Question:
If $x_{i}{ }^{\prime}$ s are the mid-points of the class intervals of grouped data, $f_{i}{ }^{\prime}$ s are the corresponding frequencies and $\bar{x}$ is
the mean, then $\Sigma\left(f_{i} x_{i}-\bar{x}\right)$ is equal to
(a) 0
(b) -1
(c) 1
(d) 2
Solution:
(a) $\because$ $\bar{x}=\frac{\Sigma f_{i} x_{i}}{n}$
$\therefore$ $\Sigma\left(f_{i} x_{i}-\bar{x}\right)=\Sigma f_{i} x_{i}-\Sigma \bar{x}$
$=n \bar{x}-n \bar{x}$ $[\because \Sigma \bar{x}=n \bar{x}]$
$=0$