Standard Deviation Formula

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

Standard deviation formula

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.

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.

An 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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