Question

Find Karl Pearson's coefficient of correlation from the following index numbers and interpret it.

Wages (₹)	:	100	101	103	102	100	99	97	98	96	95
Cost of living	:	98	99	99	97	95	92	95	94	90	91

Answer 1

Wages (X)	Cost of Living (Y)	$\left(\mathit{X}\mathit{-}\overline{X}\right)$ x	x2	$\left(\mathit{Y}\mathit{-}\overline{\mathit{Y}}\right)$ y	y2	xy
100	98	0.9	0.81	3	9	27
101	99	1.9	3.61	4	16	7.6
103	99	3.9	15.21	4	16	5.6
102	97	2.9	8.41	2	4	5.8
100	95	0.9	0.81	0	0	0
99	92	−0.1	0.01	−3	9	0.3
97	95	−2.1	4.41	0	0	0
98	94	−1.1	1.21	−1	1	1.1
96	90	−3.1	9.61	−5	25	15.5
95	91	−4.1	16.81	−4	16	16.6
ƩX = 991	ƩY = 950		Ʃx2 = 60.9		Ʃy2 = 96	Ʃxy = 65.2

N = 10

$$\begin{gathered} \overline{X}=\frac{\mathrm{\Sigma }X}{N}=\frac{991}{10}=99.1 \\ \overline{Y}=\frac{\mathrm{\Sigma }Y}{N}=\frac{950}{10}=95 \\ r=\frac{\mathrm{\Sigma }xy}{\sqrt{\mathrm{\Sigma }{x}^2}\times \sqrt{\mathrm{\Sigma }{y}^2}} \\ =\frac{65.2}{\sqrt{60.9}\times \sqrt{96}} \\ =\frac{65.2}{7.80\times 9.79} \\ =0.85 \end{gathered}$$

Wages and cost of living has high positive correlation. If wages increase by 1 unit, the cost of living increases by 0.85 units.