Understanding Activity Variance in PERT

In project management, specifically using the Program Evaluation and Review Technique (PERT), activity times are often uncertain. PERT uses three time estimates to account for this uncertainty: optimistic time, most likely time, and pessimistic time.

• Optimistic Time (\(t_o\)): The shortest possible time in which the activity can be completed. This assumes everything goes exceptionally well.
• Most Likely Time (\(t_m\)): The most probable time required to complete the activity, assuming normal conditions.
• Pessimistic Time (\(t_p\)): The longest possible time required to complete the activity. This assumes unfavorable conditions and potential problems.

PERT uses these three estimates to calculate the expected time and the variance of an activity's duration. The variance helps in understanding the range of possible completion times and assessing the risk associated with the activity duration.

Calculating Standard Deviation in PERT

The standard deviation (\(\sigma\)) of an activity time in PERT is calculated using the optimistic and pessimistic times. It represents the spread or dispersion of the activity duration around its expected time.

The formula for standard deviation is:

\(\sigma = \frac{t_p - t_o}{6}\)

This formula is based on the assumption that activity durations follow a Beta probability distribution, where the range between the pessimistic and optimistic times \((t_p - t_o)\) approximates six standard deviations.

Determining Activity Variance in PERT

The variance (\(\sigma^2\)) of an activity is the square of its standard deviation. Variance provides a measure of the uncertainty in the activity's duration. A higher variance indicates greater uncertainty.

To find the variance, we square the formula for standard deviation:

\(\text{Variance} = \sigma^2 = \left( \frac{t_p - t_o}{6} \right)^2\)

Analyzing the Options

Let's look at the given options based on our understanding of the variance formula:

1. \({\left( {\frac{{{t_p} - {t_o}}}{6}} \right)^{1/2}}\): This is the square root of the standard deviation, which is incorrect for variance. Variance is the square of the standard deviation.
2. \({\left( {\frac{{{t_p} - {t_o}}}{{12}}} \right)^2}\): This formula uses 12 in the denominator instead of 6, which is incorrect for the standard PERT calculation.
3. \({\left( {\frac{{{t_p} - {t_o}}}{6}} \right)^2}\): This formula correctly represents the square of the standard deviation \(\left( \frac{t_p - t_o}{6} \right)\). This matches the standard formula for variance in PERT.
4. \({\left( {\frac{{{t_p} - {t_o}}}{{12}}} \right)^{1/2}}\): This formula uses 12 in the denominator and takes the square root, both of which are incorrect for calculating variance in PERT.

Based on the standard PERT methodology, the variance of an activity is calculated by squaring the difference between the pessimistic and optimistic times, divided by 6.

Thus, the correct expression for the variance of an activity is \({\left( {\frac{{{t_p} - {t_o}}}{6}} \right)^2}\).