PMP Sigma (σ) factor

 

standard_deviation

While preparing for the PMP exam, there are many terms that you come across and one of them is Standard Deviation. How many of us would just like to look at it as just a formula to be applied and be done with! But actually, it’s much better to understand the reasoning behind the same so that you are better able to apply the formulas.

So what is Standard Deviation?

As per Wiki: In statistics, the standard deviation (SD, also represented by the Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.

And the formula for SD is not as simple as we see in PMP study references. But for now we will keep it simple and not jump into the deep ocean of statistics.

Before we go into SD, let’s look at the technique that’s common in PMP for estimation –

The Three Point Estimation

The Three Point Estimate technique is used for time as well as cost estimation calculations. The value that’s arrived at from these calculations provide a better range of estimates.

The Triangular/Straight line Distribution:

E = (O + M + P) / 3  ,

where: E–>Expected value, O–> Optimistic value, M–> Most likely value, P–> Pessimistic value

This formula is a simple average of the three estimates. When plotted on a chart, this would result in a sharp peak, so the name Triangular Distribution.

Beta Distribution (PERT)/Weighted Average:

E = (O + 4M + P) / 6 ,

where: E–>Expected value, O–> Optimistic value, M–> Most likely value, P–> Pessimistic value

As the name also goes this is a weighted average calculation and more weight is given to the most likely value. If this was plotted on a chart, it would result in a more uniform or bell shaped curve, called a Normal Distribution.

Normally the exam would state which formula to use either the PERT or triangular.

NOTE: In case it’s not given clearly in the exam then how do you know which formula to use? – note that for additional weight to be given to Most likely, you would need to be sure that this is almost accurate or provided by experts. So, PERT would normally be used where you have more confidence on the accuracy of your data based on past experience and data to base your estimates on or experts are providing the estimates.

And if there is not much experience or data available then you would be better off using the Triangular Distribution.

In the exam, in case the question does not specify which method to be considered and there is not enough data then solve first using PERT formula and stop if the answer matches the option, else repeat with triangular distribution formula (hopefully in this case only one answer should match).

Now, that you got the estimate value how do you arrive at the range? That’s by calculating the standard deviation and adding/subtracting the σ from the estimate value “E” obtained in the above calculation.

For the PMP exam normally the PERT is used and the SD is calculated using the simple formula for PERT. (Please note this is the SD formula for PERT estimates and NOT Triangular, triangular distribution has a more complex SD formula and hopefully not needed for PMP)

SD or (σ) = (P-O)/6

So this would give you 1σ value. Please note the below standard values for SD.

±1σ = 68.27%

±2σ = 95.45%

±3σ = 99.73%

In the exam if you calculate SD using the above formula then it says that 68% of the values fall within that range. If the exam asks for 95.45% or 99.73%  then do note that they are 2σ or 3σ and you would need to multiply your answer with the correct value (2 for 95% and 3 for 99%).

So the estimate range (R) can be calculated as

R = E  ± (n * σ)

where

E = Estimate value calculated using PERT

σ= The standard deviation calculated using (P-O)/6

n=1 if Probability is 68.27%

n=2 if Probability is 95.45%

n=3 if Probability is 99.73%

Note: Larger the standard deviation is, the  lesser will be the confidence on the estimates provided, as it would mean there is a huge range between your optimistic and pessimistic estimates.

That’s the simple calculation for SD.

Now if there is no PERT estimation done and it’s just estimate values then how do you arrive at the SD. Hopefully you won’t need this for the exam, but it is a good know-how.

In such case the SD is calculated as

SD = square root of Variance (from the formula that Variance = (SD)square)

What is variance? It is the average of the squared differences from the Mean and is calculated as follows:

Let’s say there are n values A1,A2, …An.

The mean for these would be

Mean = A1+A2+..+An/n

And the Variance is calculated as

Variance = ((VarA1)^2 + (VarA2)^2 + ..+(VarAn)^2)/n

VarA1 = A1 – Mean,

VarA2 = A2 – Mean and so on till

VarAn = An – Mean

Why are we squaring each of the variance and adding them? If we just add up the sum of the variances then the negatives might cancel the positives and you would not get a clear indicator. The same was tested with absolute values and proved to be erroneous, so the best method arrived at was to square the variances and average them out.

And once we have the Variance, SD is calculated as

Standard Deviation = Square root of the Variance

Example:

Let’s say we want to find the variance and standard deviation of the estimates provided by 5 people. The estimates were – 10, 14, 15, 16, 20

First the Mean is calculated as – Mean = (10+14+15+16+20)/5 = 15

Then for the Variance, the difference from mean is calculated for each

10 – 15 = -5

14 – 15 = -1

15 – 15 = 0

16 – 15 = 1

20 – 15 = 5

Variance = ((-5)^2 + (-1)^2 + (0)^2 + (1)^2 + (5)^2)/5 = 10.4

So variance is 10.4 and the SD is square root of variance

SD = 3.23

So this means that on average the estimates provided have a variance of 3.23

Hope this helps in clearing some points for SD.


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