Chapter 12.3, Problem 16PPS

To determine

Tocalculate : the standard deviation of the given data to the nearest tenth.

Answer to Problem 16PPS

The standard deviation of the given data is $\underset{\_}{1.6}$ .

Explanation of Solution

Given information :

Consider the selling price of each keychain given in the question:

  $\$3.25,\text{}\$\text{4}\text{.50,\$5}\text{.00,\$5}\text{.75,\$2}\text{.25,\$8}\text{.50,\$6}\text{.00,\$3}\text{.50,\$4}\text{.50,\$5}\text{.00}$

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: $\$3.25,\text{}\$\text{4}\text{.50,\$5}\text{.00,\$5}\text{.75,\$2}\text{.25,\$8}\text{.50,\$6}\text{.00,\$3}\text{.50,\$4}\text{.50,\$5}\text{.00}$

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

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

  $\begin{array}{l}\overline{x}=\frac{3.25+\text{4}\text{.50+5}\text{.00+5}\text{.75+2}\text{.25+8}\text{.50+6}\text{.00+3}\text{.50+4}\text{.50+5}\text{.00}}{10}\\ \overline{x}=\frac{48.25}{10}=4.825\end{array}$

Therefore the mean value price is $\$4.825$ .

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

  $\begin{array}{}$ {\sigma }^2=[{(4.825-3.25)}^2+{(4.825-\text{4}\text{.50)}}^2\text{+(4}\text{.825}-\text{5}{\text{.00)}}^2\text{+(4}\text{.825}-\text{5}{\text{.75)}}^2\text{+(4}\text{.825}-\text{2}{\text{.25)}}^2\text{+}$$ \text{ (4}\text{.825}-\text{8}{\text{.50)}}^2\text{+(4}\text{.825}-\text{6}{\text{.00)}}^2\text{+(4}\text{.825}-\text{3}{\text{.50)}}^2\text{+(4}\text{.825}-\text{4}{\text{.50)}}^2\text{+(4}\text{.825}-\text{\$5}{\text{.00)}}^2]÷10$$ =[{(1.575)}^2+{(0.325)}^2+{(-0.175)}^2+{(-0.925)}^2+{(2.575)}^2+$$ {(-3.675)}^2+{(-1.175)}^2+{(1.325)}^2+{(0.325)}^2+{(-0.175)}^2]÷10$$ =\frac{2.48+0.11+0.03+0.85+6.63+13.5+1.38+1.75+0.11+0.03}{10}$$ =\frac{26.87}{10}=2.687$\end{array}$

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

  $\begin{array}{l}{\sigma }^2=2.687\\ \sqrt{{\sigma }^2}=\sqrt{2.687}\\ \sigma \approx 1.6\end{array}$