Which of the following data has minimum variance?

Which of the following data has minimum variance?
  1. 1, 2, 3, 4, 5
  2. 1, 1, 2, 3, 6
  3. $1,1,2,3,5$
  4. $1,1,2,2,5$

Solution

Option (a) Data $\rightarrow 1,2,3,4,5$ $ \begin{aligned} & \text { Variance }=\sum_{i=1}^n \frac{\left(x_i-x_{\mathrm{avg}}\right)^2}{n-1}=\frac{1}{4}\left[\Sigma\left(x_i-x_{\mathrm{avg}}\right)^2\right] \\ & \quad x_{\text {avg }}=\frac{1+2+3+4+5}{5}=3 \\ & \therefore \text { Variance }=1 / 4 \\ & {\left[(1-3)^2+(2-3)^2+(3-3)^2+(4-3)^2+(5-3)^2\right]} \\ & \quad=\frac{1}{4}[4+1+0+1+4]=\frac{10}{4}=2.5 \end{aligned} $ $ \begin{aligned} & \text { Option (b) Data } \rightarrow 1,1,2,3,6 \\ & \qquad x_{\text {avg }}=\frac{1+1+2+3+6}{5}=\frac{13}{5}=2.6 \\ & \text { Variance }=\frac{1}{4}\left[(1-2.6)^2+(1-2.6)^2+\right. \\ & \qquad=4.3 \\ & \text { Option (c) Data } \rightarrow 1,1,2,3,5 \\ & \text { Variance }=2.8 \\ & \text { Option (d) Data } 1,1,2,2,5 \\ & \text { Variance }=2.7 \\ & \therefore \text { Minimum Variance is of } 1,2,3,4,5 \text {; i.e. } 2.5 . \end{aligned} $

Asked in: AP EAMCET 2021 (23 Aug Shift 1)

Practice more Statistics questions on Aicharya