Question

For the data The Karl Pearson’s coefficient is

• A. $\frac{63}{\sqrt{94 \times 66}}$
• B. 63
• C. $\frac{63}{\sqrt{94}}$
• D. $\frac{63}{\sqrt{66}}$

Answer 1

Answer: (A)

$\frac{63}{\sqrt{94 \times 66}}$

Answer 2

Take $\mathbf{A = 5,B = 5}$

xi	yi	ui=xi−5	vi=yi−5	ui2	vi2	uivi
4 7 8 3 4	5 8 6 3 5	– 1 2 3 – 2 – 1	0 3 1 – 2 0	1 9 1 4 0	0 9 1 4 0	0 6 3 4 0
Total		$$\sum_{}^{}{u_{i} = 1}$$	$$\sum_{}^{}{v_{i} = 2}$$	$$\sum_{}^{}{u_{i}^{2} = 19}$$	$$\sum_{}^{}{v_{i}^{2} = 14}$$	$$\sum_{}^{}{u_{i}v_{i} = 13}$$

$\mathbf{\because}$ $\mathbf{r(x,y) =}\frac{\sum_{}^{}{\mathbf{u}_{\mathbf{i}}\mathbf{v}_{\mathbf{i}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{n}}\sum_{}^{}{\mathbf{u}_{\mathbf{i}}\sum_{}^{}\mathbf{v}_{\mathbf{i}}}}}{\sqrt{\sum_{}^{}{\mathbf{u}_{\mathbf{i}}^{\mathbf{2}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{n}}\left( \sum_{}^{}\mathbf{u}_{\mathbf{i}} \right)^{\mathbf{2}}}}\sqrt{\sum_{}^{}{\mathbf{v}_{\mathbf{i}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{n}}\left( \sum_{}^{}\mathbf{v}_{\mathbf{i}} \right)^{\mathbf{2}}}}}$ $\mathbf{=}\frac{\mathbf{13 -}\frac{\mathbf{1 \times 2}}{\mathbf{5}}}{\sqrt{\mathbf{19 -}\frac{\mathbf{1}^{\mathbf{2}}}{\mathbf{5}}}\sqrt{\mathbf{14 -}\frac{\mathbf{2}^{\mathbf{2}}}{\mathbf{5}}}}\mathbf{=}\frac{\mathbf{63}}{\sqrt{\mathbf{94}}\sqrt{\mathbf{66}}}$.