Question

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

$\mathbf{22}\mathbf{,}\mathbf{25}\mathbf{,}\mathbf{27}\mathbf{,}\mathbf{30}$

Short answer

The mean, variance and standard variation of the given data set is$\mathbf{26}\mathbf{,}\mathbf{4}\mathbf{.}\mathbf{5}$ and$\mathbf{2}\mathbf{.}\mathbf{12}$ 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.

$\begin{array}{l}\overline{x}=\frac{22+25+27+30}{4}\\ =\frac{104}{4}\\ =26\end{array}$

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.

$\begin{array}{l}{\sigma }^2=\frac{{(22-26)}^2+{(25-26)}^2+{(27-26)}^2+{(30-30)}^2}{4}\\ =\frac{{(-4)}^2+{(-1)}^2+{\left(1\right)}^2+{\left(0\right)}^2}{4}\\ =\frac{16+1+1}{4}\\ =\frac{18}{4}\\ =4.5\end{array}$

Step 3. Find the standard deviation.

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

$\begin{array}{l}{\sigma }^2=4.5\\ \sigma =\sqrt{4.5}\\ \approx 2.12\end{array}$

Therefore, the mean, variance and standard variation of the given data set is$26,4.5$ and$2.12$ respectively.