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?
![](https://i0.wp.com/readandgain.com/wp-content/uploads/2022/07/image-18.png?resize=628%2C329&ssl=1)
![](https://i0.wp.com/readandgain.com/wp-content/uploads/2022/07/image-19.png?resize=440%2C259&ssl=1)
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.
![](https://i0.wp.com/readandgain.com/wp-content/uploads/2022/07/image-20.png?resize=900%2C282&ssl=1)
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?
![](https://i0.wp.com/readandgain.com/wp-content/uploads/2022/07/image-21.png?resize=900%2C390&ssl=1)
If we calculate the Coefficient of Variation in this case then healthcare has higher variation.
![](https://i0.wp.com/readandgain.com/wp-content/uploads/2022/07/image-22.png?resize=900%2C381&ssl=1)
For usage, upper management has to be aware of need.