Chapter 12.4, Problem 45SR

To determine

To find: the mean, variance and standard deviation of the given set.

Answer to Problem 45SR

The mean is 26, the variance is 187.4 and the standard deviation is 13.7.

Explanation of Solution

Given information:

The given data:

  $20,30,10,40,20,12,50$

Calculation:

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

  $\begin{array}{l}\overline{\text{X}}=\frac{20+30+10+40+20+12+50}{7}\\ =\frac{182}{7}\\ =26\end{array}$

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

  $\begin{array}{l}{\sigma }_{}^2=\frac{{\left(20-26\right)}^2+{\left(30-26\right)}^2+{\left(10-26\right)}^2+{\left(40-26\right)}^2+{\left(20-26\right)}^2+{\left(12-26\right)}^2+{\left(50-26\right)}^2}{6}\\ =\frac{{\left(-6\right)}^2+{\left(4\right)}^2+{\left(-16\right)}^2+{\left(14\right)}^2+{\left(-6\right)}^2+{\left(-14\right)}^2+{\left(24\right)}^2}{7}\\ =\frac{36+16+256+196+36+196+576}{7}\\ =\frac{1312}{7}\\ =187.4\end{array}$

Standard deviation is the square of the variance. To find standard deviation,

  $\begin{array}{l}{\sigma }_{}^2=\frac{1312}{7}\\ \sqrt{{\sigma }_{}^2}=\sqrt{\frac{1312}{7}}\\ \sigma \approx 13.7\left[\text{Use a calculator}\text{.}\right]\end{array}$

Hence, the mean is 26, the variance is 187.4 and the standard deviation is about 13.7.