First slide

Introduction to statistics

Question

For two data sets, each of size 5, the variance is given to be 4 and 5 and the corresponding means are given to be 2 and 4 respectively. The variance of the combined dataset is

Moderate

A

$\frac{5}{2}$
B

$\frac{11}{2}$
C

6
D

$\frac{13}{2}$

Solution

$σ_{1}^{2} = 4, n_{1} = 5, {\bar{x}}_{1} = 2$
$σ_{2}^{2} = 5, n_{1} = 5, {\bar{x}}_{2} = 4$

${\bar{x}}_{12} = \frac{n_{1} {\bar{x}}_{1} + n_{2} {\bar{x}}_{2}}{n_{1} + n_{2}} = \frac{5 \times 2 + 5 \times 4}{10} = 3$

$d_{1} = ({\bar{x}}_{1} - {\bar{x}}_{2}) = - 1, d_{2} = ({\bar{x}}_{2} - {\bar{x}}_{12}) = 1$

$σ = \sqrt{\frac{n_{1} σ_{1}^{2} + n_{2} σ_{2}^{2} + n_{1} d_{1}^{2} + n_{2} d_{2}^{2}}{n_{1} + n_{2}}}$

$= \sqrt{\frac{5.4 + 5.5 + 5.1 + 5.1}{10}} = \sqrt{\frac{55}{10}} = \sqrt{\frac{11}{2}}$

$∴ σ^{2} = \frac{11}{2}$

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