Chapter 11.7, Problem 2LC

Solution

To find: The number of standard deviations from the mean that include all the data values given.

The data falls within 2 standard deviations of the mean.

Given information:

The given data set is 12,17,15,13,9,10,12,10,15,17.

Calculation:

The mean of the data is given by:

  $\begin{array}{l}\overline{x}=\frac{{\displaystyle \sum x_n}}{n}\\ \overline{x}=\frac{12+17+15+13+9+10+12+10+15+17}{10}\\ \overline{x}=\frac{130}{10}\\ \overline{x}=13\end{array}$

To determine the variance, the following table is used:

$x$	$\overline{x}$	$x-\overline{x}$	${\left(x-\overline{x}\right)}^2$
12	13	-1	1
17	13	4	16
15	13	2	4
13	13	0	0
9	13	-4	16
10	13	-3	9
12	13	-1	1
10	13	-3	9
15	13	2	4
17	13	4	16
		Sum	76

Hence,

  $\begin{array}{l}{\sigma }^2=\frac{{\displaystyle \sum {\left(x-\overline{x}\right)}^2}}{n}\\ {\sigma }^2=\frac{76}{10}\\ {\sigma }^2=7.6\end{array}$

The standard deviation is given by:

  $\begin{array}{l}\sigma =\sqrt{{\sigma }^2}\\ \sigma =\sqrt{7.6}\\ \sigma =2.76\end{array}$

The minimum and maximum values of the data are 9 and 17, respectively.

1 standard deviation below the mean: $13-2.76=10.24$

2 standard deviations below the mean: $13-2\left(2.76\right)=7.48$

Hence the minimum value falls within two standard deviations of the mean.

1 standard deviation above the mean: $13+2.76=15.76$

2 standard deviations above the mean: $13+2\left(2.76\right)=18.52$

Hence the maximum value falls within two standard deviations of the mean.

Conclusion:

The data falls within 2 standard deviations of the mean.