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:
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.
When calculating the formula for standard deviation, you follow these steps:
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.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:
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.
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.
Standard deviation is a typical amount by which we would expect values to vary around the mean.
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.
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:
This spreadsheet formula finds the same number as the calculation you did by hand: 3.83.
Source: Adapted from Sophia tutorial by Jonathan Osters.