Question

Find Karl Pearson's correlation coefficient from the following given values of X
and Y-
X 78 89 96 69 59 79 68 61
Y 125 137 156 112 107 136 123 108


Consider 69 and 112 as assumed mean in X and Y series respectively, .​

Answer 1

The Karl Pearson's correlation coefficient is

0.977

Given:

Mean of X = 69

Mean of Y = 112

To find:

Karl Pearson's correlation coefficient

Solution:

we can calculate the Karl Pearson's correlation coefficient by following these steps:

Subtract the given mean from each value in the two variables to get the deviations. For X, the deviations are [78 - 69, 89 - 69, 96 - 69, 69 - 69, 59 - 69, 79 - 69, 68 - 69, 61 - 69] = [9, 20, 27, 0, -10, 10, -1, -8]. For Y, the deviations are [125 - 112, 137 - 112, 156 - 112, 112 - 112, 107 - 112, 136 - 112, 123 - 112, 108 - 112] = [13, 25, 44, 0, -5, 24, 11, -4].

Calculate the product of the deviations for X and Y. The product of the deviations is [913, 2025, 2744, 00, -10*-5, 1024, -111, -8*-4] = [117, 500, 1188, 0, 50, 240, -11, 32].

Calculate the sum of the products of the deviations. The sum of the products is 117 + 500 + 1188 + 0 + 50 + 240 + -11 + 32 = 2178.

Calculate the standard deviation of X and Y. The standard deviation of X is sqrt((9^2 + 20^2 + 27^2 + 0^2 + (-10)^2 + 10^2 + (-1)^2 + (-8)^2)/8) = 15.68. The standard deviation of Y is sqrt((13^2 + 25^2 + 44^2 + 0^2 + (-5)^2 + 24^2 + 11^2 + (-4)^2)/8) = 18.97.

Calculate the correlation coefficient by dividing the sum of the products of the deviations by (n-1) * standard deviation of X * standard deviation of Y. The correlation coefficient is (2178/(715.6818.97)) = 0.977.

So, the Karl Pearson's correlation coefficient for the given values of X and Y is 0.977

#SPJ1