A common way to quantify the spread of a set of data is to use the sample standard deviation. Your calculator may have a built-in standard deviation button, which typically has an *s _{x}* on it. Sometimes it’s nice to know what your calculator is doing behind the scenes.

The steps below break down the formula for a standard deviation into a process. If you're ever asked to do a problem like this on a test, know that sometimes it’s easier to remember a step-by-step process rather than memorizing a formula.

After we look at the process, we will see how to use it to calculate a standard deviation.

## The Process

- Calculate the mean of your data set.
- Subtract the mean from each of the data values and list the differences.
- Square each of the differences from the previous step and make a list of the squares.
- In other words, multiply each number by itself.
- Be careful with negatives. A negative times a negative makes a positive.

- Add the squares from the previous step together.
- Subtract one from the number of data values you started with.
- Divide the sum from step four by the number from step five.
- Take the square root of the number from the previous step. This is the standard deviation.
- You may need to use a basic calculator to find the square root.
- Be sure to use significant figures when rounding your final answer.

## A Worked Example

Suppose you're given the data set 1, 2, 2, 4, 6. Work through each of the steps to find the standard deviation.

- Calculate the mean of your data set. The mean of the data is (1+2+2+4+6)/5 = 15/5 = 3.
- Subtract the mean from each of the data values and list the differences. Subtract 3 from each of the values 1, 2, 2, 4, 6

1-3 = -2

2-3 = -1

2-3 = -1

4-3 = 1

6-3 = 3

Your list of differences is -2, -1, -1, 1, 3 - Square each of the differences from the previous step and make a list of the squares.You need to square each of the numbers -2, -1, -1, 1, 3

Your list of differences is -2, -1, -1, 1, 3

(-2)^{2}= 4

(-1)^{2 }= 1

(-1)^{2 }= 1

1^{2 }= 1

3^{2 }= 9

Your list of squares is 4, 1, 1, 1, 9 - Add the squares from the previous step together. You need to add 4+1+1+1+9 = 16
- Subtract one from the number of data values you started with. You began this process (it may seem like a while ago) with five data values. One less than this is 5-1 = 4.
- Divide the sum from step four by the number from step five. The sum was 16, and the number from the previous step was 4. You divide these two numbers 16/4 = 4.
- Take the square root of the number from the previous step. This is the standard deviation. Your standard deviation is the square root of 4, which is 2.

Tip: It’s sometimes helpful to keep everything organized in a table, like the one shown below.

Mean Data Tables | ||
---|---|---|

Data | Data-Mean | (Data-Mean)^{2} |

1 | -2 | 4 |

2 | -1 | 1 |

2 | -1 | 1 |

4 | 1 | 1 |

6 | 3 | 9 |

We next add up all of the entries in the right column. This is the sum of the squared deviations. Next divide by one less than the number of data values. Finally, we take the square root of this quotient and we are done.