 Hello and welcome to the session. In this session we are going to discuss Kyle Pearson's Coefficient of Correlation. This is a mathematical method for measuring correlation. It is also known as quantitative method of measuring correlation between two variables. It has been given by Kyle Pearson so it is known as Pearson's Coefficient of Correlation and it is denoted by the symbol rho of xy. This formula is based on the concept of covariance. So let us discuss covariance first. It will be variable x, x1, x2, x3 up to xn and another variable y takes the value of y1, y2, y3 up to yn. Then the covariance between the two variables x and y is given by covariance of xy is equal to x1 minus x bar into y1 minus y bar plus x2 minus x bar into y2 minus y bar plus and so on up to x1 minus x bar into yn minus y bar whole upon m where x bar and y bar denotes the arithmetic means of the two series. That is x bar is given by x1 plus x2 plus and so on up to xn whole upon m y bar is equal to y1 plus y2 plus and so on up to yn whole upon m. That is covariance of xy can also be written as 1 by m into summation of xi minus x bar into yi minus y bar where I go from 1 to m. Covariance of xy can also be written as 1 by m into summation of ds into dy where ds is equal to xi minus x bar and dy is equal to yi minus y bar. When we divide covariance by the product of the individual standard deviations the coefficient obtained is called the correlation coefficient. Therefore, Karl Pearson's coefficient of correlation denoted by r of rho of xy is equal to covariance of xy upon sigma x into sigma y and we know that covariance of xy can be written as summation of ds into dy by m upon sigma x into sigma y which is equal to summation of ds into dy upon n into sigma x into sigma y. Since we know that the value of sigma x is equal to square root of summation of dx square by m and the value of sigma y is equal to square root of summation of dy square by m. Therefore, Karl Pearson's coefficient of correlation at r of xy is equal to summation of dx into dy by m into square root of summation of dx square by m into square root of summation of dy square by n which is equal to summation of dx into dy upon square root of summation of dx square into summation of dy square where dx is equal to xi minus x bar and dy is equal to yi minus y bar. Are the deviations taken from the aftering mean? This formula is called first formula. It is also called product moment method. As it is based on summation of dx into dy by m that is product of the deviation of the observations from the mean. This method is useful when arithmetic means are in whole numbers or integers and the coefficient of correlation are lies from minus 1 to plus 1. Now we are going to discuss second formula for coefficient of correlation r without using deviations from mean that is by directly using the values of xi and yi. As we know that summation of xi minus x bar the whole square is equal to summation of xi square minus twice of xi into x bar plus x bar square which is equal to summation of xi square minus twice of x bar into summation of xi plus n times x bar square which is equal to summation of xi square minus 2 into x bar and x bar can be written as summation of xi by n into summation of xi plus n times x bar square which is equal to n into summation of xi by n dy square which is equal to summation of xi square minus twice of summation of xi square by n plus n into summation of xi square by n square which is equal to summation of xi square minus twice of summation of xi by square by n plus summation of xi by whole square by n which is equal to summation of xi square minus of summation of xi by whole square by n so, the value of summation of xi minus x bar by whole square is equal to summation of xi square minus summation of xi by whole square by n Similarly summation of y i minus y bar divided by square is equal to summation of y i square minus summation of y i divided by square by m. Therefore summation of x i minus x bar into y i minus y bar is equal to summation of x i into y i minus of x i into y bar minus of y i into x bar plus x bar into y bar which is equal to summation of x i into y i minus of y bar into summation of x i minus of x bar into summation of y i plus summation of x bar into y bar which is equal to n times x bar into y bar. As we know that x bar is equal to summation of x i by m which implies that summation of x i is equal to n into x bar and y bar is equal to summation of y i by m which implies that summation of y i is equal to n into y bar. So we have summation of x i into y i minus of y bar into summation of x i which is equal to n into x bar minus of x bar into summation of y i which is equal to n into y bar plus n into x bar into y bar which is equal to summation of x i into y i minus of 2 into m into x bar into y bar plus n into x bar into y bar which is equal to summation of x i into y i minus of n into x bar into y bar which is equal to summation of x i into y i minus of n into x bar which is equal to summation of x i by m into y bar which is equal to summation of y i by n. So the value of summation of x i minus x i into y i minus y i is equal to summation