# Practical Application: Calculating the Standard Deviation

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Knowing the standard deviation of a set of data is important as it provides a good basis for deciding whether or not a certain data point fits the rest of the data or not.

## Standard Deviation

When a teacher says that her students' test scores all follow a normal distribution, she means that the majority of her test scores fall within one standard deviation. This range of test scores changes for each class as it is based on the test scores for each class.

According to the lesson Calculating the Standard Deviation, the steps to calculate the standard deviation are these:

1. Find the mean or average of the data set.
2. Subtract the mean from each data point.
3. Square each of the differences.
4. Average the squares to find the variance.
5. Take the square root of the variance to find the standard deviation.

One standard deviation is found by adding and subtracting the standard deviation from the mean. Two standard deviations is found by adding and subtracting the standard deviation from the one standard deviation range.

## Questions to Consider

Practice finding the standard deviation with the following three scenarios. Use these questions as an aid when calculating the standard deviation.

• Did you square each of the differences?

After finding the difference of each data point from the average, make sure to square each difference.

• Did you perform an average calculation twice?

The first average calculation is of all the data points and the second average calculation is that of the squares of the differences.

• Are you taking the square root of the variance?

Make sure to get the variance and then take the square root of that to find the standard deviation.

## Test Scores

This first scenario is inside a classroom. Jen, the high school science teacher has just finished grading the chapter test. Find the standard deviation of Jen's science class with these results.

Test Scores
98
97
87
80
99
100
100
89
92
94

Following the steps, the calculation starts with finding the mean.

• (98 + 97 + 87 + 80 + 99 + 100 + 100 + 89 + 92 + 94) / 10 = 93.6

Then, this mean is subtracted from each data point.

 98 - 93.6 = 4.4 97 - 93.6 = 3.4 87 - 93.6 = -6.6 80 - 93.6 = -13.6 99 - 93.6 = 5.4 100 - 93.6 = 6.4 100 - 93.6 = 6.4 89 - 93.6 = -4.6 92 - 93.6 = -1.6 94 - 93.6 = 0.4

After this step, then these differences are squared.

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