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Examples 12 to 14

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(12) Find the mean deviation of  15, 18, 20, 10, 12.

Soln.  \overline{x} =   \frac{\sum x_{i}}{n}  =   \frac{15+18+20+10+12}{5}   =   15

\delta_{\overline{x}} =  \frac{\sum \left |x_{i}-\overline{x} \right |}{n}   =   \frac{0 + 3+5+5+3}{5}  =  3.2

(13) Find the mean deviation of the frequency distribution :

Class 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50
Frequency 5 7 16 11 1

Soln. We have :

Classes Mid-value x_{i} f_{i} f_{i}x_{i} \left |x_{i}- \overline{x} \right | f_{i} \left |x_{i}- \overline{x} \right |
0 – 10 5 5 25 19 95
10 – 20 15 7 105 9 63
20 – 30 25 16 400 1 16
30 – 40 35 11 385 11 121
40 – 50 45 1 45 21 21
n  =  40 \sum f_{i}x_{i}=  960 316

\therefore \overline{x}= \frac{\sum f_{i}x_{i}}{n}= \frac{960 }{40}= 24

\therefore \delta_{\overline{x}} = \frac{\sum f_{i} \left |x_{i}- \overline{x} \right |}{n} = \frac{316}{40} = 7.9

(14) Find the variance of  35, 50, 32, 26, 37.

Soln. \overline{x}  =  \frac{35+50+32+26+37}{5}  =  \frac{180}{5}  =  36.

\sigma = \sqrt{\frac{\sum \left (x_{i}- \overline{x} \right )^{2}}{n}}

Variance  =  \frac{\sum \left (x_{i}-\overline{x} \right )^{2}}{n}  =  \frac{1^{2}+14^{2}+4^{2}+10^{2}+1^{2}}{5}  =  \frac{314}{5}  =  62.8

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