Chapter 12.3, Problem 17PPS

To determine

To calculate : the standard deviation of the given data to the nearest tenth.

Answer to Problem 17PPS

The standard deviation of the given data is $\underset{\_}{5.0}$ .

Explanation of Solution

Given information :

Consider the information given in the question:

The number of hours the girls on the tennis team work at part-time jobs each week:

  $10,12,0,6,9,15,12,10,11,20$

Formula used :

Step1: Find the mean, $\overline{x}$

Step2: Find the square of the difference between each value in the set of data and the mean. Then sum the squares, and divide by the number of values in the set of data. The result is the variance.

Step3: Take the square root of the variance to find the standard deviation.

Calculation :

As per problem,

The data set: $10,12,0,6,9,15,12,10,11,20$

The number of values in the data set $\text{=10}$

Step1: To find the mean, add the hours and then divide by the number of values in the data set.

  $\begin{array}{l}\overline{x}=\frac{10+12+0+6+9+15+12+10+11+20}{10}\\ \overline{x}=\frac{105}{10}=10.5\end{array}$

Step2: To find the variance, square the difference between each value and the mean. Then add the squares, and divide by the number of values.

  $\begin{array}{}$ {\sigma }^2=[{(10.5-10)}^2+{(10.5-12\text{)}}^2\text{+(}10.5-0{\text{)}}^2\text{+(10}\text{.5}-6{\text{)}}^2\text{+(10}\text{.5}-{\text{9)}}^2\text{+}$$ \text{ (}10.5-15{\text{)}}^2\text{+(}10.5-12{\text{)}}^2\text{+(}10.5-10{\text{)}}^2\text{+(}10.5-11{\text{)}}^2\text{+(}10.5-20{\text{)}}^2]÷10$$ =[{(0.5)}^2+{(-1.5)}^2+{(10.5)}^2+{(4.5)}^2+{(1.5)}^2+$$ {(-4.5)}^2+{(-1.5)}^2+{(0.5)}^2+{(-0.5)}^2+{(-9.5)}^2]÷10$$ =\frac{0.25+2.25+110.25+20.25+2.25+20.25+2.25+0.25+0.25+90.25}{10}$$ =\frac{248.5}{10}=24.85$\end{array}$

Step 3: the standard deviation is the square root of the variance

  $\begin{array}{l}{\sigma }^2=24.85\\ \sqrt{{\sigma }^2}=\sqrt{24.85}\\ \sigma \approx 5.0\end{array}$