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Chad has taught Math for the last 9 years in Middle School. He has an M.S. in Instructional Technology and Elementary Education.

In this lesson, we will examine the meaning and process of calculating the standard deviation of a data set. Standard deviation can help to determine if the data set is a normal distribution.

What Is Standard Deviation?

Standard deviation is the measure of how closely all of the data in the data set surrounds the mean. The standard deviation helps to identify a normal distribution of data by comparing the distance of the average of each data point's variance to the mean. A normal distribution of data is represented when the majority of the data is found close to the average of the set.

Variance

To calculate the standard deviation, you will first need to find the variance. The variance of a data set is the average of each distance from the mean squared.

Let's look at an example data set and calculate the variance. In the annual fishing competition, there were 10 competitors who caught fish. Each participant weighed their total catch and recorded their weights. There were 10 competitors, and the total weights of their fish were 23 lbs., 37 lbs., 82 lbs., 49 lbs., 56 lbs., 70 lbs., 63 lbs., 72 lbs., 63 lbs. and 45 lbs.

The first step to calculate the variance is to find the mean of the data set. To calculate the mean, we will add 23 + 37 + 82+ 49 + 56 + 70 + 63 + 72 + 63 + 45, which equals 560. Then, divide 560 ÷ 10 = 56. The average weight of fish caught was 56 lbs.

The next step to calculate the variance is to subtract the mean from each value. The best way to set this up is in a table. Looking at the table, you can add a column for the mean to make the subtracting easier. To get these totals, we will now subtract:

23 - 56 = -33

37 - 56 = -19

82 - 56 = 26

49 - 56 = -7

56 - 56 = 0

70 - 56 = 14

63 - 56 = 7

72 - 56 = 16

63 - 56 = 7

45 - 56 = -11

Next, we will take each of these differences and square them. So we will calculate:

-33^2 = 1,089

-19^2 = 361

26^2 = 676

-7^2 = 49

0^2 = 0

14^2 = 196

7^2 = 49

16^2 = 256

7^2 = 49

-11^2 = 121

Finally, to calculate the variance, we will average each of these squared totals. To do so, add 1,089 + 361 + 676 + 49 + 0 + 196 + 49 + 256 + 49 + 121 = 2,846. Next, take the total, 2,846, and divide by the 10 data points. 2,846 ÷ 10 = 284.6, so the variance of this data set is 284.6.

How to Calculate the Standard Deviation

Calculating the standard deviation is simple once we've found the variance. To find the standard deviation, we will simply take the square root of the variance. In the previous example, our variance was 284.6. The square root of 284.6 is 16.9 when rounded to the tenths place. The standard deviation for the total weights of fish caught was 16.9 lbs.

Steps to Finding the Standard Deviation

As you can see, finding the standard deviation is not too difficult. There is a specific series of steps that must be carried out in order:

Find the mean of your data set.

Subtract the mean from each of the data points.

Take each of the differences and square them.

Find the variance, which is the average of the squared differences.

Calculate the square root of the variance, which is the standard deviation.

Example Part 1

Let's use these specific steps to work through another example. The owner of a local coffee shop wanted to conduct a survey to see how many cups of coffee each visitor purchased in a week. He decided to ask the next 6 customers who entered the store how many cups of coffee they had purchased. The first customer had purchased 7 cups of coffee, the second customer had purchased 12 cups of coffee, the third customer had purchased 6 cups of coffee, the fourth customer had purchased 4 cups of coffee, the fifth customer had purchased 7 cups of coffee, and the sixth customer had purchased 8 cups of coffee. The owner wants to know how many of the customers purchased within at least one standard deviation of the average number of cups purchased.

Remember that the easiest way to calculate the standard deviation is to use a table. You can see all of the information easily set up in a table.

The first step is to calculate the mean of our data. By adding the number of cups purchased by each of the different customers, we can see that there were 44 cups of coffee sold to the six customers. To find the mean we will take 44 cups and divide by 6 customers to equal 7.3 cups of coffee per customer. 44 ÷ 6 = 7.3. So the mean of our data set is 7.3 cups of coffee purchased per customer. So let's add this information to our table. As you see, we need to add a column with just our mean in it for each row.

