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

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.

#### Example

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

= 50

Hence, the variance is 50.

Standard Deviation = Square Root of Variance

Standard Deviation = Square Root of 50

= 7.07

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.

### Important Note

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.

### Summary

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.

hey there!

can you help me with the use of standard deviation of the project length for the managerial decisions?

Sorry Josephine, I can not.

Hi Fahad,

In your article under the topic “Important Note” you mean to say that if there is class of 500 students and we take 5 samples from this 500 students then to calculate variance we have to divide by N-1, ie 5-1=4. Is my understanding correct?

Right.

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..

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?

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

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

Hi,

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

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

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?

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.

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.)

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

Hope it is working now…

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.

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)

From where did you get these figures?

Here is the link for wikipedia:

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

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

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

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.

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.

Variance tells you how far your data is spread out.

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

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.

It’s Validate scope

Validate scope – Getting validation/acceptance of deliverables from stakeholders or customer post control quality process

Control scope – Monitoring and controlling so that deliverables are not out of scope by running analysis on work performance data

Right. It is validate scope in the PMBOK Guide 6th Edition.

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

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.

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)