First slide

Introduction to statistics

Question

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

Difficult

A

$\frac{5}{2}$
B

$\frac{11}{2}$
C

6
D

` $\frac{13}{2}$

Solution

$∵ σ_{x}^{2} = 4 and σ_{y}^{2} = 5$
Also, $\bar{x} = 2 and \bar{y} = 4$
${\begin{array}{l} now, \frac{{Σx}_{i}}{5} = 2 \Rightarrow {Σx}_{i} = 10; {Σy}_{i} = 20 \\ and σ_{x}^{2} = (\frac{1}{5} {Σx}_{i}^{2}) - (\bar{x})^{2} \\ \Rightarrow {Σx}_{i}^{2} = 5 (4 + 4) \Rightarrow {Σx}_{i}^{2} = 40 \\ and {σ^{2}}_{y} = (\frac{1}{5} {Σy}_{i}^{2}) - (\bar{y})^{2} \\ and {Σy}_{i}^{2} = 5 (5 + 16) \Rightarrow {Σy}_{i}^{2} = 105 \end{array}}_{}$
combined mean,
$\begin{array}{l} σ_{z}^{2} = \frac{1}{10} ({Σx}_{i}^{2} + {Σy}_{i}^{2} - {(\frac{\bar{x} + \bar{y}}{2})}^{2} \\ = \frac{1}{10} (40 + 105) - 9 = \frac{145 - 90}{10} = \frac{55}{10} = \frac{11}{2} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App