of x i into y i minus of summation of x i into summation of y i by n. I goes from 1 to n and we know that cancels in coefficient of correlation r is given by summation of dx into dy upon square root of summation of dx square into summation of dy square where dx is given by x i minus x y and dy is given by y i minus y y. Therefore cancels in coefficient of correlation r or rho of x y is given by summation of x i minus x y into y i minus y y where i goes from 1 to n whole upon square root of summation of x i minus x y square where i goes from 1 to n into square root of summation of y i minus y y by d rho square where i goes from 1 to n. Now we know that summation of x i minus x y into y i minus y y is equal to summation of x i into y i minus of summation of x i into summation of y i by n. Also summation of x i minus x y d rho square is given by summation of x i square minus summation of x i d rho square by n and summation of y i minus y by d rho square is equal to summation of y i square minus summation of y i d rho square by n. After substituting all these values in the above formula we get the coefficient of correlation r or rho x y is given by summation of x i into y i minus of summation of x i into summation of y i by n whole upon square root of summation of x i square minus summation of x i d whole square by n into square root of summation of y i square minus summation of y i d whole square by n. Simply we can write it as r or rho of x y is equal to summation of x into y minus of summation of x into summation of y by n whole upon square root of summation of x square minus of summation of x d whole square by n into square root of summation of y square minus of summation of y d whole square by n where x and y stand for the values of items in x and y series and n is the number of observations. Now we shall discuss how to calculate coefficient of correlation when the deviations are taken from an assumed mean. If the value worth are large involved fraction arithmetic mean is in fractions computation of the correlation coefficient is simplified considering the deviations from u i and v i of the value worth x i and y i that is the assumed means a and b respectively for the values of x and y that is if u i is equal to x i minus a is equal to y i minus b then rho of x y which is given by summation of x i into y i minus of summation of x i into summation of y i by n whole upon square root of summation of x i square minus of summation of x i d whole square by n into square root of summation of y i square minus of summation of y i be square by m. u i is equal to x i minus a implies that the value of x i is equal to u i plus a. Similarly, v i is equal to y i minus b implies that the value of y i is equal to v i plus b. Now substituting the values of x i and y i in this formula we get. Now we know that x i is equal to u i plus a and y i is equal to v i plus b. Therefore summation of x i into v i is equal to summation of u i plus a into v i plus b which is equal to summation of u i into v i plus b into u i plus a into v i plus a into b which is equal to summation of u i into v i plus b into summation of u i plus a into summation of v i plus summation of AB which is equal to N into AB. Also we have summation of xi into summation of yi which is equal to summation of ui plus a into summation of vi plus b which is equal to summation of ui plus summation of a that is N into a into summation of vi plus summation of b which is equal to n into b. Therefore we get summation of ui into vi plus n into b into summation of ui plus n into a into summation of vi plus n square into a into b. The value of summation of xi into summation of yi by n is equal to summation of ui into vi by n plus b into summation of ui plus a into summation of vi plus n into a into b. Therefore the value of the expression summation of xi into yi minus summation of xi into summation of yi by n is equal to summation of ui into vi minus summation of ui into vi by n. And similarly we can find the value of the expression square root of summation of xi square minus summation of xi y square by n into square root of summation of yi square minus summation of yi p whole square by n. And hence we get the value of rho of u vi summation of ui into vi where i goes from 1 to n minus rho 1 by n into summation of ui where i goes from 1 to n into summation of vi where i goes from 1 to n whole upon square root of summation of ui square where i goes from 1 to n minus rho 1 by n into summation of ui the whole square where i goes from 1 to n into square root of summation of vi square where i goes from 1 to n minus of 1 by n into summation of vi the whole square where i goes from 1 to n which can also be written as rho of ui is equal to summation of u into vi minus of summation of u into summation of vi by n whole upon square root of summation of u square minus summation of u the whole square by n into square root of summation of v square minus of summation of v the whole square by n where u is equal to x minus a and v is equal to y minus b where a and b are the assumed means and n is the number of observations. This completes our session. Hope you enjoyed this session.