# Rayleigh Distribution Calculator

URL: https://speedycalculator.com/calculators/statistics/rayleigh-distribution/
Last updated: 2026-10-30

> The Rayleigh distribution calculator evaluates the distribution of a magnitude such as wind speed or wave height from its scale σ: density, cumulative and tail probabilities, ranges, quantiles and random values, plus the mean, median, mode, variance, skewness and kurtosis.

## Inputs

- Scale parameter σ
- What to calculate
- Value x ≥ 0 (x₁ for a range)
- Upper value x₂
- Cumulative probability p (%)
- How many random values

## Results

- Result
- Rayleigh density with the requested area shaded
- Probability density f(x)
- P(X ≤ x) – cumulative (CDF)
- P(X > x) – survival
- P(x₁ ≤ X ≤ x₂)
- P(X < x₁ or X > x₂)
- Quantile x
- Random sample
- Mean (expected value)
- Median
- Mode (peak of the density)
- Variance
- Standard deviation
- Skewness
- Excess kurtosis

## Formula

```
f(x) = (x/σ²) e^(−x²/(2σ²)) · P(X ≤ x) = 1 − e^(−x²/(2σ²)) · quantile = σ√(−2 ln(1 − p)) · mean = σ√(π/2) · median = σ√(2 ln 2) · mode = σ · variance = (4 − π)σ²/2
```

## Explanation

The Rayleigh distribution describes the length of a two-dimensional vector whose two components are independent normal variables with mean 0 and the same standard deviation σ: R = √(X² + Y²). Whenever a magnitude is built from two random, perpendicular, equally variable parts, the Rayleigh distribution is a natural model.

It is used for wind speed (east–west and north–south components), the heights of ocean waves, the amplitude of a radio signal with scattered paths (Rayleigh fading), the miss distance of shots around a target and the noise envelope in radar and communications. It is the Weibull distribution with shape k = 2, and the square of a Rayleigh variable is exponential.

The single parameter σ is also the mode – the most likely value. The calculator gives the probability below, above, between or outside chosen values, quantiles from the closed-form inverse, random values and all the usual summary measures. A larger σ stretches the curve to the right and flattens it, but the shape – a mild right skew – never changes.

### How to use the Rayleigh Distribution Calculator

- **Scale parameter σ**
- **What to calculate**
- **Value x ≥ 0 (x₁ for a range)**
- **Upper value x₂**
- **Cumulative probability p**
- **How many random values**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

All probabilities use the closed-form CDF 1 − e^(−x²/(2σ²)); quantiles invert it exactly as σ√(−2 ln(1 − p)), which also generates random values. Choose what to calculate: a cumulative probability, an upper tail, the probability between or outside two values, a quantile (inverse CDF) or a random sample generated by inverse-transform sampling. The chart shades exactly the area you asked for.

### Worked example

With σ = 2 the probability of a value up to 3 is 1 − e^(−3²/(2 × 4)) = 1 − e^(−1.125) = 67.53%. The mean is 2√(π/2) = 2.5066, the median 2√(2 ln 2) = 2.3548, the mode 2, the variance (4 − π)/2 × 4 = 1.7168 and the 90th percentile 2√(−2 ln 0.1) = 4.2919.

Wind example: if hourly wind speed follows a Rayleigh distribution with σ = 6 m/s, the chance of more than 15 m/s is e^(−225/72) = 4.39%.

### Summary measures

MeasureFormulaValue for σ = 1Meanσ√(π/2)1.2533Medianσ√(2 ln 2)1.1774Modeσ1Variance(4 − π)σ²/20.4292Skewness2√π(π − 3)/(4 − π)^1.50.6311Excess kurtosis−(6π² − 24π + 16)/(4 − π)²0.2451

## Worked example

Inputs — Scale parameter σ: 2; What to calculate: P(X ≤ x) – probability at or below x; Value x ≥ 0 (x₁ for a range): 3.

Results — **Result = P(X ≤ 3) = 0.675348 (67.5348%)**; Probability density f(x) = 0.243489; P(X ≤ x) – cumulative (CDF) = 67.5348 %; P(X > x) – survival = 32.4652 %; Mean (expected value) = 2.5066; Median = 2.3548.

## Key terms

- **Scale parameter (σ)**: The standard deviation of each normal component; also the mode of the Rayleigh distribution.
- **Magnitude**: The length √(X² + Y²) of a vector with components X and Y.
- **Rayleigh fading**: Random variation of a radio signal's strength when there is no direct line of sight.

## FAQ

**What is the Rayleigh distribution?**
The distribution of √(X² + Y²) when X and Y are independent N(0, σ²) – the length of a random 2-D vector.

**What is P(X ≤ 3) for σ = 2?**
1 − e^(−9/8) = 67.53%.

**What are the mean and mode?**
The mode is σ and the mean σ√(π/2) ≈ 1.2533σ; for σ = 2 they are 2 and 2.5066.

**How is it related to the Weibull distribution?**
It is a Weibull distribution with shape k = 2 and scale λ = σ√2.

**Where is it used?**
Wind speeds, significant wave heights, Rayleigh fading in wireless channels, radar noise, and the radial error of shots or GPS fixes.

**How do I estimate σ from data?**
The maximum-likelihood estimate is σ̂ = √(Σxᵢ² / (2n)).


## Sources

- [Wolfram MathWorld – Probability and statistics](https://mathworld.wolfram.com/topics/ProbabilityandStatistics.html)
- [NIST/SEMATECH e-Handbook – Gallery of distributions](https://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm)
- [Penn State STAT 414 – Introduction to Probability Theory](https://online.stat.psu.edu/stat414/)