Chapter 10.3, Problem 3CYU

To determine

To calculate standard deviation and interpret the standard deviation.

Answer to Problem 3CYU

Standard deviation is 5.14

Explanation of Solution

Given information:

Number of hours girls work part-time in a week,

10,12,0,6,9,15,12,10,11,20

Formula used: Mean is,

  $\overline{x}=\frac{{\displaystyle \sum x}}{n}$

Standard deviation is,

  $SD=\sqrt{\frac{{\displaystyle \sum }{(x-\overline{x})}^2}{n}}$

Where, SD= standard deviation,

x = value

  $\overline{x}$ = mean

N= total number of item

Calculation:

For mean,

Substituting the values,

  $\overline{x}=\frac{10+12+0+6+9+15+12+10+11+20}{10}$

So,

  $\overline{x}=\frac{105}{10}$

On solving,

  $\overline{x}=10.5$

Hence, mean is 10.5

For standard deviation

Substituting the values,

  $SD=\sqrt{\frac{\begin{array}{l}{(}^{10}+{(}^{12}+{(}^0+{(}^6+{(}^9+{(}^{15}\\ +{(}^{12}+{(}^{10}+{(}^{11}+{(}^{20}\end{array}}{10}}$

So,

  $SD=\sqrt{\frac{\begin{array}{l}{(}^{-}+{(}^{1.5}+{(}^{10.5}+{(}^{-}+{(}^{-}+{(}^{5.5}+{(}^{2.5}+{(}^{-}+{(}^{1.5}\\ +{(}^{9.5}\end{array}}{10}}$

So,

  $SD=\sqrt{\frac{0.25+2.25+110.25+20.25+2.25+30.25+6.25+0.25+2.25+90.25}{10}}$

On solving,

  $SD=\sqrt{\frac{264.5}{10}}$

So,

  $SD=\sqrt{26.45}$

On removing the under-root,

  $SD=5.14$

Hence, the standard deviation is 5.14

A low standard deviation indicates that the data points tend to be very close to the mean and a high standard deviation indicates that the data points are spread out over a large range of values.

In this case, the standard deviation is low, which shows, the data points tend to be very close to the mean.