Calculating Standard Deviation

1. Standard Deviation

Standard deviation is a measure of variation that we use quite often in statistics. Standard deviation measures spread. You will interpret the standard deviation as the typical amount that you would expect data to be within the mean.

The full name for standard deviation is "standard deviation from the mean." If you break that down, "standard" just means it's typical, "deviation" means that you expect it to be off from the mean, just by chance. So "standard deviation from the mean" states that the data will be away from the mean.

Here is what the formula looks like:

formula to know

Standard Deviation

As complicated as the formula looks, it's actually the one preferred most, provided that the distribution that you’re looking for is roughly symmetric and doesn’t have outliers. If the data is not symmetric or has outliers, you will use a different measure of spread, the interquartile range.

term to know

Standard Deviation

A typical amount by which we would expect a data point to differ from the mean. Typically, about half to two-thirds of the data points fall within one standard deviation of the mean.

2. Calculating Standard Deviation

When calculating the formula for standard deviation, you follow these steps:

step by step

Step 1: Subtract the mean from each value.
Step 2: Square those values, resulting in (x minus mean)².
Step 3: Use the sigma notation, which is the same as summation notation, to add these values.
Step 4: Divide that sum by (n minus 1).
Step 5: Take the square root.

EXAMPLE

These are the heights of the Chicago Bulls basketball team.

When considering standard deviation, each of these items is a Xᵢ (X sub 1) in the original formula. Let's follow the five steps to calculate the standard deviation:

step by step

Step 1: The first step is to subtract the mean, which in this case, means you have to first calculate the mean. A thorough explanation of how to calculate the mean is covered in another tutorial, so for today, know that the mean is around 78.33 inches. Therefore, subtract 78.33 from each of these height values.

84 minus 78.33 is 5.67, 81 minus 78.33 is 2.67, etc.

84	5.67
81	2.67
79	0.67
79	0.67
81	2.67
81	2.67
79	0.67
74	-4.33
79	0.67
71	-7.33
83	4.67
75	-3.33
81	2.67
74	-4.33
74	-4.33

Step 2: Square those values, resulting in (x minus mean)².

5.67² is 32.15, 2.67 ² is 7.13, etc.

84	5.67	32.15
81	2.67	7.13
79	0.67	0.45
79	0.67	0.45
81	2.67	7.13
81	2.67	7.13
79	0.67	0.45
74	-4.33	18.75
79	0.67	0.45
71	-7.33	53.73
83	4.67	21.81
75	-3.33	11.09
81	2.67	7.13
74	-4.33	18.75
74	-4.33	18.75

Step 3: Use this sigma notation, which is the same as summation notation, to add these values up. They sum up to 205.35.

84	5.67	32.15
81	2.67	7.13
79	0.67	0.45
79	0.67	0.45
81	2.67	7.13
81	2.67	7.13
79	0.67	0.45
74	-4.33	18.75
79	0.67	0.45
71	-7.33	53.73
83	4.67	21.81
75	-3.33	11.09
81	2.67	7.13
74	-4.33	18.75
74	-4.33	18.75
Sum		205.35

Step 4: Divide that sum by n minus 1. In this case, n is 15 because there were 15 players; therefore n minus 1 equals 14. Dividing our sum by 14 equals 14.67.

hint

If you stopped here, this 14.67 number would measure another kind of variation called "variance." Variance is equivalent to the square of the standard deviation. Variance isn’t used very often, mainly because it is still a squared value. The measurement for variance, in this case, would be 14.67 inches squared, versus inches.

formula to know

Variance

Step 5: The final step is to take the square root of that number, which equals 3.83.

Here is an overview of the entire calculation:

84	5.67	32.15
81	2.67	7.13
79	0.67	0.45
79	0.67	0.45
81	2.67	7.13
81	2.67	7.13
79	0.67	0.45
74	-4.33	18.75
79	0.67	0.45
71	-7.33	53.73
83	4.67	21.81
75	-3.33	11.09
81	2.67	7.13
74	-4.33	18.75
74	-4.33	18.75
Sum		205.35

For this data set, you would expect a good portion of the heights to be within 3.83 inches of the mean, 78.33.

big idea

Variance and standard deviation share an important relationship. The variance is the square of the standard deviation, and the standard deviation is the square root of the variance.

term to know

Variance

The square of standard deviation. While it has some uses in statistics, it is not a practical unit of measurement. It is calculated the same way as standard deviation, but without the square root.

3. Interpreting Standard Deviation

Standard deviation is a typical amount by which we would expect values to vary around the mean.

try it

Considering the original list of heights, which fall into one standard deviation of the mean, meaning their heights are either 3.83 above or below 78.33?

Height of Chicago Bulls Players
Omer Asik	84	✘
Carlos Boozer	81	✔
Ronnie Brewer	79	✔
Jimmy Butler	79	✔
Luol Deng	81	✔
Taj Gibson	81	✔
Richard Hamilton	79	✔
Mike James	74	✘
Kyle Korver	79	✔
John Lucas III	71	✘
Joakim Noah	83	✘
Derrick Rose	75	✔
Brian Scalabrine	81	✔
Marquis Teague	74	✘
C.J. Watson	74	✘

You'll notice about ⅔ of the players had heights in this range. That is how you interpret the standard deviation.

4. Using Technology

As you may have noticed, calculating the standard deviation can require many steps, which could result in calculation errors. Because of this, the standard deviation is almost always found on a calculator or a spreadsheet, or some kind of applet on the internet. Typically, it is not solved by hand.

If you're frustrated with the calculation you just practiced, you can use your calculator or a spreadsheet in the future.

Here is how to use spreadsheets to calculate standard deviation:

step by step

Step 1: Enter your list of data into the spreadsheet.

Step 2: For the formula, type “=stdev(”.

Step 3: Highlight the list that you want to find the standard deviation for, close the parentheses, and hit "Enter."

This spreadsheet formula finds the same number as the calculation you did by hand: 3.83.

big idea

If you have Excel or another spreadsheet program, there should be a standard deviation formula that you can use.