Rayleigh Distribution Calculator
Calculate Rayleigh probabilities, density values, quantiles, and common measures from the scale parameter sigma.
Model magnitudes from two independent components
The Rayleigh distribution is a nonnegative continuous distribution often used for magnitudes such as signal amplitude, wind speed, wave height, and radial error. If two independent zero-mean normal components have the same variance, the magnitude sqrt(X^2 + Y^2) follows a Rayleigh distribution.
Choose a mode, enter the scale parameter sigma, and click Calculate Rayleigh Distribution. Results stay hidden until you calculate, so the page never displays an old answer after inputs change.
Rayleigh result
Enter values to calculate.
Sigma
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Probability
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Mean
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Std. deviation
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Calculation breakdown
Formula, inputs, and derived values.
Interpretation
How to read the result in context.
Common Rayleigh measures
All measures are based on the sigma value you entered.
| Measure | Formula | Value | Use |
|---|
How to Use This Calculator
Pick the question you need answered first, then enter the Rayleigh scale parameter sigma and any mode-specific inputs.
1. Choose a mode
Use cumulative probability for P(X <= x), survival for P(X > x), interval probability for a range, PDF for density, quantile for percentiles, or common measures for summaries.
2. Enter sigma
Sigma is the Rayleigh scale parameter. Larger sigma values spread the distribution farther to the right and increase the mean, median, and standard deviation.
3. Add the mode input
Enter x for CDF, survival, or PDF; enter lower and upper bounds for interval probability; or enter p as a percent for a quantile.
4. Review the breakdown
The result includes the formula path, a probability marker when relevant, and common measures for checking scale and interpretation.
Rayleigh Distribution Formulas
These formulas use a scale parameter sigma greater than 0 and a nonnegative random variable X.
Cumulative distribution
F(x) = 1 - exp(-x^2 / (2 * sigma^2))
Upper-tail probability
P(X > x) = exp(-x^2 / (2 * sigma^2))
Quantile function
x = sigma * sqrt(-2 * ln(1 - p))
Common measures
mean = sigma * sqrt(pi / 2), variance = sigma^2 * (4 - pi) / 2
Technical reference: SciPy documents the Rayleigh distribution as a continuous random variable with PDF, CDF, survival function, percent point function, and summary-statistic methods in scipy.stats.rayleigh.
Interesting Fact
Rayleigh averages sit above the peak
For the standard Rayleigh distribution, the mean is about 1.2533 and the standard deviation is about 0.6551. Scaled Rayleigh distributions keep those same ratios, so the mean is about 1.2533 * sigma and the standard deviation is about 0.6551 * sigma. That helps explain why the average value can sit to the right of the curve's peak even though the mode equals sigma. Random Services reports these standard Rayleigh moment values in The Rayleigh Distribution.
Worked Examples
These examples show how sigma and the selected mode change the Rayleigh result.
| Question | Inputs | Formula path | Result |
|---|---|---|---|
| Probability below 8 | sigma = 5, x = 8 | 1 - exp(-64 / 50) | about 72.19% |
| Probability above 6 | sigma = 3, x = 6 | exp(-36 / 18) | about 13.53% |
| 90th percentile | sigma = 5, p = 90% | 5 * sqrt(-2 ln(0.10)) | about 10.73 |
What Sigma Means in a Rayleigh Model
Sigma controls the distribution's scale. It is not the same as the standard deviation of X, but the standard deviation is directly proportional to sigma.
Location on the curve
The mode equals sigma
The peak of the Rayleigh PDF occurs at x = sigma. If sigma doubles, the peak shifts right and the curve becomes wider.
Average value
Mean is about 1.253 sigma
A quick mental check is mean ~= 1.253 * sigma and standard deviation ~= 0.655 * sigma.
Tail behavior
Large values fade exponentially
The upper tail is exp(-x^2 / (2 sigma^2)), so values several sigma above zero become increasingly unlikely.
Source: Wolfram MathWorld summarizes the Rayleigh PDF, CDF, raw moments, mean, variance, skewness, and related formula references in Rayleigh Distribution.
Which Rayleigh Calculation Should You Use?
