Rayleigh Distribution Calculator

How to use this Rayleigh distribution calculator

What this calculator does

This calculator evaluates a one-parameter Rayleigh distribution for nonnegative values. It can compute a tail or interval probability, the cumulative distribution function (CDF), the probability density function (PDF), a quantile, the distribution's common measures, or a finite random sample. The model uses the standard scale-parameter convention σ > 0, with density f(x) = x/σ² × exp( – x²/(2σ²)) for x ≥ 0. It is a mathematical distribution tool: it does not decide whether a real dataset actually follows a Rayleigh law, and it does not replace goodness-of-fit analysis.

Use it when you need to translate a known Rayleigh scale into probabilities or thresholds, check distribution measures used in engineering or signal models, compare how changing σ stretches the distribution, or generate a small sample for demonstrations and testing. The NIST Rayleigh PDF reference describes the same standard family and its relationship to the chi and Weibull distributions.

How to calculate

1. The calculator opens with a complete demonstration: Mode is Probability, Probability type is P(X ≤ x), Scale parameter σ is 2.4, and Argument, x is 3. The results, plot, table, and example XLSX are immediately available.
2. Choose the Mode that matches your task. Probability uses Probability type; CDF and PDF use Argument, x; Quantile uses Cumulative probability, p; Common measures needs only σ; Sample additionally uses Sample size. Replace the demonstration values with your own valid values and read the live primary result and distribution measures.
3. Use Download Excel to export the current validated model as an XLSX workbook. Reset clears the demonstration and entered data rather than restoring it; after Reset, export is disabled until a complete valid state is entered again.

Input guide

• Mode is a required selector. Probability returns a tail or interval probability; CDF returns F(x); PDF returns f(x); Quantile solves x from p; Common measures focuses on summary statistics; Sample draws values. Changing Mode changes which additional fields are required.
• Probability type is required only in Probability mode. Choose P(X ≤ x), P(X ≥ x), or P(a ≤ X ≤ b). For example, P(X ≤ x) with x = 3 is the startup case. A common mistake is selecting an interval while leaving one bound incomplete.
• Scale parameter σ is required in every mode and must be a finite number greater than 0. Plain decimals such as 2.4 and correctly comma-grouped values such as 1,000.5 are accepted; decimal-comma forms such as 2,4 and scientific notation are rejected to prevent silent reinterpretation. Larger σ stretches the distribution to larger x values.
• Argument, x is required for lower-tail/upper-tail Probability, CDF, and PDF modes. It is a nonnegative plain decimal; 3 is a valid example. Increasing x raises the CDF and lowers the upper-tail probability, while PDF behavior depends on whether x is below or above the mode σ.
• Lower bound, a and Upper bound, b are required for interval probability. Both must be nonnegative and b must be at least a. Values a = 1 and b = 4 are valid. Reversing the bounds is rejected instead of silently swapping them.
• Cumulative probability, p is required for Quantile mode and must be strictly between 0 and 1, entered as a decimal such as 0.75. Values 0 and 1 correspond to boundary/infinite quantiles and are therefore rejected. The NIST percent-point reference documents the inverse-CDF concept for the Rayleigh family.
• Sample size is required only in Sample mode and must be a whole number from 1 through 200. A value of 12 is a practical small example. The calculator draws a fresh sample when σ or Sample size changes in Sample mode; the current drawn sample is then held stable for display and export until another relevant edit occurs.

Output guide

Primary result changes with Mode: a probability or CDF is a dimensionless number from 0 to 1; PDF is a density whose units are the reciprocal of x's unit; Quantile is an x-value; Common measures uses the mean as the primary summary; Sample uses the arithmetic mean of the current sample. The secondary cards report Mean, Median, Distribution mode, Variance, Standard deviation, and Skewness. Mean, median, mode, and standard deviation scale linearly with σ; variance scales with σ²; Rayleigh skewness is constant because changing σ only rescales the distribution.

The Rayleigh cumulative distribution plot shows CDF values for the current σ in most modes; PDF mode instead shows the density curve. The Curve data table lists the exact keyed x and y values used to draw that line. In Sample mode, the detail table changes to Sample values with an index and the exact generated value. These rows are analytical or sampled model data, not decorative chart geometry.

Worked example

With σ = 2.4 and x = 3, the startup lower-tail probability is F(3) = 1 – exp( – 3²/(2 × 2.4²)) = 0.54216664. The same model gives a mean of 3.007954, median 2.825784, mode 2.400000, variance 2.472213, standard deviation 1.572327, and skewness 0.631111. Those first-open values are also the workbook checkpoints used for the downloadable example.