Chapter 12.4, Problem 46SR

To determine

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

Answer to Problem 46SR

The mean is 67, the variance is 559.7 and the standard deviation is 23.7.

Explanation of Solution

Given information:

The given data:

  $44,34,64,74,94,104,55$

Calculation:

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

  $\begin{array}{l}\overline{\text{X}}=\frac{44+34+64+74+94+104+55}{7}\\ =\frac{496}{7}\\ =67\end{array}$

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

  $\begin{array}{l}{\sigma }_{}^2=\frac{{\left(44-67\right)}^2+{\left(34-67\right)}^2+{\left(64-67\right)}^2+{\left(74-67\right)}^2+{\left(94-67\right)}^2+{\left(104-67\right)}^2+{\left(55-67\right)}^2}{6}\\ =\frac{{\left(-23\right)}^2+{\left(-33\right)}^2+{\left(-3\right)}^2+{\left(7\right)}^2+{\left(27\right)}^2+{\left(37\right)}^2+{\left(-12\right)}^2}{7}\\ =\frac{529+1089+9+49+729+1369+144}{7}\\ =\frac{3918}{7}\\ =559.7\end{array}$

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

  $\begin{array}{l}{\sigma }_{}^2=\frac{3918}{7}\\ \sqrt{{\sigma }_{}^2}=\sqrt{\frac{3918}{7}}\\ \sigma \approx 23.7\left[\text{Use a calculator}\text{.}\right]\end{array}$

Hence, the mean is 67, the variance is 559.7 and the standard deviation is 23.7.