Rayleigh questions often sound similar, but the correct calculator mode depends on whether you need a probability, a density, a cutoff value, or a distribution summary.
| User question | Use this mode | Inputs | Output means | Common wording |
|---|---|---|---|---|
| How likely is X to be at or below a threshold? | P(X <= x) cumulative probability | sigma and x | Area under the curve from 0 to x. | less than, at most, below, not exceeding |
| How likely is X to exceed a limit? | P(X > x) upper-tail probability | sigma and x | Area under the curve to the right of x. | greater than, exceedance, over, threshold risk |
| How likely is X to fall within a range? | P(a <= X <= b) interval probability | sigma, lower bound, upper bound | Difference between two CDF values: F(b) - F(a). | between, within, from a to b, acceptance band |
| What is the curve height at one point? | Probability density f(x) | sigma and x | Density value, not point probability. | PDF, likelihood curve, density at x |
| What cutoff captures a selected percentage? | Quantile x from probability p | sigma and p | The x value where the CDF equals p. | percentile, cutoff, design value, quantile |
If your wording includes "exactly x," avoid treating that as a nonzero probability. For a continuous distribution such as Rayleigh, probabilities come from areas over intervals, while the PDF gives the curve height at a point.
Source: Random Services provides a detailed Rayleigh distribution reference covering the CDF, PDF, quantile function, reliability function, and scale-parameter form in The Rayleigh Distribution.
How to Estimate Sigma from Sample Data
This calculator needs sigma as an input. If you have observed Rayleigh-like magnitudes, estimate sigma first, then use that estimate for probability, percentile, or density calculations.
Step 1
Use magnitude data only
Use nonnegative observations such as amplitudes, speeds, radial errors, or distances from an origin. Do not mix component values with magnitude values.
Step 2
Square each observation
Add the squared observations, divide by twice the sample size, then take the square root.
Step 3
Check units and fit
The estimated sigma has the same unit as the observations. Check the distribution shape before using it for high-stakes thresholds.
| Sample data | Sum of squares | Calculation | Estimated sigma |
|---|---|---|---|
| 3, 4, 7, 8 | 9 + 16 + 49 + 64 = 138 | sqrt(138 / 8) | about 4.153 |
| 10 observations | sum of x^2 = 520 | sqrt(520 / 20) | about 5.099 |
Practical caution: estimates become more reliable as sample size grows. If the data is censored, rounded heavily, truncated by a measurement limit, or mixed from different operating conditions, fit quality matters more than the formula alone.
Source: M. M. Siddiqui's NIST Journal of Research paper discusses statistical inference for Rayleigh distributions and related parameter-estimation work in Statistical Inference for Rayleigh Distributions.
When Rayleigh Is the Right Model
Rayleigh is powerful when its assumptions match the data, but related distributions may be better when there is a dominant signal, unequal component variation, or a heavier tail.
| Data pattern | Better model to consider | Why it matters | Quick check |
|---|---|---|---|
| Magnitude from two zero-mean, equal-spread components | Rayleigh | This is the classic Rayleigh setup. | Histogram starts near 0, rises to one peak, then tapers right. |
| There is a strong line-of-sight or baseline component | Rice distribution | A nonzero underlying component shifts the magnitude pattern. | Data rarely appears near 0 even when noise is present. |
| Tail is heavier or lighter than Rayleigh predicts | Weibull, gamma, or lognormal | These models can adapt to different tail shapes. | Observed high percentiles differ strongly from Rayleigh quantiles. |
| One signed normal component was converted to an absolute value | Half-normal distribution | That is a one-dimensional magnitude, not a two-dimensional one. | Measurements are absolute values of one axis or one error term. |
A useful workflow is to estimate sigma, compare predicted median and 90th percentile against observed values, then inspect the upper tail. If the fitted Rayleigh model misses the tail, threshold probabilities and design cutoffs can be misleading.
Where the Rayleigh Distribution Is Useful
Rayleigh models are most useful when you are modeling a magnitude formed from two orthogonal random components with similar spread.
Wireless and signal amplitude
In communication systems, Rayleigh fading is used when a received signal is the sum of many scattered paths and no dominant line-of-sight component is present.
