Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y).

$\begin{matrix} X & 65 & 66 & 57 & 67 & 68 & 69 & 70 & 72 Y & 67 & 56 & 65 & 68 & 72 & 72 & 69 & 71 \end{matrix}$

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Solution

$\begin{matrix} X & Y & X Y & X^{2} & Y^{2} 65 & 67 & 4355 & 4255 & 4489 66 & 56 & 3696 & 4356 & 3136 57 & 65 & 3705 & 3249 & 4255 67 & 68 & 4556 & 4489 & 4624 68 & 72 & 4896 & 4624 & 5184 69 & 72 & 4968 & 4761 & 5184 70 & 69 & 4830 & 4900 & 4761 72 & 71 & 5112 & 5184 & 5041 \sum X = 534 & \sum Y = 540 & \sum X Y = 36118 & \sum X^{2} = 35788 & \sum Y^{2} = 36644 \end{matrix}$

$r = \frac{\sum X Y - \frac{(\sum X) (\sum Y)}{N}}{\sqrt{\sum X^{2} - \frac{(\sum X)^{2}}{N}} \sqrt{\sum Y^{2} - \frac{(\sum Y)^{2}}{N}}}$

$= \frac{36118 - \frac{534 \times 540}{8}}{\sqrt{35788 - \frac{(534)^{2}}{8}} \sqrt{36644 - \frac{(540)^{2}}{8}}}$

$= \frac{36118 - \frac{288360}{8}}{\sqrt{35788 - \frac{285156}{8}} \sqrt{36644 - \frac{291600}{8}}}$

$= \frac{36118 - 36045}{\sqrt{35788 - 356445} \sqrt{36644 - 36450}}$

$= \frac{73}{\sqrt{143.5} \sqrt{194}} = \frac{73}{11.98 \times 13.93} = \frac{73}{166.88} = 0.44$

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Similar questions

X	65	66	57	67	68	69	70	72
Y	67	56	65	68	72	72	69	71

Height of fathers (inches)	66	68	69	72	65	59	62	67	61	71
Height of Sons (inches)	65	64	67	69	64	60	59	68	60	64

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

Height (in inches)	60−61	61−62	62−63	63−64	64−65	65−66	66−67	67−68	68−69
Frequency	4	16	8	24	35	18	19	16	10

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