What this calculator does
The Rayleigh distribution describes the magnitude of a two-dimensional vector whose components are independent and normally distributed with the same variance. It arises naturally whenever two perpendicular random quantities are combined into a single length.
It has only one parameter, and the three measures of centre sit in a fixed order because the distribution is right-skewed. With σ = 2 the mode is exactly 2, the median 2.355 and the mean 2.507. The mode always equals σ, which makes the parameter easy to read off a histogram.
The formula
The density is x over σ² times e to the minus x² over 2σ², and the cumulative distribution is 1 minus that exponential term. The mean is σ times the square root of π/2, the median is σ times the square root of 2 ln 2, and the mode is σ itself. All three scale linearly with the single parameter.
| Term | Meaning |
|---|---|
| σ (scale) | The single parameter, which is also the mode of the distribution. |
| Mode | Exactly σ, the most likely value. |
| Median | σ√(2 ln 2), about 1.177σ. |
| Mean | σ√(π/2), about 1.253σ. |
The inputs explained
| Field | What to enter |
|---|---|
| x value | The value at which to evaluate. |
| Scale (σ) | Scale parameter, which equals the mode. It is not the standard deviation of the Rayleigh distribution itself. |
When to use it
Wind speed modelling
Wind speed is the magnitude of two perpendicular components, which makes Rayleigh the natural first model. It is the Weibull with shape exactly 2.
Signal processing
The envelope of a narrowband signal with random phase follows a Rayleigh distribution, which is the basis of Rayleigh fading in radio.
Scatter and targeting
The radial miss distance from a target, when errors in each axis are independent and normal, is Rayleigh distributed.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do the probabilities build up?
The density and cumulative probability at a range of values.
| x value | PDF at x | CDF, P(X ≤ x) | Mode |
|---|---|---|---|
| x = 1 | 0.220624 | 0.117503 | 2.000 |
| x = 2 | 0.303265 | 0.393469 | 2.000 |
| x = 3 | 0.243489 | 0.675348 | 2.000 |
| x = 5 | 0.054921 | 0.956063 | 2.000 |
Questions
What is the Rayleigh distribution used for?
Modelling the magnitude of a two-dimensional vector with independent normal components: wind speed from its two horizontal components, radial miss distance from independent aiming errors, and signal envelopes in radio propagation.
How does it relate to the Weibull distribution?
It is exactly the Weibull with shape parameter 2, with the scale parameters related by a factor of √2. That is why Rayleigh appears as the default in wind resource work, where the general Weibull is fitted and the shape often comes out near 2.
Is sigma the standard deviation?
No, and the notation is genuinely misleading. σ is the standard deviation of the underlying normal components, not of the Rayleigh distribution itself. The Rayleigh standard deviation is σ√((4−π)/2), about 0.655σ, which is smaller.
Why is the mean above the median?
Because the distribution is right-skewed, with a long upper tail and a hard floor at zero. That ordering is fixed: the mode at σ, then the median at about 1.177σ, then the mean at about 1.253σ, for every value of the parameter.
For the general case, see the Weibull distribution calculator. For normal probabilities, see the normal distribution calculator.