Question

Find Karl Pearson's coefficient of correlation for the following data:

Fertilizers used ( in tons )	:	15	18	20	24	30	35	40	50
Productivity ( in tons)	:	85	90	95	105	120	130	150	160

Answer 1

Fertilizers (X)	Productivity (Y)	$\left(\mathit{X}\mathit{-}\overline{X}\right)$ x	x2	$\left(\mathit{Y}\mathit{-}\overline{\mathit{Y}}\right)$ y	y2	xy
15	85	−14	196	−32.25	1040.06	451.5
18	93	−11	121	−24.25	588.06	266.75
20	95	−9	81	−22.25	495.06	200.25
24	105	−5	25	−12.25	150.06	61.25
30	120	1	1	2.75	7.56	2.75
35	130	6	36	12.75	162.56	76.5
40	150	11	121	32.75	1072.56	360.25
50	160	21	441	42.75	1827.56	897.75
ƩX =232	ƩY = 938		Ʃx2 =1022		Ʃy2 =5343.48	Ʃxy =2317

N = 8

$$\begin{gathered} \overline{X}=\frac{\mathrm{\Sigma }X}{N}=\frac{232}{8}=29 \\ \overline{Y}=\frac{\mathrm{\Sigma }Y}{N}=\frac{938}{8}=117.25 \\ r=\frac{\mathrm{\Sigma }xy}{\sqrt{\mathrm{\Sigma }{x}^2}\sqrt{\mathrm{\Sigma }{y}^2}} \\ =\frac{2317}{\sqrt{1022}\sqrt{5343.48}} \\ =\frac{2317}{31.96\times 73.09} \\ =\frac{2317}{2335.95} \\ =0.99 \end{gathered}$$
Thus, there exists high positive correlation between the amount of fertilizers used and the productivity.