Answer to Question #104478 in Statistics and Probability for Amra Musharraf

Question

Twenty-five pairs of value of variates X and Y led to the following results: N=25, ∑X=127, ∑Y=100, ∑X^2=760, ∑Y^2=449 and∑XY=500
A subsequent scrutiny showed that two pairs of values were copied down as:
X Y instead of X Y
8 14 8 12
8 6 6 8

Obtain the correct value of the correlation coefficient

Answer 1

Twenty-five pairs of values of variates X and Y led to the following results: 

$n=25, \sum{X}=127, \sum{Y}=100, \sum{X^2}=760,$

$\sum{Y^2}=449, \sum{XY}=500$

A subsequent scrutiny showed that two pairs of values were copied down as  $\begin{matrix} X & Y \\ 8 & 14 \\ 8 & 6 \end{matrix}$ while the correct ones are  $\begin{matrix} X & Y \\ 8 & 12 \\ 6 & 8 \end{matrix}$ . Obtain the correct value of the correlation coefficient. 

New values

$n=25$

$\sum{X}=127-8-8+8+6=125$

$\sum{Y}=100-14-6+12+8=100$

$\sum{X^2}=760-8^2-8^2+8^2+6^2=732$

$\sum{Y^2}=449-14^2-6^2+12^2+8^2=425$

$\sum{XY}=500-8(14)-8(6)+8(12)+6(8)=484$

$$r={n\sum{XY}-\sum{X}\sum{Y} \over \sqrt{n\sum{X^2}-(\sum{X})^2}\sqrt{n\sum{Y^2}-(\sum{Y})^2}}$$

$$r={25\cdot484-125\cdot100 \over \sqrt{25\cdot732-(125)^2}\sqrt{25\cdot425-(100)^2}}\approx-0.309356$$