In a study, following observation were made: n=12, $\sum x$=100, ${\sum x ^2} = 954$, $\sum y$=103, ${\sum y ^2} = 1079$, $\sum x y = 918$. On checking it was found that one pair of observation recorded as (X,Y)=(5,7) was wrongly entered and should be discarded. Calculate the corrected correlation coefficient.

Solution:

\[\begin{array}{l}Corrected\;{\rm{ }}\sum x  = 100 - 5 = 95\\Corrected\;{\rm{ }}\sum y  = 103 - 7 = 96\\Corrected\;{\rm{ }}{\sum x ^2} = 954 - {5^2} = 929\\Corrected\;{\rm{ }}{\sum y ^2} = 1079 - 49 = 1030\\Corrected\;{\rm{ }}\sum x y = 918 - 5 \times 7 = 883\\Corrected\;{\rm{ Correlation\;(r) = }}\frac{{(n - 1)\sum x y - \sum x \sum y }}{{\sqrt {\left[ {(n - 1)\left( {{{\sum x }^2} - {{\left( {\sum x } \right)}^2}} \right)} \right]\left[ {(n - 1)\left( {{{\sum y }^2} - {{\left( {\sum y } \right)}^2}} \right)} \right]} }}\\ = \frac{{11 \times 883 - 95 \times 96}}{{\sqrt {\left[ {11 \times 927 - {{95}^2}} \right]\left[ {11 \times 1030 - {{96}^2}} \right]} }}\\ = \frac{{593}}{{\sqrt {1457 \times 2114} }} = 0.37\end{array}\]