Interactive module guide

Triangular Distribution Calculator & Guide

This triangular distribution calculator uses minimum, most likely, and maximum estimates in a piecewise-linear model distinct from beta-based PERT.

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The triangular distribution converts three expert judgments—minimum a, most likely c, and maximum b—into a complete probability model with a piecewise-linear density that rises from a to the peak at c and falls to b.

It is the standard three-point model of project estimation and scarce-data simulation: transparent enough to explain in one sentence, yet a genuine distribution with exact intervals, quantiles, and moments.

This page computes all of them in closed form, including the mean (a+b+c)/3 and the piecewise square-root quantile function.

Is this the right module for my question?

Use it when

Use it to turn three-point estimates into probability statements: once a task owner supplies best case, most likely, and worst case, questions like “chance of finishing within two weeks” become exact CDF evaluations.
Use it in Monte Carlo models when data are scarce but bounds are defensible; a triangular input is transparent to reviewers, and its closed-form quantile function makes sampling trivial.
Use it to teach how little the “most likely” value determines: moving c inside fixed bounds visibly shifts mass and skew while the mean stays the simple average of the three points.

Choose another method when

Avoid it when the bounds themselves are soft: the model assigns exactly zero probability outside [a, b], and real overruns beyond a guessed maximum are precisely what it cannot represent.
Avoid it when data exist in quantity—fitted flexible families (beta on rescaled support, gamma, log-normal) will represent shape better than any three-point elicitation.
Avoid the classic elicitation trap of over-narrow bounds: experts systematically underestimate ranges, and a triangular model inherits that overconfidence without complaint.

Interactive tool

The calculator loads as you approach this section so the guide remains fast on mobile connections.

Example preview

Three-point task estimate

Inputs: X ~ Triangular(a=0, c=1, b=4); evaluate finishing within 2.
Representative result: P(X≤2) = 1 − 4/12 ≈ 0.6667 from the closed-form triangular CDF.

Illustrative only. Load the interactive tool to enter your own values and review assumptions.

Watch the explanation

3:36 min

The player loads only after you press play. You can also watch on YouTube.

Read the complete transcript

The Triangular Distribution. How can best case, most likely, and worst case become a probability model without pretending that most likely means average? Triangular uses lower bound a, mode c, and upper bound b, with c ordered between two distinct bounds. Density rises linearly from a to c, then falls linearly to b. Peak height is two divided by b minus a. The cumulative function uses a squared lower ratio before c, a complementary squared ratio after c, and hard zero-one limits outside. At the mode, cumulative probability is not generally one half. It is c minus a over b minus a. Invert the two cumulative branches to get a piecewise square-root quantile that switches at the mode probability.

The geometry normalizes itself: total base b minus a times peak height two over b minus a, divided by two, equals one. Mode is c, but mean is the simple average of a, b, and c. Asymmetry pulls average away from the peak. Median uses one of two square-root branches, selected by whether c lies left or right of the interval midpoint. Variance is a squared plus b squared plus c squared minus all three pairwise products, divided by eighteen. Move c left of midpoint and the long right side creates positive skew. Move it right and skew reverses. Excess kurtosis is always negative zero point six. Hard bounds make this family thin-tailed by construction. The mode may equal either bound. These valid edge cases are one-sided limiting triangles, not literal divisions by zero. Use triangular inputs for project durations, costs, demand, or simulation when data are scarce and bounds are defensible. Triangular is not beta-PERT. Its sides are straight and its mean weights all three points equally; classic PERT weights the mode four times.

Suppose a software task has best case zero weeks, most likely one week, and worst case four weeks. The mode sits one quarter across the support, so F of one is one quarter, not one half. Two weeks is on the falling branch. One minus four over twelve gives completion probability two thirds. Mean is zero plus one plus four, divided by three: one point six six seven weeks, above the one-week mode. Because c lies left of midpoint two, median is four minus square root six, about one point five five one weeks. A ninety-five percent commitment is four minus square root zero point six, or about three point two two five weeks.

Pressure-test the hard bounds. The model assigns exactly zero chance to finishing before a or after b. Check the peak at c, the straight slopes on both sides, and branch probability F of c. Reject nonfinite inputs, require a below b and c within bounds, and keep quantile probabilities strictly between zero and one. A triangular-looking histogram does not validate elicited bounds. With enough data, fit a more flexible family instead. For this right-skewed task, mode one is below median one point five five one, below mean one point six six seven. Report all three inputs, units, event boundary, population, and whether the values were elicited or fitted. Explore triangular probabilities, quantiles, moments, and exact three-point scenarios free at Distri Scope dot com.

