How this instrument works
The Rayleigh distribution describes the magnitude of a two-dimensional vector whose two perpendicular components are each independent, normally distributed, and centered at zero with the same spread. Picture wind blowing with a random north-south component and a random east-west component, each individually bell-shaped around zero: the overall wind speed — the length of the combined vector, regardless of direction — follows a Rayleigh distribution. The same structure shows up in ocean wave heights, in the magnitude of noise in radio and radar signals, and in certain particle-speed models in physics.
The distribution has a single scale parameter, σ, which controls how spread out the magnitudes tend to be. The density function, f(x) = (x/σ²) × e^(−x²/(2σ²)), is zero at x = 0 (a magnitude of exactly zero requires both perpendicular components to be exactly zero simultaneously, an essentially impossible coincidence), rises to a peak, then decays. The cumulative function, F(x) = 1 − e^(−x²/(2σ²)), gives the probability that the magnitude is x or less.
The mean, E[X] = σ × √(π/2), is always somewhat larger than σ itself — about 1.2533 times σ — which is a useful sanity check: if you're told a wind-speed data set has a Rayleigh-distributed magnitude with σ = 5 m/s, you should expect an average wind speed around 6.27 m/s, not 5 m/s. The Rayleigh distribution is also a special case of the more general Weibull distribution, corresponding to a Weibull shape parameter of exactly 2 — this site's Weibull Distribution calculator can reproduce the same PDF and CDF shape with the right parameter choice, though the two instruments use different scale-parameter conventions.
- Enter the distribution's spread into Scale parameter (σ), a positive number in whatever units your magnitude is measured in.
- Enter the magnitude you want to evaluate into x (value, x ≥ 0) — magnitudes can never be negative.
- Read Probability density f(x) — the density at that x, useful for comparing shapes rather than as a direct probability.
- Read Cumulative probability F(x) = P(X ≤ x) — the actual probability that the magnitude is x or less.
- Read Mean, E[X] — the expected long-run average magnitude for this σ, always about 1.2533 × σ regardless of what x you entered.
Worked example — magnitude x = 1 with scale σ = 1
Enter σ = 1 into Scale parameter and x = 1 into x — the standard Rayleigh distribution, evaluated at a magnitude of 1. The instrument first computes x²/(2σ²) = 1/2 = 0.5, then e^(−0.5) ≈ 0.60653066.
Probability density f(x) reads 0.606530659713, computed as (1/1) × 0.60653066. Cumulative probability F(x) reads 0.393469340287, computed as 1 − 0.60653066 — so there's about a 39.3% chance the magnitude is 1 or less, and about a 60.7% chance it exceeds 1. Mean, E[X], reads 1.25331413732, computed as 1 × √(π/2) — notably larger than σ = 1 itself, since the mean of a Rayleigh distribution always sits above its scale parameter.
Questions
Why is the density exactly zero at x = 0?
Because a magnitude of zero requires both underlying perpendicular components to land at exactly zero at the same time — an event with essentially zero probability for continuous, normally distributed components. As x moves away from zero, there are more and more combinations of the two components that produce that magnitude, so the density rises before eventually decaying again for large x.
How is the Rayleigh distribution related to the Weibull distribution?
The Rayleigh distribution is a special case of the Weibull distribution with shape parameter k = 2. This site's Weibull Distribution calculator uses a scale parameter λ that relates to this instrument's σ by λ = σ√2, since the two distributions parameterize the exponent slightly differently — Weibull's exponent is (x/λ)^k while Rayleigh's is x²/(2σ²). Plugging in matched parameters produces identical PDF and CDF curves under either instrument.
Why is the mean larger than the scale parameter σ?
Because the mean, σ × √(π/2) ≈ 1.2533σ, accounts for the full shape of the distribution, not just its scale. σ itself is the parameter that stretches or compresses the distribution, but the actual average magnitude, once you integrate over the whole distribution's shape, always lands about 25% above σ — a fixed ratio that holds for every value of σ, since π/2 is a constant.
What real-world quantities follow a Rayleigh distribution?
Wind speed (combining independent north-south and east-west gusts), significant wave height in oceanography, the envelope of a radio or radar signal corrupted by Gaussian noise, and the miss-distance from a target when horizontal and vertical aiming errors are each independently normal. Any situation combining two independent, zero-centered, equal-spread normal components into a single magnitude is a candidate.
Can x or σ be negative?
No. σ must be strictly positive — a scale of zero or less would make the distribution's spread undefined or nonsensical. x, the magnitude, must be zero or greater, since it represents a vector length or distance, which by definition cannot be negative.