Triangular Distribution Calculator

Minimum

a

Maximum

b

Mode

c

Probability Type:

X Value (

x_{1}

Upper Bound (

x_{2}

Output type: PDF Chart CDF Chart Data Table

Decimal Places: 2 3 5 8

Clear Reset

Introduction

The triangular distribution calculator is a tool designed to analyse continuous probability for datasets defined by a known range and a most likely value. It allows researchers to evaluate the probability of a random variable $X$ falling within specific intervals when data is limited but the minimum, maximum, and mode are identified, providing essential insights into probability density and cumulative distribution functions.

What this calculator does

This calculator computes the probability density function (PDF), cumulative distribution function (CDF), mean, variance, and standard deviation based on three user-defined parameters. By entering a lower bound, an upper bound, and a peak value, the tool determines the likelihood of values being less than, greater than, or between specified points. It generates numerical summaries alongside visualisations like PDF and CDF charts or detailed data tables for academic review.

Formula used

The calculation relies on the triangular distribution parameters: minimum $a$ , maximum $b$ , and mode $c$ . The mean is the average of these three points. The CDF for $a \leq x < c$ and $c \leq x < b$ determines the area under the triangular curve to find specific probabilities.

M e a n = \frac{a + b + c}{3}

V a r i a n c e = \frac{a^{2} + b^{2} + c^{2} - a b - a c - b c}{18}

How to use this calculator

1. Enter the minimum value $a$ , maximum value $b$ , and the mode $c$ .
2. Select the probability type: less than, greater than, or between specific bounds.
3. Input the target $x$ value or values for the interval analysis.
4. Select the desired output format and execute the calculation to view the statistical results and charts.

Example calculation

Scenario: A student in environmental science is modelling the daily rainfall in a region where the minimum is 1mm, the maximum is 5mm, and the mode is 3mm.

Inputs: $a = 1$ , $b = 5$ , $c = 3$ , and $x = 2$ for $P (X \leq 2)$ .

Working:

Step 1: $F (x) = \frac{{(x - a)}^{2}}{(b - a) (c - a)}$

Step 2: $F (2) = \frac{{(2 - 1)}^{2}}{(5 - 1) (3 - 1)}$

Step 3: $F (2) = \frac{1^{2}}{4 \times 2}$

Step 4: $F (2) = \frac{1}{8}$

Result: 0.125

Interpretation: There is a 12.5% probability that the rainfall will be 2mm or less.

Summary: The model effectively quantifies likelihood within the lower segment of the distribution.

Understanding the result

The result represents the cumulative probability of a random variable falling within the specified range. A value closer to 1.00 indicates a high likelihood, while the mean and standard deviation provide a central tendency and measure of dispersion, helping to characterise the shape and spread of the data around the mode.

Assumptions and limitations

This model assumes that the underlying data follows a continuous distribution defined strictly by three points. It requires that the minimum is less than the maximum and the mode lies within that range. It is not suitable for discrete datasets or distributions with multiple peaks.

Common mistakes to avoid

A common error is entering a mode value that sits outside the range of the minimum and maximum parameters. Additionally, researchers may confuse the triangular mean with the mode; in this distribution, the mean is only equal to the mode if the triangle is perfectly symmetrical.

Sensitivity and robustness

The triangular distribution is sensitive to the placement of the mode relative to the boundaries. Small shifts in the mode parameter significantly alter the skewness of the distribution and the resulting cumulative probabilities. The calculation remains stable as long as the inputs maintain the logical order of minimum, mode, and maximum.

Troubleshooting

If the calculator returns an error, ensure that the minimum value is strictly less than the maximum. Check that all inputs are numeric and do not exceed the permitted educational range. If a "between" probability returns zero, verify that the upper bound is greater than the lower bound.

Frequently asked questions

What is the difference between the PDF and CDF charts?

The PDF chart displays the height of the probability density at any point, while the CDF chart shows the accumulated probability from the minimum value up to that point.

Can the mode be equal to the minimum or maximum?

Yes, the mode can be equal to the minimum or maximum, resulting in a right-angled triangular distribution instead of a scalene or isosceles shape.

How is the standard deviation calculated?

The standard deviation is derived by taking the square root of the variance, which is calculated using the squared values and products of the three main parameters.

Where this calculation is used

The triangular distribution is frequently utilised in population studies and social research when historical data is scarce, allowing analysts to model uncertainty based on expert estimates of best-case, worst-case, and most-likely scenarios. In sports analysis, it helps in predicting performance ranges when only extremes and a typical value are known. Educational settings use it to teach the fundamentals of continuous probability density and the geometric interpretation of statistical areas under a curve in introductory probability theory.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.