PERT distribution in Excel: formula, mean and when to use it

Guide · Updated October 2026

A PERT distribution turns a three-point estimate (min, most likely, max) into a beta distribution stretched from min to max, with mean (min + 4 × most likely + max) / 6: for a building’s structure estimated at 760, 820 and 1,010 (USD thousands), that mean is 841.7. Excel has no PERT function, but =BETA.INV(RAND(), α, β, min, max) draws from one.

Its shape parameters are α = 1 + 4 × (most likely − min) / (max − min) and β = 1 + 4 × (max − most likely) / (max − min). A triangular distribution with the same three numbers averages 863.3 and spreads wider.

What a PERT distribution is

A three-point estimate gives the lowest value you think possible, the most likely one and the highest. A PERT distribution turns these three numbers into a full range of outcomes: nothing below the min or above the max, a peak at the most likely value, and a smooth fall toward both ends, so values near the limits are possible but rare.

Mathematically it is a beta distribution, a family of distributions on a fixed interval whose shape is set by two positive numbers, α (alpha) and β (beta), stretched from the interval 0 to 1 onto min to max. PERT picks α and β so that the peak falls on the most likely value and the mean is (min + 4 × most likely + max) / 6: the most likely value counts four times, each limit once. The weight 4 comes from PERT (Program Evaluation and Review Technique), the project scheduling method developed for the US Navy’s Polaris missile program in the late 1950s, which estimated each activity’s expected duration as (optimistic + 4 × most likely + pessimistic) / 6. Project managers still call that expression the PERT or three-point estimate; it is the mean of this distribution.

The mean, 841.7, lies above the most likely value because the range reaches further above it (820 to 1,010) than below it (760 to 820). Cost and duration estimates are usually skewed this way, which is why a total of most likely costs is optimistic, as the project contingency example shows. The older PERT shortcut for the standard deviation, (max − min) / 6, is only an approximation: 41.7 here, against 44.3 from the formula above.

The PERT formula in Excel

Excel has no PERT function, but BETA.INV(probability, alpha, beta, A, B) returns values of a beta distribution stretched onto any interval from A to B. With the min, most likely and max of an estimate in cells B2, C2 and D2:

With RAND() as the probability, every recalculation draws a new value. With a fixed probability, BETA.INV returns that percentile directly, with no simulation: 0.9 gives 904.1 for the Structure item and 0.1 gives 786.9. BETAINV, the name in Excel 2007 and earlier, takes the same arguments. The min and max must differ, since both shape formulas divide by max − min.

A percentile of one item is not a percentile of a total: the P90s of several items do not add up to the P90 of their sum (see P50, P80 and P90). For a total you need a simulation, and in plain Excel that means collecting thousands of recalculations, for example with a data table, as Monte Carlo simulation in Excel explains.

PERT vs triangular: same three numbers, different answers

A triangular distribution takes the same min, most likely and max, but its density rises in a straight line from the min to the most likely value and falls in a straight line to the max. Its mean is (min + most likely + max) / 3: the most likely value counts once instead of four times, so the far limit pulls the mean further. For the Structure item that is 2,590 / 3 = 863.3, about 22 above the PERT’s 841.7. Its standard deviation, √((min² + ml² + max² − min × ml − min × max − ml × max) / 18), is 53.3 against the PERT’s 44.3.

To see what that means in a model, we ran the building example’s 5,000 trials twice with the same seed: once as published, with Structure as a PERT, and once with Structure switched to a triangular distribution with the same three numbers. Every other input draws exactly the same values in both runs, and each trial draws the same random number for Structure, so the runs differ only in the shape of that one distribution.

The draws agree with the formulas to the decimal shown: their means are 841.7 and 863.3. Latin Hypercube sampling, xellstorm’s default, helps here: it gives each input exactly one draw in each of 5,000 equally likely slices of its range. The triangular is wider, with a P10 to P90 range 142.4 wide against 117.2, and 23.6% of its draws exceed the PERT’s P90 of 904.1, where the PERT’s own draws do so in 10%. It is not wider on both sides, though: near the min the PERT has more draws, and the triangular’s P10 is higher, 798.7 against 786.9. With the most likely value near the low end, the triangular moves weight into the long upper tail; for a symmetric estimate it would give both ends more weight.

