# PMP Exam Prep: Standard Deviation (SD)

Standard deviation, from a project management lens, is used most frequently in the manage and control quality processes. Project Managers seeking to enhance their skill set or who are preparing for the Project Management Institute (PMI)’s Project Management Professional (PMP)® certification exam, should know the basic formula for calculating Standard Deviation (SD), understand its role in establishing confidence intervals, and know how it can be used in schedule and cost processes.

**On this page:**

- Standard Deviation (SD) Defined
- Standard Deviation Formula
- Pessimistic Estimate
- Optimistic Estimate
- Standard Deviation, PERT, and Probablitiy
- Standard Deviation and PERT Formulas
- Examples
- Deductions from Standard Deviation
- Standard Deviation in Context of Project Management
- Standard Deviation in the PMP® Certification Exam
- PMP® Certification Exam Question Examples

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## Standard Deviation (SD) Defined

Math textbooks and websites provide the classic definition of standard deviation. In project management work, a simplified formula is used for what is sometimes referred to as “Standard deviation PMP” or the “Standard deviation formula PMP.” The mathematical component is the same, the difference is the values used for the calculation in a project context. Note the PMI.org online lexicon does not have a definition for Standard Deviation (SD) at this time, but it has been included in previous editions of the PMI’s *A* *Guide to Project Management Body of Knowledge* (*PMBOK*® *Guide*).

Mathematical Definition | Noted by the lower-case Greek letter Sigma (σ), it is the square root of the statistical variance and indicates the spread of distribution (curve)– Source: https://www.mathsisfun.com/data/standard-deviation.html |

Project-Focused Definition | A statistical concept that gives a measure of the ‘spread’ of the values of a random variable around the mean of a distribution, the more the variation, the more the uncertainty or risk in the process.By calculating the mean and standard deviation of the project duration estimate, one can calculate the probability of completing the project within a given duration. – Source: https://www.deepfriedbrainproject.com/2010/08/standard-deviation-project-estimates.html |

Project Managers may have questions about how the SD formula could be used in business settings. ”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.” Manufacturing provides a great example, specifically in clothing manufacturing. When determining how many of what size shirts to create, the standard deviation of a population (say female customers from Canada) can be used to determine the mean (average size) and the distance from it (variance) of other sizes. With that insight, the distribution of each size shirt to manufacture can be determined, in other words, how many shirts of each size to manufacture to meet expected needs.

## Standard Deviation Formula

The “standard deviation formula PMP” is straightforward math: (P – O) / 6. As noted below, P is for the pessimistic estimate (“worst-case”) and O for the optimistic estimate (“best-case”).

## Pessimistic Estimate

The “Pessimistic estimate” is represented as “P” in project management formulas, including SD.

Pessimistic Estimate (P) | Estimate for all unfavorable conditions with all negative risks occurring and no mitigation of negative risks |

It is the opposite of the Optimistic estimate in concept. The Pessimistic Estimate means it is the “worst-case” and thus longest duration, or highest cost, to complete the work.

## Optimistic Estimate

The “Optimistic estimate” is represented as “O” in project management formulas, including SD.

Optimistic Estimate (O) | Estimate for all favorable conditions with no risks or changes |

The Optimistic Estimate is the “best-case” and thus shortest duration, or lowest cost, to complete the work.

Values from the SD formula calculations can be graphed to help understand this concept. With graphing, there is a visual representation of the mean and the distance from it (variance). In project management, SD graphs are what is known as “normal curve” or “bell curve” given the even distribution of values.

“The mean shows the height of the curve, and the standard deviation determines the width of the curve. A narrow curve has a relatively low standard deviation. A flatter distribution has a relatively greater standard deviation.” | |

Source: https://www.dummies.com/careers/project-management/pmp-certification/basic-statistics-you-should-know-for-the-pmp-certification-exam/ |

Determining Standard Deviation involves obtaining the Pessimistic Estimate (P), the Optimistic Estimate (O), calculating the SD, graphing the data, and interpreting it to inform project decisions.

## Standard Deviation, PERT and Probability

The Program Evaluation and Review Technique (PERT) is used to find the estimated time for activities to be completed when there are many unknown factors. For the PMP® exam and PMI® purposes, PERT is defined as a technique used to estimate project duration through a weighted average of optimistic, pessimistic, and most likely activity durations when there is uncertainty with the individual activity estimates. When learning about PERT as part of PMP® exam prep, it is important to know it is one type of three-point estimating.

Project managers who calculate the PERT estimate (mean) and the standard deviation of the project duration estimate, can determine the probability of completing the project within a given duration. It is a probability of course, and the future is relatively unknown. SD and normal distribution fit into a probability with the basis of using the past to predict the future.

