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

= [100+0+100+25+25]/5

= 250/5

= 50

Hence, the variance is 50

And, standard deviation = square root of variance

Standard deviation = square root of 50

= 7.07

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.

#### Important Note:

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.

### Summary

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.

mehrdad najmi says

hi fahad

your explantion is crystal clear.

please compare the computation as presented above to those performed in rita mulcahy,s PMP exam preperation:

P=pesimestic

O=optemistic

activty standard deviation=(P-O)/6

activity varience=((P-O)/6)^2

if we were to follow above calculation we had for your example:

P=170

O=150

then the varience that we would have come up with using above formula would be:

((170-150)/6)^2= 2.6

please let me know where am i doing wrong.

thanks and regards

mehrdad

Fahad Usmani says

In my example, there were five students and I have calculated the standard deviation for this population.

In your example, you are calculating the standard deviation for a single activity which has three possible estimate durations.

Ben says

In Rita’s book the formula (P-O)/6 is the beta activity standard deviation and you can only use it for the beta distribution estimate which is (P+4M+O)/6.

You cannot use it for anything else, not even for the triangular distribution which has its own standard deviation formula which I do not think is required for PMP exam.

For information triangular distrib std deviation is:

=SQRT([(O – P)^2 + (M – O) * (M – P)]/18)

http://www.super-business.net/Quantitative-Methods/1055.html

Sangeeta says

Hi Fahad –

Can you help me understand the difference between Scope Statement and Scope Baseline. When I see these 2 as choices to a question I get confused. Also, Verify Scope and Control Scope.

Thanks

Fahad Usmani says

Scope Baseline = Scope Statement + WBS + WBS Dictionary

Scope Statement = detail description of project scope + details of assumption and constraints + Acceptance Criteria, etc.

For verify scope and control scope, refer study notes.

Ajay Bagati says

Dear Fahad,

Though the significance of Standard Deviation is clear, I want to know the significance or utility of Variance. Is it needed just for finding the value of Standard Deviation?

Thanks in advance.

Fahad Usmani says

Variance tells you how far your data is spread out.

José says

Hello buddy.

Thanks for this wonderful and valuable site.

Regarding your great explanation about standard deviation.

How can I calculate when it includes 2 or 3 Sigma? How does it works? And what Sigmas exists?

Thanks and sorry if this is a basic question.

Fahad Usmani says

Sigma is a measure of quality which strives for near perfection.

In six sigma 99.99966% perfection is needed.

In three sigma 93.3% accuracy is required, and in two sigma 69% accuracy is desired.

khaled abdel baki says

Thanks Fahad, however for accuracy seek, in six sigma secure error up to 3.4 per million, in 3 sigma accuracy up to 99.73%, in 2 sigma up to 95.45%, and in 1 sigma is 68.27% (source wikipedia)

Fahad Usmani says

From where did you get these figures?

Here is the link for wikipedia:

http://en.wikipedia.org/wiki/Six_Sigma

S says

Hi Fahad, please refer to this article: http://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule

Fahad Usmani says

The first link is talking about the Six Sigma while the link provided by you is talking about the “68–95–99.7” rule.

Mike says

Fahad,

It may be a problem with my computer but I am unable to view the diagram that you refer to. Would you mind checking to see that the embedded link is still working?

Thanks,

Mike

Fahad Usmani says

Hope it is working now…

AchMil says

Dear Fahad,

Thanks for your explanations which are very helpful, even when compared to some of the real life examples provided in the study guides. The difference between Quality Assurance and Quality Control is finally conquered in my mind.Thanks, A

(quick question… are the sigma references accurate, as I think the rounding off at 1sigma is 68.38pct and 2sigma is 95.5 etc.)

Kim Tijerina says

I’m trying to better understand the term “prelimenary scope statement” Vs project scope statement. Is this the output term coming from Development of the Project Charter from the statement of work and business need? Thus being projected into the Scope mgmt plan?

Fahad Usmani says

Perliminary scope statement was more like a project charter, and these are explained in more detail in project scope statement, therefore, the term “preliminary scope statement” is no longer used in the PMBOK Guide.

Hope it helps.

Mayura says

Hi,

What is the difference between Beta and Triangular distribution.. how do we know which formula to use when?

Fahad Usmani says

Beta and triangular distributions are different. You can read about them here:

http://en.wikipedia.org/wiki/Beta_distribution

http://en.wikipedia.org/wiki/Triangular_distribution

Fahad

Sushil says

Hi Fahad,

I am not getting proper results using below values to calculate Variance and Standard Deviations.

This is due to square root for values less than 1.

H1 10

H2 10.50

H3 10.25

H4 9.5

H5 11

(Mean or Average) = 51.25/5 = 10.25

Variance = ((10-10.25)^2 + (10.50-10.25)^2 + (10.25-10.25)^2 + (9.5-10.25)^2 + (11-10.25)^2)/5

Variance = (0.0625+0.0625+0+0.5625+0.5625)/5=0.25

SD=SQRT(0.25)=0.5

Range = Mean +- SD = 9.75 to 10.75

So for variances less than 1, the calculations does not yields expected results..

laura says

hi I am still not clear on how to attribute the SD to the Sigma. Using the problem given the students are between 153cm and 167 cm. How do I get to wether it is 1 sigma or 6 sigma?

Fahad Usmani says

Six sigma is about quality. Follow below given link to read more on it:

https://pmstudycircle.com/2016/10/six-sigma/