Variance , Standard deviation & Coefficient of Variance

Hi folks! Today we will learn about the variance (σ²), Standard deviation (σ) and coefficient of variance. Objective of all the three terms is to determine the variation and to compare with different data set These tools are used in Six Sigma and other TQM initiatives.

Let us understand with some examples. Suppose ABC is a plastic water bottle manufacturing company.

Operator is operating the machine and manufacturing of bottles is going on. Quality inspector comes and examines the weight of individual bottle and observes the following:-

Bottle	Weight in gm
1	120
2	130
3	115
4	125
5	130
6	125

we need to find mean weight of above data set .  

Mean weight =  (120+130+115+125+130+125)/6 = 125 gm.

Bottle	Weight in gm	Mean wt.	Difference from mean	Difference
1	120	125	120-125	-5
2	130	125	130-125	5
3	115	125	115-125	-10
4	125	125	125-125	0
5	130	125	130-125	5
6	125	125	125-125	0

Now square the difference and sum it.

Bottle	Weight in gm	Mean wt.	Difference from mean	Difference	Square of difference
1	120	125	120-125	-5	25
2	130	125	130-125	5	25
3	115	125	115-125	-10	100
4	125	125	125-125	0	0
5	130	125	130-125	5	25
6	125	125	125-125	0	0
				Sum	175

Again divide with the Nos of date set, which is  6 in this case, thus variance will be = 175/6 = 29.2

Bottle	Weight in gm	Mean wt.	Difference from mean	Difference	Square of difference	Variance(σ2 )
1	120	125	120-125	-5	25	= 175/6 =29.2
2	130	125	130-125	5	25
3	115	125	115-125	-10	100
4	125	125	125-125	0	0
5	130	125	130-125	5	25
6	125	125	125-125	0	0
				Sum	175

Standard Deviation = Square root of variance = Variance^.5

Bottle	Weight in gm	Mean wt.	Difference from mean	Difference	Square of difference	Variance(σ2 )	Standard Deviation (σ )
1	120	125	120-125	-5	25	= 175/6 =29.2	= Square root of variance   = 29.2^.5   = 5.4
2	130	125	130-125	5	25
3	115	125	115-125	-10	100
4	125	125	125-125	0	0
5	130	125	130-125	5	25
6	125	125	125-125	0	0
				Sum	175

Note – square root of m can be written in this form.       √m = m^.5    

Let us understand with some more example.

Which one has higher variation – Red or Blue?

Calculate the variation of above date set.

Range = Maximum – Minimum

In case of blue line = Max = 5, Min = 1

Range = 5-1= 4

In case of Red line = Max = 5, Min= 1

Range = 5-1 = 4

If we calculate the variation by Range, it is observed that in both the cases Range is the same.  Therefore we cannot actually differentiate.

But if we calculate the standard deviation, then we can  see variation and blue data set has higher variation.

Coefficient of Variation = (Standard deviation/ Mean) *100

Bottle	Weight in gm	Mean wt.	Difference from mean	Difference	Square of difference	Variance (σ2 )	Standard Deviation (σ )	Coefficient of Variation
1	120	125	120-125	-5	25	= 175/6 =29.2	= Square root of variance   = 29.2^.5   = 5.4	=(5.4/125)*100 = 4.32
2	130	125	130-125	5	25
3	115	125	115-125	-10	100
4	125	125	125-125	0	0
5	130	125	130-125	5	25
6	125	125	125-125	0	0
				Sum	175

In the following chart, which one has Higher variation – Red or Blue?

If we calculate the Coefficient of Variation in this case then healthcare has higher variation.