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# Correlation Coefficient Calculator

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Correlation coefficients determines how two variables are associated. Correlation coefficient is denoted by "r". Correlation coefficient r ranges between 1 and -1. If r = 1, then its perfect positive correlation and when r = -1, then its perfect negative correlation. If r = 0, then there is no correlation coefficient.

Correlation Coefficient Calculator finds the correlation coefficient when you plug in the two sets of variables into the calculator.

## Steps

Step 1 : Observe and note the number of sample (n).

Step 2 : Determine the value of XY, $X^2$ and $Y^2$.

Step 3 : Find the sum($\Sigma$) of X, $X^2$, Y, $Y^2$ and XY.

Step 4 : Plug in these values into the correlation coefficient formula given below and you will get the correlation value(r).
$r = \frac{n(\Sigma XY)-(\Sigma X)(\Sigma Y)}{\sqrt{[n(\Sigma X^{2})-(\Sigma X)^{2}][n(\Sigma Y^{2})-(\Sigma Y)^{2}]}}$

## Problems

Given below are the problems based on correlation coefficient.

### Solved Examples

Question 1: Find the value of correlation coefficient for

 X value Y value 21 65 42 75 59 81 43 98 26 79

Solution:

Solution :
Step 1 : Number of sample, n = 5

Step 2 :
 X values Y values XY $X^2$ $Y^2$ 21 65 1365 441 4225 42 75 3150 1764 5625 59 81 4779 3481 6561 43 98 4214 1849 9604 26 79 2054 676 6241

Step 3 :
$\Sigma X$ = 21+42+59+43+26 = 191
$\Sigma Y$ = 65+75+81+98+79 = 398
$\Sigma XY$ = 1365+3150+4779+4214+2054 = 15562
$\Sigma X^2$ = 441+1764+3481+1849+676 = 8211
$\Sigma Y^2$ = 4225+5625+6561+9604+6241 = 32256

Step 4 :
Correlation r = $\frac{n(\Sigma XY)-(\Sigma X)(\Sigma Y)}{\sqrt{[n(\Sigma X^{2})-(\Sigma X)^{2}][n(\Sigma Y^{2})-(\Sigma Y)^{2}]}}$

r = $\frac{5(15562)-(191)(398)}{\sqrt{[5(8211)-(191)^{2}][5(32256)-(398)^{2}]}}$

$r = 0.4941$

Question 2: Find the value of correlation coefficient for

 X value Y value 1 2 3 4 5 6 7 8 9 0

Solution:

Solution :
Step 1 : Number of sample, n = 5

Step 2 :
 X values Y values XY $X^2$ $Y^2$ 1 2 2 1 4 3 4 12 9 16 5 6 30 25 36 7 8 56 49 64 9 0 0 81 0

Step 3 :
$\Sigma X$ =1+3+5+7+9 = 25
$\Sigma Y$ = 2+4+6+8+0 = 20
$\Sigma XY$ = 2+12+30+56+0 = 100
$\Sigma X^2$ = 1+9+25+49+81 = 165
$\Sigma Y^2$ = 4+16+36+64+0 =120

Step 4 :
Correlation r = $\frac{n(\Sigma XY)-(\Sigma X)(\Sigma Y)}{\sqrt{[n(\Sigma X^{2})-(\Sigma X)^{2}][n(\Sigma Y^{2})-(\Sigma Y)^{2}]}}$

r = $\frac{5(100)-(25)(20)}{\sqrt{[5(165)-(25)^{2}][5(120)-(20)^{2}]}}$

$r = 0$