While deriving the simplified form of variance

MATHEMATICS · Single correct · CBSE · Mathematics · Statistics · Statistics

Problem

While deriving the simplified form of variance, a student reaches the step

\sigma^2=\frac{1}{n}\Bigl[\sum x_i^2-2\bar{x}\sum x_i+n\bar{x}^2\Bigr].

If he now substitutes

\sum x_i=n\bar{x}

, which of the following expressions should correctly follow? (a)

\displaystyle \sigma^2=\frac{\sum x_i^2}{n}-2\bar{x}^2

(b)

\displaystyle \sigma^2=\frac{\sum x_i^2}{n}+\bar{x}^2

(c)

\displaystyle \sigma^2=\frac{\sum x_i^2}{n}-\bar{x}^2

(d)

\displaystyle \sigma^2=\frac{\sum x_i^2}{n}