How to calculate standard deviation

In this article we will learn how to calculate Standard Deviation by hand. Before that let’s see what standard deviation is in simple terms.

The word deviation itself says how the numbers are deviated or spread from the rest.

STANDARD DEVIATION is a number that is used to tell how measurements for a group are spread out/deviated from the mean/average.

It is denoted by the greek letter sigma. Its symbol is σ.

The formula we use for standard deviation depends on whether the data is being considered as a whole population or just a sample taken from the whole population.

·      

   I  If the data is considered as a whole population we divide the number of data points by N.

·      If the data is a sample from large population we divide the number of data points by n-1.

 

Formula for standard deviation: Population standard deviation

 

Formula for standard deviation: Sample standard deviation

Both these formula’s are the same except for the denominator wherein we divide one number less than the number of data points in sample deviation..

 Let’s break the formula into 5 steps based on the above formula. Let’s learn it through a simple example.

Example : Population standard deviation

Four students were comparing there marks scored in a class test: 6,3,2,1

Step 1: Find the mean µ

[Mean=sum of numbers/number of terms]

Mean µ = (6+3+2+1)/4      

             µ  = 3              

 Step 2: Find the square of the distance from each data point  to the mean | x - µ |

| x - µ |²

X	| x - µ |2
6	|6 – 3|2	32	9
3	|3 – 3|2	02	0
2	|2 – 3|2	12	1
1	|1 – 3|2	22	4

Step 3: Find the summation of | x - µ |²                                                    

∑| x - µ |²

= 9 + 0 + 1 + 4

= 14

Step 4 and 5: Find √ ∑| x - µ |² / N

= √14/ 4

= √ 3.5              

                                     

                                       

Example : Sample Standard Deviation

A sample of 4 kids were taken to see how many balls each had:

2,2,5,7

Step 1: Find the mean µ

[Mean=sum of numbers/number of terms]

Mean µ = (2+2+5+7)/4      

             µ  = 4             

Step 2: Find the square of the distance from each data point  to the mean | x - µ |

| x - µ |²

X	| x - µ |2
2	|2 – 4|2	22	4
2	|2 – 4|2	22	4
5	|5 – 4|2	12	1
7	|7 – 4|2	32	9

Step 3: Find the summation of | x - µ |²                                                    

∑| x - µ |²

= 4 + 4 + 1 + 9

= 18

Step 4 and 5: Find √ ∑| x - µ |² / N-1               

[Divide the sum one less than the number of data points]

= √18/ 4-1

= √ 18/3

=   √6 ≈2.45               

 

A trick: How to identify that the problem is sample deviation or population deviation. It depends on why you are calculating standard deviation. If the problem involves randomly selecting data points then it goes with sample deviation. Incase of population deviation you are calculating 100% data of the population involved.                

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