This tutorial will explain the concept of standard deviation. Our discussion breaks down as follows:
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 to find the standard deviation of a sample looks like:
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- Standard Deviation of a Sample
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.
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- 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:
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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.
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EXAMPLE
These are the heights of the Chicago Bulls basketball team.
|
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
|
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:
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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.
5.67
2 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.
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.
- Variance of a Sample
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.
-
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.
-
- 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.
-
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 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.
-
If you have Excel or another spreadsheet program, there should be a standard deviation formula that you can use.
Interpreting standard deviation is important, but it can be difficult to calculate. For this reason, calculating this number is typically done using technology. It's a measure of how far we would expect a typical data point to be from the mean. Standard deviation is the square root of a value called variance, which is a little bit easier to calculate but not as useful as a standard deviation number. Since the standard deviation is based on the mean, the standard deviation should only be reported as the measure of spread when you're reporting the measure of center to be the mean. You shouldn't mix and match the standard deviation with the median or the interquartile range.
Good luck!
