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Write a detailed note on Carl's Correlation Coefficient Algorithm. Justify the necessary of Algorithm by giving suitable example.
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Carl Pearson Coefficient of correlation is a measure of linear relationship between two variables. It lies between -1 and 1.

The closer it is to 1(or -1), the stronger the positive(or negative) linear relationship between the two variables. If it is close to 0, there is no linear relation.

Example :

X(height - cm) Y(weight - kg)
174 61
175 65
176 67
177 68
178 72
182 74
183 80
186 87
189 92
193 95

Number of cases: n = 10

Sr. No. X Y $X^2$ $Y^2$ XY
1 174 61 30276 3721 10614
2 175 65 30625 4225 11375
3 176 67 30976 4489 11792
4 177 68 31329 4624 12036
5 178 72 31684 5184 12816
6 182 74 33124 5476 13468
7 183 80 33489 6400 14640
8 186 87 34596 7569 16182
9 189 92 35721 8464 17388
10 193 95 37249 9025 18335
1813 761 329069 59177 138646

∑ X = 1813

∑ Y = 761

∑ $X^2$= 329069

∑ $Y^2$ = 59177

∑ XY = 138646

∑ X ∑ Y = 138646

$r=\dfrac{n(∑xy)-(∑x)(∑y)}{\sqrt{[n∑x^2 -(∑x)^2 ][n∑y^2 - (∑y)^2 ]} }$

$\therefore$ Correlation coefficient: r = 0.9864