Standard Deviation Formula

The standard deviation formula that you will use to find the standard deviation is shown below.

As you can see, x represents a set of numbers. For example, x could be {5, 6, 14, 1, 6, 10}.


The mean is the average of the set of numbers.

The symbol Σ refers to an addition.

n is the size of the list or set. For example, for the set above or {5, 6, 14, 1, 6, 10}, n = 6.

σ is the symbol used for standard deviation.
Standard deviation formula

Steps to follow when calculating the standard deviation


Step 1: Find the mean of the set of data (mean).

Step 2: Find the difference between each value and the mean (x - mean).

Step 3: Square each difference or each answer you got in step 2 (x - mean)2.

Step 4: Add all the answers you got in step 3 and then divide by the number of answers to get the average.

Σ(x - mean)2 / n

Step 5: Take the square root of the average found in step 4.

A good example that shows how to use the standard deviation formula to find the standard deviation.


Using the standard deviation formula, I will now show you how to get the standard deviation step by step.

Let S = {4, 6, 8, 2, 5}

Step 1: Find the mean of S = {4, 6, 8, 2, 5}

mean =
4 + 6 + 8 + 2 + 5 / 5

mean =
25 / 5
= 5

Step 2: Find the difference between each value in S = {4, 6, 8, 2, 5} and 5

4 - 5 = -1
6 - 5 = 1
8 - 5 = 3
2 - 5 = -3
5 - 5 = 0

Step 3: Square each difference or each answer you got in step 2

-12 = 1
12 = 1
32 = 9
-32 = 9
02 = 0

Step 4: Add 1, 1 9, 9, and 0. Then divide by 5

1 + 1 + 9 + 9 + 0 / 5

20 / 5
= 4

Step 5: Take the square root of the average found in step 4. That is the standard deviation


Standard deviation = σ =   4 = 2


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