Enter the minimum (a), maximum (b), peak (c), and random variable (x) values to analyze a triangular distribution. The calculator returns the PDF probability, CDF cumulative probability, mean, median, mode, and variance — giving you a full statistical picture of your distribution in one step.
CDF — Cumulative Probability P(X ≤ x)
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PDF — Probability Density f(x)
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Mean (μ)
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Median (M)
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Mode
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Variance (σ²)
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Standard Deviation (σ)
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A triangular distribution is a continuous probability distribution defined by three parameters: a minimum value (a), a maximum value (b), and a peak (most likely) value (c). It gets its name from the triangular shape of its probability density function (PDF). It is commonly used in project management, risk analysis, and simulations when only limited data is available.
The PDF (Probability Density Function) f(x) describes the relative likelihood of the random variable taking a specific value x. The CDF (Cumulative Distribution Function) F(x) gives the probability that the random variable is less than or equal to x. The CDF is the integral of the PDF from the minimum value up to x.
The mean (μ) of a triangular distribution is calculated as (a + b + c) / 3, where a is the minimum, b is the maximum, and c is the peak value. This is the simple average of the three defining parameters.
The variance (σ²) is calculated using the formula: (a² + b² + c² − ab − ac − bc) / 18. The standard deviation σ is the square root of the variance. The triangular distribution's variance depends on the spread and asymmetry defined by a, b, and c.
The median depends on where the peak c falls relative to the midpoint of [a, b]. If c ≥ (a + b) / 2, the median is a + √((b − a)(c − a) / 2). If c < (a + b) / 2, the median is b − √((b − a)(b − c) / 2). These formulas find the point where the CDF equals 0.5.
The three parameters must satisfy a < b (minimum must be strictly less than maximum) and a ≤ c ≤ b (the peak must be between the minimum and maximum, inclusive). If these conditions are not met, the triangular distribution is undefined.
The triangular distribution is widely used in project management for three-point estimation (optimistic, pessimistic, and most likely durations or costs), in Monte Carlo simulations, in risk analysis, and in any scenario where you need a simple model but only have minimum, maximum, and most-likely estimates rather than full historical data.
At x = c, the PDF reaches its maximum value of 2 / (b − a). This is the highest point of the triangular distribution's PDF, representing the most likely outcome. For x values outside [a, b], the PDF is 0.