Weibull Distribution Calculator
Enter the Weibull shape and scale to get the probability that a lifetime falls below, above or between values, or the time that matches a given probability. The mean, median, mode and skewness come with a shaded density chart.
Below 1: failures fall over time. 1: constant. Above 1: wear-out
The time by which 63.2% of items have failed
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What the Weibull distribution models
The Weibull distribution is the workhorse of reliability engineering: it describes the time until a part fails, a patient relapses or a storm exceeds a wind speed. Its shape parameter k says how the risk of failure changes with age, which is why one family covers very different behaviour.
F(x) = 1 − exp(−(x/λ)^k) reliability R(x) = exp(−(x/λ)^k)
f(x) = (k/λ) · (x/λ)^(k−1) · exp(−(x/λ)^k), for x ≥ 0
Mean = λ · Γ(1 + 1/k) Median = λ · (ln 2)^(1/k) Mode = λ · ((k − 1)/k)^(1/k) for k > 1
Variance = λ² · [Γ(1 + 2/k) − Γ(1 + 1/k)²]
- k < 1: the failure rate decreases with time (early-life failures, or a population in which weak items fail first).
- k = 1: the failure rate is constant and the Weibull is the exponential distribution.
- k > 1: the failure rate increases with time (wear-out); k = 2 is the Rayleigh distribution and k ≈ 3.6 is close to symmetric.
Worked example
A bearing has a Weibull life with shape k = 1.5 and scale λ = 100 hours. The chance it has failed by 120 hours is
F(120) = 1 − exp(−(120/100)^1.5) = 1 − exp(−1.314534) = 0.731401
Load example shows this value and the complement 0.268599, which is the reliability at 120 hours. The mean life is 90.2745 hours, the median 78.322 hours and the most likely failure time (the mode) 48.075 hours: the distribution is skewed to the right, so a few long-lived parts pull the mean above the median.
Software equivalents
Check the order of the two parameters: MATLAB takes the scale first.
| Software | Cumulative probability | Parameters |
|---|---|---|
| Excel / Sheets | WEIBULL.DIST(x, alpha, beta, TRUE) | alpha = shape k, beta = scale λ |
| R | pweibull(x, shape = k, scale = λ) | shape, then scale |
| Python (SciPy) | scipy.stats.weibull_min.cdf(x, c=k, scale=λ) | c = shape |
| MATLAB | wblcdf(x, lambda, k) | scale first, then shape |
Assumptions and limits
This page uses the two-parameter Weibull, which starts at zero. If your parts cannot fail before a guaranteed life, a three-parameter Weibull with a location shift is the better fit. The shape must be between 0 and 1000 so that the mean, variance and skewness stay accurate. To fit k and λ to data you need a regression or maximum-likelihood tool; this calculator works from parameters you already have.
Related guides and calculators
With shape 1 the Weibull distribution is the exponential distribution. Compare it with the gamma and lognormal distributions, and with the Poisson distribution, which counts events in the same kind of process. Read probability distributions for an overview.
Frequently Asked Questions
How do I calculate Weibull reliability?
Reliability at time x is P(X > x) = exp(−(x/λ)^k). Choose the probability above x and enter x; the result card labelled Probability is the reliability.
What does the Weibull shape parameter mean?
It describes how the failure rate changes with age: below 1 it decreases, at 1 it is constant (exponential), above 1 it increases. Values around 1.5 to 3 are typical of mechanical wear-out.
What is the difference between the Weibull scale and the mean?
The scale λ is the characteristic life: about 63.2% of items fail before x = λ for any shape. The mean is λ times Γ(1 + 1/k), which equals λ only when k = 1.
How do I find the time by which 10% of parts have failed (B10 life)?
Choose Find x from a lower-tail probability and enter 0.1. The x value is the B10 life; for the example shape and scale it is about 22.3 hours.
Is the Weibull distribution the same as the exponential distribution?
Only when the shape is 1. Then the failure rate is constant and the scale equals the mean. For any other shape the Weibull has an increasing or decreasing failure rate.
Which parameter comes first in Excel, R, SciPy and MATLAB?
Excel's WEIBULL.DIST, R's pweibull and SciPy's weibull_min take the shape first. MATLAB's wblcdf takes the scale first, then the shape, so swap them when you compare.
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