class=\"p157 ft2\">Probable Error of Correlation Coefficient (r):<\/p>\n<p class=\"p158 ft25\" style=\"text-align: justify\">We can also calculate the range of degree of correlation for the entire population with the help of \u2018probable error\u2019.<\/p>\n<p class=\"p161 ft99\"><span class=\"ft96\">\u01bf<\/span><span class=\"ft97\">\u00b1\u00a0<\/span>P.E.<span class=\"ft98\">r<\/span><\/p>\n<p class=\"p162 ft10\">Lower limit of population correlation coefficient =\u00a0<span class=\"ft100\">\u01bf &#8211;\u00a0<\/span><span class=\"ft99\">P.E.<\/span><span class=\"ft98\">r<\/span><\/p>\n<p class=\"p163 ft10\">Upper limit of population correlation coefficient =\u00a0<span class=\"ft100\">\u01bf +\u00a0<\/span><span class=\"ft99\">P.E.<\/span><span class=\"ft98\">r<\/span><\/p>\n<p class=\"p164 ft2\">5. Coefficient of Determination (r\u00b2):<\/p>\n<p class=\"p34 ft20\" style=\"text-align: justify\">The most widely used and convenient way to interpret the value of correlation coefficient between two variables is the coefficient of determination. The square of r is called as Coefficient of Determination (r<span class=\"ft101\">2<\/span>) that measures the common variations in both the variables. This is very useful measure of linear covariation of two variables.<\/p>\n<p class=\"p33 ft103\">Coefficient of Determination (r<span class=\"ft77\">2<\/span>) =\u00a0<span class=\"ft102\">\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd<\/span><\/p>\n<p class=\"p33 ft103\">\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd \ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd<\/p>\n<\/div>\n<div id=\"page_16\">\n<p>&nbsp;<\/p>\n<p class=\"p166 ft35\">For example, if value of r = 0.50, then r<span class=\"ft105\">2\u00a0<\/span>= 0.25 i.e r<span class=\"ft105\">2\u00a0<\/span>= 25\/ 100<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p167 ft20\" style=\"text-align: justify\">It may be interpreted as out of total variations (100 percent), 25 percent variations in dependent variable may be explained by variations in independent variable. The value of r<span class=\"ft106\">2\u00a0<\/span>is ranging from 0 to 1.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p168 ft20\" style=\"text-align: justify\">Coefficient of determination is widely used in regression analysis to assess the goodness of fit of the regression model. The greater the value of r<span class=\"ft106\">2\u00a0<\/span>the better is the fit and the more useful the regression equation as a predictive instrument.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"p33 ft2\"><strong>6. Limitations of Correlation analysis:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p class=\"p34 ft20\" style=\"text-align: justify\">From the discussion made so far, one can easily understand that correlation analysis is a statistical tool that is used to find out the association\/ relationship between the variables. It must be used very carefully to avoid misleading conclusions\/interpretations. The most common mistakes that are usually made by us are as hereunder:<\/p>\n<ul>\n<li class=\"p169 ft20\" style=\"text-align: justify\"><span class=\"ft27\">Correlation coefficient (r) gives us an idea about the\u00a0<\/span><span class=\"ft19\">linear relationship\u00a0<\/span>between the variables. As the value of r increase from 0 to 1, it means linear association between the variables also increases. A value of r = 0 does not show the absence of relationship. In this case, both the variables are not linearly related to one another but they may be associated in any other manner.<\/li>\n<li class=\"p169 ft20\" style=\"text-align: justify\">For the value of r = 0.8 and r = 0.4; we cannot say that in first case degree of relationship between the variables is just double\/two times as compared to second case. It means correlation coefficient does not follow the principle of proportionality.<\/li>\n<li class=\"p169 ft20\" style=\"text-align: justify\">A value of r = 0.6 does not mean that correlation explains 60% variations of total variations. Rather, r = 0.6 means that correlation explains only 36% variations in both the variables (as r\u00b2 = 0.36).<\/li>\n<li class=\"p169 ft20\" style=\"text-align: justify\">One may consider presence of strong correlation between the variables as\u00a0<span style=\"text-align: initial;font-size: 1em\">cause-effectrelationship (causation). Actually, correlation speaks nothing about the causation. In case of two variables, causation may be established only with the help of simple regression analysis.<\/span><\/li>\n<li class=\"p169 ft20\">A very common mistake in interpretation of correlation coefficient takes place when we conclude a strong relationship between the variables but actually they are not related. For example, sales of Hero bikes in New Delhi and number of accidents in Mumbai. Both the variables seem to be correlated as they show similar movements but actually, it is not possible to link them.<\/li>\n<\/ul>\n<\/div>\n<div id=\"page_17\">\n<p>&nbsp;<\/p>\n<p class=\"p172 ft2\"><strong><span class=\"ft2\">7.<\/span><span class=\"ft108\">Summary:<\/span><\/strong><\/p>\n<p class=\"p173 ft18\" style=\"text-align: justify\">Correlation analysis is a very good technique to find out the association or relationship between the two or more variables. In simple correlation we consider only two variables. In multiple correlations we try to find out the correlation among more than two variables. In partial correlation, we try to find out the correlation between two variables by partial out the effect of other variables. Pearson\u2019s coefficient of correlation is most widely used technique for finding out the correlation between the variables (recorded using interval\/ ratio scale). This assumes a linear relationship among the variables and normal distribution of variables. The calculation of correlation coefficient is little complex and is affected by the extreme values. For the rank data, Spearman\u2019s correlation coefficient is used. It is distribution free coefficient. Significance of Pearson\u2019s coefficient of correlation can be found out using\u00a0t-test.\u00a0Coefficient of determination (r\u00b2) is widely used in determining the common variations between the related variables. Existence of correlation between the variable does not signify any\u00a0cause-effect\u00a0relationship. Causation may be established by simple regression analysis.<\/p>\n<\/div>\n","protected":false},"author":3,"menu_order":34,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-376","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/376","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":3,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/376\/revisions"}],"predecessor-version":[{"id":388,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapters\/376\/revisions\/388"}],"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\/376\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/media?parent=376"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/pressbooks\/v2\/chapter-type?post=376"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/contributor?post=376"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/wp-json\/wp\/v2\/license?post=376"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}