How to read the result

The mean weighs the three points equally—(a+b+c)/3—so an extreme “worst case” moves the mean by a third of its distance, a fact worth showing to stakeholders who anchor on the most likely value.

Skewness is read off the geometry: mode left of midpoint means right skew and mean above median; the summary panel confirms it numerically.

Fixed excess kurtosis −0.6 marks the family as thin-tailed—by construction it can never produce surprises beyond its stated bounds.

Inputs and parameter meaning

Three parameters with the ordering a ≤ c ≤ b enforced; parameter combinations violating it produce an explicit error rather than a silent adjustment.

Setting c = a or c = b produces the right- and left-triangle limiting shapes with the peak at a boundary.

The plotted range extends slightly beyond [a, b] to make the hard support boundaries visible as exact zeros.

Formula and available parameters

Density or probability mass

f(x) = \begin{cases} \dfrac{2(x-a)}{(b-a)(c-a)} & a \le x < c \\ \dfrac{2(b-x)}{(b-a)(b-c)} & c \le x \le b \end{cases}

Mean

\mu = \frac{a+b+c}{3}

Variance

\sigma^2 = \frac{a^2+b^2+c^2-ab-ac-bc}{18}

Lower bound (a): interactive range -10 to 10; default 0.
Mode (c): interactive range -10 to 10; default 1.
Upper bound (b): interactive range -10 to 10; default 4.

What the method is doing

Density, CDF, and quantiles are exact piecewise formulas; the quantile switches branches at p = (c−a)/(b−a), the CDF value at the mode.

Moments are closed-form: mean (a+b+c)/3, the symmetric variance expression (a²+b²+c²−ab−ac−bc)/18, a sign-tracking skewness formula, and excess kurtosis fixed at −3/5.

The median takes one of two square-root forms depending on whether the mode lies left or right of the interval midpoint.

Worked example: a three-point task estimate

A task owner estimates a software task at best case a = 0, most likely c = 1, and worst case b = 4 (in weeks). Model duration as Triangular(0, 1, 4). Find the probability of finishing within 2 weeks, the mean and median duration, and the 95th-percentile commitment date.

1Within 2 weeks: 2 is above the mode, so F(2) = 1 − (b−x)²/((b−a)(b−c)) = 1 − 4/12 ≈ 0.667.
2Mean: (0+1+4)/3 ≈ 1.667 weeks—two thirds of a week above the “most likely” answer of 1, pulled by the long right side.
3Median: the mode (1) is left of the midpoint (2), so the median uses the upper branch: 4 − √((4)(3)/2) = 4 − √6 ≈ 1.551 weeks.
4Commitment at 95%: Q(0.95) = 4 − √(0.05 × 4 × 3) ≈ 4 − 0.775 ≈ 3.23 weeks.

Interpretation

The gap between the most likely value (1 week), the median (1.55), the mean (1.67), and the 95% commitment (3.23) is the whole story of right-skewed estimates: what the owner calls “most likely” is neither the average outcome nor a safe promise. Committing at the 95th percentile requires more than three times the modal estimate—arithmetic that three-point models make impossible to hide.

Common mistakes

Do not present the most likely value c as the expected duration; the mean (a+b+c)/3 differs whenever the triangle is asymmetric, and skewed tasks are the norm.
Do not chain optimistic bounds through a project plan; zero probability beyond b in every task compounds into severe overconfidence at the portfolio level.
Do not confuse triangular with PERT-beta: the classic PERT mean (a+4c+b)/6 weights the mode more heavily and comes from a different (beta) model.

Limits and independent validation

Hard bounds are structural: the model cannot express any chance of exceeding b, which is often the riskiest region in practice.

The linear density is an elicitation convenience, not an empirical claim; with real data, fitted smooth families are preferable.

Only the three-parameter symmetric-support version is implemented; PERT-beta and trapezoidal variants are out of scope.

Before using the result

Check the branch point: F(c) must equal (c−a)/(b−a)—here F(1) = 1/4—directly verifiable with the point calculator.
Recompute one quantile by hand with the square-root formula and compare with the quantile lookup.
Pressure-test the elicitation: ask the estimator for the realized durations of past similar tasks and check how often they landed outside the stated [a, b].

uniform distribution →See the bounds-only model the triangular refines with a most-likely point.beta distribution →Move to the smooth bounded family behind PERT-style estimates.data fitting →Replace elicited shapes with fitted ones once real durations accumulate.

← Rayleigh Distribution Discrete Uniform Distribution →

See every option in the Probability Distribution Explorer or review the DistriScope methodology. Educational information; last reviewed 2026-08-09. Verify consequential calculations independently.