Consider a data set consisting of observations with arithmetic mean

MATHEMATICS · Subjective · CBSE · Mathematics · Statistics · Statistics

Problem

Consider a data set consisting of

n

observations

x_1, x_2,\ldots ,x_n

with arithmetic mean

\bar{x}

. Answer the following: (i) State the definition of the population variance

\sigma^2

for ungrouped data. (ii) Prove that

\displaystyle \sum_{i=1}^{n}(x_i-\bar{x}) = 0

. (iii) Explain clearly how the result obtained in part (ii) is used in simplifying the expression

\displaystyle \sum_{i=1}^{n}(x_i-\bar{x})^2

while arriving at the usual variance formula.