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Calculate R(pearman's rank correlation) and r(karl-pearson's) from the following data. interpret our result.

$\begin{array}{|c|c|c|c|} \hline X& 12 & 17 & 22&27&32\\ \hline Y& 113& 119&117&115&121\\ \hline \end{array}$

Interprete the result.

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Solution:

1)Spearmen's Rank (R)

$\begin{array}{|c|c|c|c|} \hline sr.no&X&R1 & y & R2&D^2=(R1-R2)^2\\ \hline 1&12&5&113&5&0\\ \hline 2&17&4&119&2&4\\ \hline 3&22&3&117&3&0\\ \hline 4&27&2&115&4&4\\ \hline 5&32&1&121&1&0\\ \hline N=5&\sum x=110&&\sum y=585&&\sum D^2=8\\ \hline \end{array}$

$R=1- \frac{6 \sum D^2}{N^3-N}$

=$1- \frac{6 (8)}{5^3-5}$

=$1- \frac{48}{125-5}$

=$\frac {120-48}{120}$

R=0.6

2) Carl pearson(r):

$\begin{array}{|c|c|c|c|} \hline sr.no&X&y & (X-\bar x)^2 & (y-\bar y)^2 &(X-\bar x)(y-\bar y)\\ \hline 1&12&113&100&16&40\\ \hline 2&17&119&25&4&-10\\ \hline 3&22=\bar x&117&0&0&0\\ \hline 4&27&115&25&4&-10\\ \hline 5&32&121&100&16&40\\ \hline N=5&\sum x=110&\sum y=585&\sum (X-\bar x)^2=250& (y-\bar y)^2=40& \sum (X-\bar x)(y-\bar y)=60\\ \hline \end{array}$

$r= \frac {\sum (X-\bar x)(y-\bar y)}{\sqrt{(\sum (x-\bar x)^2) (\sum(y-\bar y)^2)}}$

$r= \frac {60}{\sqrt{(250\times 40})}$

r=0.6

Thus the values of R and r are equal. The values of X increases by 5 and values of y, When arranged in ascending order also increases by the same amount 2 every time.

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