Chapter 10.3, Problem 9PPS

To determine

To calculate standard deviation.

Answer to Problem 9PPS

Standard deviation is 4.34.

Explanation of Solution

Given information: Data of Samantha and her friend’s charge per hour,

$8.00, $8.50, $9.00, $7.50, $15.00, $8.25, $8.75

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

Calculation:

Re-arranging the values in an ascending order,

$7.50, $8.00, $8.50, $8.25, $8.75, $9.00, $15.00

For Mean,

Substituting the values in mean formula,

  $\overline{x}=\frac{7.50+8+8.25+8.50+8.75+9+15}{7}$

So,

  $\overline{x}=\frac{65}{7}$

On solving,

  $\overline{x}=9.28$

Hence, Mean is 9.28

For standard deviation,

Substituting the values in standard deviation formula,

  $SD=\sqrt{\frac{\begin{array}{}$ {(7.5-9.28)}^2+{(8-9.28)}^2+{(8.25-9.28)}^2+{(8.5-9.28)}^2+{(8.75-9.28)}^2+{(9-9.28)}^2$$ +{(15-9.28)}^2$\end{array}}{7}}$

So,

  $SD=\sqrt{\frac{{(-9.78)}^2+{(-1.28)}^2+{(-1.03)}^2+{(-0.78)}^2+{(-0.53)}^2+{(-0.28)}^2+{(5.72)}^2}{7}}$

On solving,

  $SD=\sqrt{\frac{132.03}{7}}$

So,

  $SD=\sqrt{18.86}$

On removing the under-root,

  $SD=4.34$

Hence, the standard deviation is 4.34.

The standard deviation of the given data set is low, that means, the points are closer to mean.