## Standard Deviation and PERT Formulas

A PERT estimate is a schedule network diagram, not a statistical distribution; this is often a confusing point given there are diagrams for both. But they are in fact two concepts that work together but are not interchangeable. Here are the values and formulas for SD and PERT:

## PERT Example

Here is a basic PERT example related to the duration of an activity.

*Find the time required to go from point A to point B.*

*Optimistic value (O) = 45 minutes**PERT = (O+P+4M)/6**Pessimistic value (P) = 225 minutes**PERT = (45+225+4×90)/6**Most Likely value (M) = 90 minutes**PERT = 105 minutes*

*It means there is a fair chance of completing the task (going from point A to point B) in 105 minutes.*

## Standard Deviation and PERT Example

In this SD and PERT example, the calculations have already been determined. The data has been graphed with a normal distribution.

*Using the three-point estimating technique, our well-trained project manager interviews the work-package owner or subject matter expert to obtain time or cost estimates. In return, the respondent provides their most optimistic (O), most pessimistic (P), and Most Likely (ML) estimates for work results. Source: **https://www.interfacett.com/blogs/three-point-estimates-in-six-sigma-and-pmi/*

## Standard Deviation Example Problems

From Project Management Academy content, here are examples of standard deviation PMP questions:

Note the problems assume the project manager’s knowledge of pessimistic, optimistic, PERT, and Standard Deviation.

**Studying for the PMP Exam?**

## Deductions from Standard Deviation

Standard deviation is easy from a formula perspective, but more challenging from a project data interpretation or application lens. Consider these points:

*low value for standard deviation indicates the data points tend to be close to the mean**high value for standard deviation indicates the data points are spread out over a wider range**concept of SD fits into probability and using the past to predict the future**concept of SD is most effective with large quantities of similar items**concept of SD is not very effective in settings with single or very unique items*

If unreliable data is going into the SD, the output will also be unreliable. Using software can make the graphing process easier, but if poor data is entered, the data analysis will suffer.

## Standard Deviation in Context of Project Management

Think of standard deviation as “the mean of the mean.” It can be used for quality, cost estimating, duration estimation, and risk questions on the PMP® certification exam. Use standard deviation to analyze data and inform project decisions.

In a manufacturing or factory setting, in which matching items are created in the same way, standard deviation is a powerful tool. If projects are in a creative setting (say an agency doing a social media campaign) with work products used for a single time, it is not prudent to try and determine standard deviation.

## Standard Deviation in the PMP® Certification Exam

It can be helpful to know these distribution populations from the *PMBOK*® *Guide*:

+ 1 σ | 68.3% of the data points fall within 1 SD | 34% on either side of the mean |

+ 2 σ | 95.5% of the data points fall within 2 SD | 34%+13.5% = 47.5% on either side of the mean |

+ 3 σ | 99.7% of the data points fall within 3 SD | 34%+13.5% + 2.5% = 49% on either side of the mean |

When mapping the standard deviation, know for a normal distribution, almost all values lie within 3 standard deviations of the mean.

## PMP® Certification Exam Question Examples

Question | A | B | C | D |

You are a project manager working on a newly assigned construction project. Your project has an optimistic estimate of 20 weeks, a most likely estimate of 25 weeks, and a pessimistic estimate of 38 weeks. What is the standard deviation of the estimate? | 26.3 | 13.8 | 3 | 7.5 |

Assuming a PERT weighted average computation, what is the probability of completing the project within plus or minus 3 standard deviations of the mean? | 68.26% | 99.73% | 95.44% | 75% |

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**Answers**

**C.**Standard Deviation = (Pessimistic – Optimistic) / 6; Standard Deviation = (38-20) / 6; Standard Deviation = 3**B.**Applying normal probability analysis (bell curve), the work will finish within plus or minus 3 standard deviations (SD) 99.73% of time. Work will finish within plus or minus 2 SD’s 95.44% of the time. Work will finish within plus or minus 1 SD 68.26% of the time. With each SD you add you are making the ranger wider, so that the probability of completing the work in that time frame becomes higher.

## Conclusion

Project managers working in regulated or manufacturing settings with large, consistent data will find the most value in the standard deviation tool. SD should be used in conjunction with estimating tools (like PERT) and recognized for what it does: provides insights into possible futures. The quality of data used in the calculations impacts the validity of the data analysis for the standard deviation. For PMP® certification exam prep work, project managers should know SD is used in quality processes, know what the SD formula is, and understand how SD can be used in certain project settings.