One item of six is enough to move the total. With Structure as a triangular distribution, the mean total cost rises from 2,785 to 2,806, the same step as the item’s mean (every other draw is the same); the P90 rises by 27, and the chance of going over the 2,900 budget from 16.1% to 20.8%.

Neither shape is right or wrong: they are two readings of the same three numbers. Choose one on purpose, and say which one a budget rests on.

When to use PERT, triangular or lognormal

• PERT for expert estimates where the most likely value is the number you trust most: costs, durations and quantities with a known floor and ceiling. Its mean stays close to the most likely value and the limits are rarely reached, which suits a min and a max meant as bounds you do not expect to hit.
• Triangular when values near the limits are realistic, or when you want a more cautious spread from the same three numbers: on a skewed estimate it puts more weight on the long side, so its mean and upper percentiles are higher. It is also easy to explain, since its density is two straight lines.
• Lognormal when there is no firm maximum: costs, losses or durations that can overrun by a multiple, and other quantities that cannot fall below zero but have a long right tail. It is set by a typical value and a spread rather than by limits.
• A fitted distribution when the three numbers are not limits at all. If an expert’s “low” and “high” are one-in-ten cases (P10 and P90), a PERT that uses them as min and max leaves out the 20% of outcomes beyond them; fit a distribution to the percentiles instead. With historical data, fit it to the data.

Symmetric variation around a target, such as a machined dimension, is usually normal, as in the tolerance stack-up example.

Entering a PERT distribution in xellstorm

xellstorm calculates your workbook in the browser and keeps the distributions next to it, so the workbook needs no BETA.INV formulas:

1. Open the workbook, click a cell you are unsure of on the model map and choose Make input. New inputs start as a PERT from 90% to 110% of the cell’s saved value.
2. On the Distributions step, keep PERT in the Distribution column and type the min, likely and max. The Shape, Mean and P10 – P90 columns update as you type, so you see what the three numbers imply before you run.
3. If the estimates are already in the workbook, in cells labeled Min, Likely and Max (or Low, Base and High) next to the input, the side panel offers “Link to these cells”: the PERT then reads them before every run, so edits to the sheet carry over when you open the edited workbook and apply your saved project. The risk register example works this way.
4. If your low and high are P10 and P90 rather than limits, type them with the typical value as P50 into “Fit from estimates” in the side panel and pick PERT, Normal, Lognormal or Triangular as the shape; xellstorm finds the closest distribution of that shape and shows its fit error. The “From data” tab fits one to historical values instead.
5. To compare shapes, add a scenario that changes the input’s distribution to Triangular and type the same three numbers. Scenarios run on the same random draws, and the Results step compares each one with the base case.

Workbooks built for @RISK, ModelRisk or Analytic Solver often hold PERT functions (RiskPert, VosePERT, PsiPert). “Import from workbook” turns a standalone function, or one with a single positive multiplier followed by at most one offset, into an input; VosePERT(E7,1,F7)*D7 is one example. Arguments and multipliers that use cells stay linked, including their arithmetic, and are read again after each scenario’s fixed cells are applied. The BETA.INV formula above also converts into an equivalent beta input: its two shape calculations and its min and max remain linked to the cells. Its explicit divisions still require different min and max. A PERT input whose min, most likely and max are all equal draws that one value.

Questions

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• Guide

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  Three ways to run a Monte Carlo simulation on an Excel model: RAND() and a data table, an add-in, or a browser tool with no add-in. PERT formula included.
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  Worked example: a building estimate’s most likely costs are exceeded in 94.2% of simulated outcomes. Size the construction cost contingency with Monte Carlo.
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xellstorm is a browser-based Monte Carlo simulation tool for Excel models: no add-in, and the workbook never leaves your computer.