A few days ago a visitor of my blog asked me a question regarding the Standard Deviation while she was reading the blog post about the quality control and verify scope. She was having problem with calculating the standard deviation and asking for my help.
It was a simple question and even I typed the reply. Although, I was fulfilling her quest but I knew that I was not giving her a fair reply.
To calculate the standard deviation, you have to go through many steps, and more importantly, you must know why you’re doing this. Once you know the practical application of it, you will develop an interest, and you will always remember it.
So, here is the blog post about the standard deviation.
Standard deviation is the “Mean of the Mean”; it tells you how the data are spread.
However, before moving to the standard deviation, we need to understand the Mean, and the Variance.
Since this is a mathematical concept, I believe it will be better to start directly from an example.
Okay, let’s get started…
Assuming in your class you have 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
Now we will calculate the Mean, Variance, and Standard Deviation.
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 all together, and then take the average.
Variance = [(150-160)2 + (160-160)2 +(170-160)2 + (165-160)2 + (155-160)2]/5
Hence, the variance is 50
And, standard deviation = square root of variance
Standard deviation = square root of 50
Hence, the standard deviation is 7.07 cm
Now, you might be thinking: What is the use of these data?
These data are very important, they give you following information.
- The average height of students is 160 cm (mean).
- The height of most of the students varies from 152.93 (160 – 7.07) to 167.07 (160 + 7.07).
The exhibit above shows the graph for standard deviation. 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 maroon lines are representing 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 is lies between these two maroon lines. In other words you can say that the height of most of the students is varies between 152.93 cm to 167.07 cm.
Let’s revise the whole procedure once again:
- Calculate the average height of all students.
- Then subtract the average height from the height of each student, and square it.
- Add all of them together and take the average.
- Take the square root.
In my example I used population based data; by this, I mean there were only five students in the class.
However, if you select a Sample Data, this means you select a few random numbers from a large data pool; in this case you would have to divide variance by (N-1), where N is the number of sample data. In other words, if there was a class of five hundred students in our example, you would have to divide Variance by (5 – 1); i.e. 4.
You also might be thinking that since we have taken the square of difference, and then take the square root of it, why are we squaring a number if we are going to take the square root of it?
There is a reason for this calculation – if we simply add the difference, positive and negative numbers will cancel each other out.
Application of Standard Deviation
Standard deviation is used frequently in analyzing data. It is a very important tool for industries, especially for the fabric manufacturing industry.
Standard deviation gives information about what size is small, normal, medium, large, or extra-large. Based on the result, the manufacturer would be able to fix the size of pants, shirts, t-shirts, etc.
Standard deviation is a very important concept. It is a statistical analysis tool and helps industries to come up with a general understanding about any parameter for the whole population just by analysing a sample of data. Although this technique involves mathematical calculation, the concept is very simple. Standard deviation tells you how your data are spread, and based on this information you can develop and market your product.
image credit => artur84 / FreeDigitalPhotos.net