Question

Find the mean, variance, and standard deviation to the nearest tenth for each set of data.

$\mathbf{10}\mathbf{,}\mathbf{10}\mathbf{,}\mathbf{12}\mathbf{,}\mathbf{14}$

Short answer

The mean, variance and standard variation of the given data set is$\mathbf{11}\mathbf{.}\mathbf{5}\mathbf{,}\mathbf{2}\mathbf{.}\mathbf{75}$ and$\mathbf{1}\mathbf{.}\mathbf{658}$ respectively.

Step-by-step solution

Step 1. Find the mean.

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

$\begin{array}{l}\overline{x}=\frac{10+10+12+14}{4}\\ =\frac{46}{4}\\ =11.5\end{array}$

Step 2. Find the variance.

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

$\begin{array}{l}{\sigma }^2=\frac{{(10-11.5)}^2+{(10-11.5)}^2+{(12-11.5)}^2+{(14-11.5)}^2}{4}\\ =\frac{{(-1.5)}^2+{(-1.5)}^2+{\left(0.5\right)}^2+{\left(2.5\right)}^2}{4}\\ =\frac{2.25+2.25+0.25+6.25}{4}\\ =\frac{11}{4}\\ =2.75\end{array}$

Step 3. Find the standard deviation.

To find the standard deviation, take square root of the variance.

$\begin{array}{c}{\sigma }^2=2.75\\ \sigma =\sqrt{2.75}\\ \approx 1.658\end{array}$

Therefore, the mean, variance and standard variation of the given data set is$11.5,2.75$ and$1.685$ respectively.