Question

For $10$ points of observations on two variables $X$ and $Y$, the following data are available:
$\sum (x-2)=30,\sum y-5=40,\sum (x-2{)}^2=900$
$\sum (y-5{)}^2=800,\sum (x-2\left)\right(y-5)=480$.
Find the correlation coefficient between $X$ and $Y$.

Answer 1

The basic formula for calculating correlation coefficient

$\text{corr}(X,Y)=\frac{n\sum xy-\sum x\sum y}{\sqrt{(n\sum x^2-(\sum x{)}^2\left)\right(n\sum y^2-(\sum y{)}^2)}}$ .....(1)

where $n$ $=$ no. of observations and $n=10$ in our case.

We need to find out each of the terms in equation (1).

From the given data,

$\sum (x-2)=30$

$⟹\sum x-2\sum 1=30$

$⟹\sum x-2n=30$

$⟹\sum x-20=30$

$\therefore \sum x=10$ ....(2)

$\sum y-5=40$

$\therefore \sum y=45$ .....(3)

$\sum (x-2{)}^2=900$

$⟹\sum (x^2-4x-4)=900$

$⟹\sum x^2-4\sum x-4\sum 1=900$

$⟹\sum x^2-4\times 10+4\times 10=900$.... from $\left(2\right)$

$\therefore \sum x^2=900$ ------ $\left(4\right)$

$\sum (y-5{)}^2=800$

$⟹\sum (y^2-10y+25)=800$

$⟹\sum y^2-10\sum y+25\sum 1=800$

$⟹\sum y^2-10\times 45+25\times 10=800$..... from $\left(3\right)$

$\therefore \sum y^2=1000$ --------$\left(5\right)$

$\sum (x-2)(y-5)=480$

$⟹\sum (xy-5x-2y+10)=480$

$⟹\sum xy-5\sum x-2\sum y+10\sum 1=480$

$⟹\sum xy-5\times 10-2\times 45+10\times 10=480$.....from $\left(2\right)\&\left(3\right)$

$\therefore \sum xy=520$---------$\left(6\right)$

Substituting $\left(2\right),\left(3\right),\left(4\right),\left(5\right),\left(6\right)$ in $\left(1\right)$ gives us

$corr(X,Y)=\frac{10\times 520-10\times 45}{\sqrt{(10\times 900-{10}^2)(10\times 1000-{45}^2)}}$

$corr(X,Y)=0.564$