Wind, wave, and vibration magnitudes
The distribution can approximate positive magnitudes such as wind speed, wave height, vibration amplitude, and radial displacement when the component assumptions are reasonable.
Radial error and distance from origin
If horizontal and vertical errors are independent normal errors with the same standard deviation, the radial error distance follows a Rayleigh distribution.
Reliability and lifetime screening
Some reliability workflows use Rayleigh-shaped lifetime models, but the fit should be checked against data before using the result for decisions.
Common Rayleigh Distribution Mistakes
Most errors come from mixing up probabilities, densities, and the meaning of sigma.
Treating PDF as probability
A PDF value is a density at one point, not the probability of exactly that point. Use the CDF or an interval probability for actual probabilities.
Calling sigma the standard deviation
Sigma is the scale parameter. The Rayleigh standard deviation is sigma * sqrt((4 - pi) / 2), which is about 0.655 sigma.
Ignoring the nonnegative support
Rayleigh random variables start at 0. Negative x values have CDF 0 and PDF 0 under the standard model.
Rayleigh distribution questions
Frequently Asked Questions
Quick answers about Rayleigh probability, CDF, PDF, sigma, expected value, reliability, signal fading, and model assumptions.
What is a Rayleigh distribution?
A Rayleigh distribution is a continuous probability distribution for a nonnegative random variable. It often appears when a magnitude is formed from two independent normal distribution components with equal variance, such as radial distance, signal amplitude, noise magnitude, or a simplified wind speed model.
How do I use this Rayleigh distribution calculator for cumulative probability?
Enter the scale parameter sigma and the x value, then choose the cumulative probability mode. The calculator uses the cumulative distribution function, or CDF, F(x) = 1 - exp(-x^2 / (2 * sigma^2)) and returns P(X <= x) as both a decimal and a percentage. In that mode, it works like a focused probability calculator for Rayleigh questions.
What does the Rayleigh PDF or probability density function tell me?
The PDF gives the probability density function value f(x) at a specific x value. It helps describe the shape of the distribution and where the curve is highest, but it is not the probability of getting exactly x. For probability over a range, use the interval mode instead.
How do I find a Rayleigh percentile, median, or quantile?
Use x = sigma * sqrt(-2 * ln(1 - p)), where p is the cumulative probability as a decimal. For example, p = 0.90 gives the 90th percentile, meaning 90% of the distribution is at or below that x value. The median is the same quantile formula with p = 0.50.
Is sigma the standard deviation?
No. Sigma is the scale parameter of the Rayleigh distribution, not the standard deviation itself. The standard deviation equals sigma * sqrt((4 - pi) / 2), the variance equals sigma^2 * (4 - pi) / 2, the mode equals sigma, and the mean or expected value equals sigma * sqrt(pi / 2). Those summary values are why this page can also serve as a compact statistics calculator for a Rayleigh model.
When should I use upper-tail probability or the survival function?
Use upper-tail probability when you want the chance that X exceeds a threshold. In a Rayleigh model, P(X > x) = exp(-x^2 / (2 * sigma^2)), which is also called the survival function. This is useful for reliability checks, exceedance questions, and fading thresholds where large magnitudes matter.
Can this calculator fit sigma from sample data?
This version calculates probabilities and measures from a sigma you provide. If you have sample data, estimate the parameter first using a suitable statistical method, then enter that estimate here to evaluate the distribution. For a simulation study, estimate sigma from generated or observed magnitudes, run the calculator, and compare the predicted CDF or quantiles with the simulated results.
Can I use a Rayleigh model for signal fading, noise, or wind speed?
Sometimes. A Rayleigh model can be reasonable for signal fading when many scattered paths combine and there is no dominant line-of-sight component. It can also describe the magnitude of two independent noise components or approximate wind speed in simplified two-component models, but you should compare the fitted distribution with real data before relying on the result.
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Disclaimer
This calculator is for educational, statistical, and planning use. It evaluates standard Rayleigh distribution formulas from the values you enter and does not confirm that the Rayleigh model is appropriate for your dataset or application.
Before using results for research, engineering, safety, finance, medical, legal, or business decisions, check model assumptions, measurement units, sample quality, independence, equal-variance component assumptions, and uncertainty with a qualified professional.
Last updated: June 18, 2026