How to Calculate Standard Deviation

In statistics, the standard deviation of the given set of data is the square root of its variance, which is used to show the variation from the mean. Low standard deviation suggests that the data points are close to the mean, while high standard deviation means that the data points are spread over a large range. Though it is quite simple, not many people are aware about how to calculate standard deviation. Given below is the step by step process of calculating standard deviation which will give you a rough idea about how to solve such problems.

How to Calculate Standard Deviation: Methods

Step #1: The foremost thing to do when calculating standard deviation is to determine the mean of the given data. The mean, also referred to as the arithmetic mean, is basically the sum of the given numbers divided by the quantity of numbers in the list.

Step #2: The second step to calculate standard deviation is to subtract the arithmetic mean you get from each number. After executing this step you will get a list of deviations. (You will also get negative numbers after executing this step, which is perfectly normal, as you would be subtracting higher values from lower values).

Step #3: As the third step, you will have to square the list of deviations you get, and add each of these squares to get the total sum, and divide that sum by a number one less than the actual quantity in the list. For e.g if the actual quantity was 5, you have to divide by 4, and if the actual quantity was 8 you have to divide by 7.

Step #4: In the fourth and the final step, you will have to calculate the square root of the sum you get after dividing the squared deviations with a number one less than the actual quantity. This square root that you will get will be the standard deviation. Read more on how to find standard deviation.

How to Calculate Standard Deviation: An Example

If that was a bit confusing, here is an example of how to calculate average standard deviation which will make things easier for you to understand.

Execution of Step #1: If the numbers listed in a given set of data are 10, 20, 25, 40, 65 and 80, its arithmetic mean will be calculated by dividing 240 (i.e. the sum of these numbers) by 6 (i.e. the quantity of items in the list), and hence the answer will be 40.

Execution of Step #2: According to the second step, we will have to subtract the mean. The list of deviations we get after subtracting the mean will be -30 (10 - 40), -20 (20 - 40), -15 (25 - 40), 0 (40 - 40), 15 (65 - 40) and 40 (80 - 40).

Execution of Step #3: Going by the third step we will have to square the deviations, and the results we get after doing it will be 900, 400, 225, 0, 225 and 1600. The sum we get after adding all these squares is 3350. The total numbers of items taken initially were 6, so we will have to divide the number by 5, i.e. 3350 divided by 5, and the answer is 670.

Execution of Step #4: According to the final step, the square root of the sum you get in the third step is the standard deviation. The sum we get in this example is 670, and therefore the square root of 670, i.e. 25.88 will be the standard deviation.

This was the brief explanation of how to calculate standard deviation of a given set of data. Today, the practical use of standard deviation has spread to various other fields including finance, sports and weather, and therefore having the basic knowledge pertaining to calculation of average standard deviation is always an advantage.