Question

The height of fathers and sons are given below:

Height of fathers (in inches)	:	65	66	67	67	68	69	71	73
Height of sons (in inches)	:	67	68	64	68	72	70	69	70

Calculate Karl Pearson's coefficient of correlation .

Answer 1

Father's Height (X)	Son's Height (Y)	$\left(\mathit{X}\mathit{-}\overline{X}\right)$ x	x2	$\left(\mathit{Y}\mathit{-}\overline{\mathit{Y}}\right)$ y	y2	xy
65	67	−3.25	10.56	−1.5	2.25	4.875
66	68	−2.25	5.06	−0.5	0.25	1.125
67	64	−1.25	1.56	−4.5	20.25	5.625
67	68	−1.25	1.56	−0.5	0.25	0.625
68	72	−0.25	0.06	3.5	12.25	−0.875
69	70	0 .75	0.56	1.5	2.25	1.125
74	69	2.75	7.56	0.5	0.25	1.375
73	70	4.75	22.56	1.5	2.25	7.125
ƩX = 546	ƩY = 548		Ʃx2 = 49.48		Ʃy2 = 40	Ʃxy = 21

N = 8

$$\begin{gathered} \overline{X}=\frac{\mathrm{\Sigma }X}{N}=\frac{546}{8}=68.25 \\ \overline{Y}=\frac{\mathrm{\Sigma }Y}{N}=\frac{548}{8}=68.5 \\ r=\frac{\mathrm{\Sigma }xy}{\sqrt{\mathrm{\Sigma }{x}^2}\times \sqrt{\mathrm{\Sigma }{y}^2}}=\frac{21}{\sqrt{49.48}\times \sqrt{40}} \\ r=\frac{21}{7.034\times 6.324}=0.472 \\ \mathrm{Thus},\mathrm{the}\mathrm{correlation}\mathrm{coefficient}\mathrm{between}\mathrm{height}\mathrm{of}\mathrm{father}\mathrm{and}\mathrm{height}\mathrm{of}\mathrm{son}\mathrm{is}0.472. \end{gathered}$$