The second step is to now subtract the mean from the number of cups purchased by each customer:

7 - 7.3 = -0.3

12 - 7.3 = 4.7

6 - 7.3 = -1.3

4 - 7.3 = -3.3

7 - 7.3 = -0.3

8 - 7.3 = 0.7

Record these values in your table.

The third step is to square each of the differences:

-0.3 ^2 = 0.09

4.7 ^2 = 22.09

-1.3 ^2 = 1.69

-3.3 ^2 = 10.89

-0.3 ^2 = 0.09

0.7 ^2 = 0.49

Record these values in your table.

The next step is to calculate the variance. To do so, we will average these squared differences. First, we will add these squared differences: 0.09 + 22.09 + 1.69 + 10.89 + 0.09 + 0.49 = 35.34. Then we take the 35.34 and divide by 6, which equals 5.89. The variance of this set of data is 5.89.

The last step to calculate the standard deviation is to find the square root of the variance. The square root of 5.89 is 2.4 to the nearest tenths place. So the standard deviation of cups of coffee purchased is 2.4.

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What Can Standard Deviation Tell Me About My Data?

The standard deviation can help us determine if our data is a normal distribution. In a normal distribution, most of your data will fall within one standard deviation of your mean. To calculate this range, you will add and subtract the standard deviation to the mean.

In the first example, the average weight of the fish caught was 56 lbs. The standard deviation was 16.9. So to find the range of where most of the information will be, we will add and subtract the standard deviation to the mean: 56 + 16.9 = 72.9 and 56 - 16.9 = 39.1. This tells us that the majority of data for this set will be between 72.9 and 39.1, which represents one standard deviation of the mean.

Let's look at the data for the problem again and see if the majority of the data is within one standard deviation. Looking at the number line, you can see that any number that falls between 39.1 and 72.9 would be within one standard deviation of our mean.

Let's compare the total weights of the fish to see which values fall within one standard deviation. The first participant caught 23 lbs. of fish, which would not lie within one standard deviation. The second participant caught 37 lbs. of fish, which would also not fall within one standard deviation. The third participant caught 82, which is greater than 72.9, meaning that it is not within one standard deviation. However, the rest of our weights (49lbs., 56 lbs., 70 lbs., 63 lbs., 72 lbs., 63 lbs. and 45 lbs.) all fall between the values 39.1 and 72.9, making them within one standard deviation.

To find the range for two standard deviations, you would add and subtract the standard deviation from the range. To calculate the range for two standard deviations you would take 39.1 - 16.9 = 22.2 and 72.9 + 16.9 = 89.8. The range for two standard deviations would be 22.2 through 89.8. You can see this easily on this number line.

Example Part 2

Let's look at the second example and see what range would represent one standard deviation from our mean. In the second example, our coffee shop owner wanted to find the average number of cups of coffee purchased by each customer in a week. The mean number of cups sold based on the survey was 7.3. The owners also found a standard deviation of 2.4.

To find the range for one standard deviation, we will add and subtract the standard deviation to the mean: 7.3 + 2.4 = 9.7 and 7.3 - 2.4 = 4.9. So the range of one standard deviation of data is from 4.9 to 9.7.

We can see all of the data points, except the 12 cups of coffee and 4 cups of coffee, fall within one standard deviation. To calculate the range for two standard deviations, we would add/subtract the standard deviation to our existing range: 9.7 + 2.4 = 12.1 and 4.9 - 2.4 = 2.5. So the range for two standard deviations is from 12.1 to 2.5.

Lesson Summary

Even though standard deviation can seem difficult, you should now have the tools to calculate the standard deviation for any set of data. Remember, standard deviation is the measure of how closely all of the data in the data set surrounds the mean. By following these five steps, you can easily calculate the standard deviation of any set of data:

Find the mean of your data set.

Subtract the mean from each of the data points.

Take each of the differences and square them.

Find the variance, which is the average of the squared differences.

Calculate the square root of the variance, which is the standard deviation.

To check to see if a number is within one standard deviation of your mean, you will need to add and subtract the standard deviation to the mean. This will give you a specific range.

Learning Outcome

Upon viewing this lesson, students will be confident in their ability to use the steps to find the standard deviation of any data set.

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