## Introduction: Standard Deviation

Standard Deviation is a useful statistical measurement to determine where certain numerical values lie in a large group of numbers. The standard deviation is especially helpful during tests where students test scores are ranked among their peers to see how well their placements are. Some tests even give grades based on the standard deviation percentile rather than the test score.

## Step 1: Step 1: Determine the Mean

When given a set of number values to find the standard deviation, the first thing to do is find the mean of the given set of number values. The mean is represented by the Greek "mu" symbol (μ) and it will be helpful later.

To find the mean, add up all numerical values in a set and divide it by the total number of values in the set.

As an example, Mrs. Green's class of sixth graders are given an assignment to find the standard deviation of the length of their feet (in inches) in a large group. The class of 30 splits up into three groups of 10 students each. Group A's measurements of their feet (in inches) are given below:

8, 9, 7, 7, 11, 10, 8, 8, 12, 10

Find the standard deviation of Group A's feet sizes to share with the class.

First, the students need to find the sum of their measurements:

8 + 9 + 7 + 7 + 11 + 10 + 8 + 8 + 12 + 10 = 90

Next, the students find the mean by dividing the sum by the total number of measurements in the given set:

90 / 10 = 9

## Step 2: Step 2: Subtract the Mean From Each Value

Now that the mean has been found, the second step will commence. Each value in a given number set must subtract the mean to find a new value, and then square that value.

Continuing with the example of Mrs. Green's class, Group A will subtract the mean from each measurement and square the difference.

7 - 9 = -2 => Squaring -2 will result in a positive 4 since two negatives make a positive (-2 x -2) = 4 (Two measurements)

Continue this process for each measurement.

8 - 9 = (-1)^2 = 1 (Three measurements)

9 - 9 = (0)^2 = 0

10 - 9 = (1)^2 = 1 (Two measurements)

11 - 9 = (2)^2 = 4

12 - 9 = (3)^2 = 9

## Step 3: Step 3: Determine the Mean of the Squared Values

After finding the squared values of the differences between the mean and the measurements, find the mean of the squared values. The sigma symbol (Σ) represents continuous addition of a formula by the total number (n above symbol) of values in a given set. The n below the horizontal bar will divide the sum of the squared values by the total number of values.

In the example of Mrs. Green's class, Group A has found the squared values of their measurements. They will find the sum of the squared values:

1 + 0 + 4 + 4 + 4 + 1 + 1 + 1 + 9 + 1 = 26

Group A will then divide the sum of the squared values by the total number of measurements, which is 10, to find a new mean:

26 / 10 = 2.6

## Step 4: Step 4: Square Root the Mean

The new mean of the squared values has been found. All that is left is to find the standard deviation by finding the square root of the mean.

In the example of Mrs. Green's class, the students of Group A will square root the mean:

√ 2.6 = 1.61245155

The students will get a number with many digits behind the decimal place. Depending on how the teacher wants the students to round, the value can go to different decimal places. In the case of this example, Mrs. Green wants the value rounded to the nearest tenth:

1.61245155 = 1.6

Congratulations! You have found the standard deviation of the given set of measurements. The standard deviation can help find the different deviations away from the mean to determine where certain measurements fall in the total amount of values and locate any possible outliers.