Analysis of Variance - Statistics

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Question

Calculate the standard deviation of the following data set:

$3, 7, 2, 9, 8, \textup{and }10$

Answer

The standard deviation is found by using the following formula:

$s=\sqrt{\frac{\sum \left | x-\bar{x}\right |^2}{n}}$

We will start by identifying the components of this equation:

$\sum=\textup{sum}$

$x=\textup{variable in the data set}$

$\bar{x}=\textup{mean of the data set}$

$n=\textup{total number of variables in the data set}$

Now, we can use this information to calculate the standard deviation.

First, we need to identify and $\bar{x}$ . There are six variables in the data set; therefore, we can write the following:

Next, we need to calculate the mean:

$\bar{x}=\frac{\sum x}{n}$

$\bar{x}=\frac{3+7+2+9+8+10}{6}$

$\bar{x}=\frac{39}{6}$

$\bar{x}=6.5$

Second, we need to find the square of the absolute of each variable minus the mean: $\left | x-\bar{x}\right |^2$

$\textup{For 3}=\left | 3-6.5\right |^2=12.25$

$\textup{For 7}=\left | 7-6.5\right |^2=0.25$

$\textup{For 2}=\left | 2-6.5\right |^2=20.25$

$\textup{For 9}=\left | 9-6.5\right |^2=6.25$

$\textup{For 8}=\left | 8-6.5\right |^2=2.25$

$\textup{For 10}=\left | 10-6.5\right |^2=12.25$

Last, let's substitute these values into the standard deviation formula and solve.

$s=\sqrt{\frac{\sum \left | x-\bar{x}\right |^2}{n}}$

$s=\sqrt{\frac{(12.25+0.25+20.25+6.25+2.25+12.25)}{6}}$

$s=\sqrt{\frac{53.5}{6}}$

Solve.

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