A few days ago, a visitor to my blog asked me a question on standard deviation while she was reading about quality control and verify scope. She was having a problem calculating the standard deviation.
I responded to help her.
Many professionals ignore it as it involves mathematical calculations. This is an important concept and you should understand its significance. Once you understand the practical application of standard deviation and know the calculations, you will not forget it.
Standard deviation is the “Mean of the Mean.” It tells you how the data are spread.
The mean is the average of the given numbers.
Now, let’s see a mathematical example of this concept.
Your class has five students, and the height of each student is as follows:
First student = 150 cm
Second student = 160 cm
Third student = 170 cm
Fourth student = 165 cm
Fifth student = 155 cm
Calculate the standard deviation.
To calculate the standard deviation, you need the mean and variance.
Mean = (150 + 160 + 170 + 165 + 155) / 5
= 160 cm
To find the variance, subtract this ‘mean height’ from the height of each student, square it, add them together, and then take the average.
Variance = [(150 – 160)2 + (160 – 160)2 + (170 – 160)2 + (165 – 160)2 + (155 – 160)2] / 5
= [100 + 0 + 100 + 25 + 25] / 5
= 250 / 5
Hence, the variance is 50.
Standard Deviation = Square Root of Variance
Standard Deviation = Square Root of 50
Hence, the standard deviation is 7.07 cm.
You might wonder how useful this data is.
These data are essential because they give you the following information:
- The average height of students is 160 cm (mean).
- The height of most of the students varies from 152.93 cm (160 – 7.07) to 167.07 cm (160 + 7.07).
The image above shows the standard deviation for five students. The vertical lines show the height of each student (e.g., 150 cm, 160 cm, etc.). The blue line is the average (or mean) line, and the maroon lines represent the standard deviation.
You can see that the standard deviation lines are drawn above and below the average line, and the height of most of the students lies between these two maroon lines.
In other words, you can say that the height of most of the students is between 152.93 cm to 167.07 cm.
Let’s revise the whole procedure once again:
- Calculate the average height of the students.
- Subtract the average height from the height of each student and then square it.
- Add them together and take the average.
- Now take the square root.
I used population-based data in my example; by this, I mean there were only five students in the class.
However, if you select sample data, i.e. you choose a few random numbers from a large data pool, you would have to divide the variance by (N-1), where N is the number of samples. In other words, if there was a class with hundreds of students, and you select five, you would have to divide the variance by (5 – 1) or 4.
You may wonder why we are squaring a number if we are going to take the square root of it.
There is a reason for this calculation; if we add the difference, the positive and negative numbers cancel each other out.
Application of Standard Deviation
Standard deviation is used in analyzing data. It is a vital tool for industries, especially for clothing manufacturing.
Standard deviation provides information about what size is small, normal, medium, large, or extra-large. Based on the result, the manufacturer sets the size of pants, shirts, t-shirts, etc.
Standard deviation is a statistical analysis tool that helps industries have a general understanding of parameters for the whole population, just by analyzing a sample of data. Although this technique involves mathematical calculation, the concept is straightforward. Standard deviation tells you how your data are spread. Based on this information, you can develop and market your product.
How is the standard deviation used in your work? Please share your thoughts in the comments section.
Standard deviation is an essential concept from a PMP perspective. You may see a question from this topic on your exam.