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,

  $SD=\sqrt{\frac{\begin{array}{}$ {(10-10.5)}^2+{(12-10.5)}^2+{(0-10.5)}^2+{(6-10.5)}^2+{(9-10.5)}^2+{(15-10.5)}^2$$ +{(12-10.5)}^2+{(10-10.5)}^2+{(11-10.5)}^2+{(20-10.5)}^2$\end{array}}{10}}$

So,

  $SD=\sqrt{\frac{\begin{array}{}$ {(-0.5)}^2+{(1.5)}^2+{(10.5)}^2+{(-4.5)}^2+{(-1.5)}^2+{(5.5)}^2+{(2.5)}^2+{(-0.5)}^2+{(1.5)}^2$$ +{(9.5)}^2$\end{array}}{10}}$

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.