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Calculate the correlation coefficient between x and y from the following data: $N=10$, $\sum x=140$, $\sum y=150$, $\sum (x-10)^{2}=180$, $\sum (y-15)^{2}=215$, $\sum (x-10) (y-15) =60$.

Topic: Maths 4

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Solution:

$\sum (x-10)^{2} = 180=\sum dx^{2}=180$

$\sum (y-15)^{2} = 215=\sum dy^{2}=215$

$\sum (x-10) (y-15) = 60=\sum dx dy=60$

Now,

\begin{aligned} \bar{x} &= A+ \cfrac{\sum dx}{N} \\ \therefore 14 &= 10+\cfrac{\sum dx}{10} \\ \therefore \sum dx &= 40 \end{aligned}

Similarly,

\begin{aligned} \bar{y} &= B+\cfrac{\sum dy}{N} \\ \therefore 15 &= 15+\cfrac{\sum dy}{10} \\ \therefore \sum dy &= 0 \end{aligned}

$\gamma = \cfrac{\sum dx dy - \cfrac{\sum dx dy}{N}}{\sqrt{\sum dx^{2}-\cfrac{(\sum dx)^{2}}{N}} \sqrt{\sum dy^{2}- \cfrac{(\sum dy)^{2}}{N}}}$

$\gamma = \cfrac{60-\cfrac{40 \times 0}{10}}{\sqrt{180-\cfrac{(40)^{2}}{10}}\sqrt{215-\cfrac{(0)^{2}}{10}}}$

$= \cfrac{60}{\sqrt{20}\sqrt{215}}=0.915$