Question

Calculate Karl Pearson's coefficient of correlation between income and expenditure of 10 families from the following data:

Income (in '000)	59	55	58	60	65	63	54	56	66	64
Expenditure (in '000)	52	53	55	58	57	66	59	54	52	54

Answer 1

Income (X)	Expenditure (Y)	X2	Y2	XY
59 55 58 60 65 63 54 56 66 64	52 53 55 58 57 66 59 54 52 54	3481 3025 3364 3600 4225 3969 2916 3136 4356 4096	2704 2809 3025 3364 3249 4356 3481 2916 2704 2916	3068 2915 3190 3480 3705 4158 3186 3024 3432 3456
$\mathit{\Sigma }\mathit{X}$=600	$\mathit{\Sigma }\mathit{Y}$=560	$\mathit{\Sigma }{\mathit{X}}^{\mathbf{2}}$=36168	$\mathit{\Sigma }{\mathit{Y}}^{\mathbf{2}}$=31524	$\mathit{\Sigma }\mathit{X}\mathit{Y}$=33614

N = 10

$$\begin{gathered} r=\frac{N\Sigma XY-\Sigma X\mathrm{\Sigma }Y}{\sqrt{N\Sigma X^2-{\left(\Sigma X\right)}^2}\sqrt{N\Sigma Y^2-{\left(\Sigma \mathrm{Y}\right)}^2}} \\ \mathrm{or},r=\frac{10\times 33614-600\times 560}{\sqrt{10\times 36168-{\left(600\right)}^2}\sqrt{10\times 31524-{\left(560\right)}^2}} \\ \mathrm{or},r=\frac{140}{40.98\times 40.49} \\ \Rightarrow r=0.0843 \end{gathered}$$

Thus, coefficient of correlation = 0.0843