Chapter 13, Problem 11RE

(a)

To determine

Mean and standard deviation of the data set 10,11,11,13,13,14

Answer to Problem 11RE

Mean $=12$ , Standard deviation $=1.54919$

Explanation of Solution

Given:

The data set 10,11,11,13,13,14

Formula used:Mean $=\overline{x}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i}$

Standard deviation $=\sqrt{\frac{{\displaystyle \sum _{i=1}^n(x_i-\overline{x}}{)}^2}{n-1}}$

Where $n=$ number of data

  $x_i=$ individual value of data

  $\overline{x}=$ Mean

Calculation:

Here, $n=6$

Then, mean $=\overline{x}$

  $\begin{array}{l}=\frac{10+11+11+13+13+14}{6}\\ =\frac{72}{6}\\ =12\end{array}$

Standard Deviation $=\sqrt{\frac{{\displaystyle \sum _{i=1}^n(x_i-\overline{x}}{)}^2}{n-1}}$

  $\begin{array}{l}=\sqrt{\frac{{(10-12)}^2+{(11-12)}^2+{(11-12)}^2+{(13-12)}^{12}+{(13-12)}^2+{(14-12)}^2}{6-1}}\\ =\sqrt{\frac{{(-2)}^2+{(-1)}^2+{(-1)}^2+1^2+1^2+2^2}{5}}\end{array}$

  $\begin{array}{l}=\sqrt{\frac{4+1+1+1+1+4}{5}}\\ =\sqrt{\frac{12}{5}}\\ =\sqrt{2.4}\\ =1.54919\end{array}$

Conclusion:

Thus, Mean $=12$ , Standard deviation $=1.54919$

(b)

To determine

Mean and standard deviation of the data set 7,8,8,10,10,11

Answer to Problem 11RE

Mean $=9$ , Standard deviation $=1.54919$

Explanation of Solution

Given:

The data set 7,8,8,10,10,11

Formula used:Mean $=\overline{x}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i}$

Standard deviation $=\sqrt{\frac{{\displaystyle \sum _{i=1}^n(x_i-\overline{x}}{)}^2}{n-1}}$

Where $n=$ number of data

  $x_i=$ individual value of data

  $\overline{x}=$ Mean

Calculation:

Here, $n=6$

Then, mean $=\overline{x}$

  $\begin{array}{l}=\frac{7+8+8+10+10+11}{6}\\ =\frac{54}{6}\\ =9\end{array}$

Standard Deviation $=\sqrt{\frac{{\displaystyle \sum _{i=1}^n(x_i-\overline{x}}{)}^2}{n-1}}$

  $\begin{array}{l}=\sqrt{\frac{{(7-9)}^2+{(8-9)}^2+{(8-9)}^2+{(10-9)}^{12}+{(10-9)}^2+{(11-9)}^2}{6-1}}\\ =\sqrt{\frac{{(-2)}^2+{(-1)}^2+{(-1)}^2+1^2+1^2+2^2}{5}}\end{array}$

  $\begin{array}{l}=\sqrt{\frac{4+1+1+1+1+4}{5}}\\ =\sqrt{\frac{12}{5}}\\ =\sqrt{2.4}\\ =1.54919\end{array}$

Conclusion:

Thus, Mean $=9$ , Standard deviation $=1.54919$