Weibull Distribution Calculator

The Weibull distribution is a continuous probability distribution defined for non-negative values, most often used to model the time until a machine fails, a light bulb burns out or a component wears through. Unlike the normal distribution it can be strongly skewed, and unlike the exponential distribution it does not insist on a constant failure rate.

It has just two parameters. The scale λ has the same units as the data and acts as a yardstick; the shape k is dimensionless and decides the behaviour of the tail and of the failure rate.

Every Weibull variable is built from the same skeleton: raise x/λ to the power k, then take the exponential. Changing k reshapes the curve — k = 1 recovers the exponential distribution, k = 2 the Rayleigh distribution, and k ≈ 3.6 a bell curve almost indistinguishable from a normal one.

The cumulative distribution gives the probability of failing at or before x, and its complement, the survival function, gives the probability of lasting longer than x. The density is that survival function multiplied by the hazard rate, which works out to the tidy (k/λ)(x/λ)^(k−1).

The quantile formula simply rearranges the same expression, so every probability and every percentile are two views of one relationship. The median is λ(ln 2)^(1/k) and the mode, when k > 1, is λ((k−1)/k)^(1/k).

The mean and variance come from the gamma function rather than from integration: mean = λΓ(1 + 1/k) and variance = λ²[Γ(1 + 2/k) − Γ(1 + 1/k)²]. Because Γ(1 + 1/k) exceeds 1 when k < 1 and falls below it when k > 1, the mean sits above the median for heavy-tailed shapes and below it for stretched ones.

A probability such as 89.5% means that in the long run 89.5% of units would fail at or before that point. It is a property of the fitted distribution, not a promise about any single unit.

The hazard rate is the conditional failure rate: the chance of failing in the next unit of time given survival so far. Its trend across x is what tells you whether the model describes wear-out or infant mortality.

The mean can sit far from the median. With k < 1 a few very large values pull the mean up; with k > 3.6 the values huddle near the mode and the mean falls below the median.

The sample mode is a single realisation, so its mean will differ from the theoretical mean. That gap shrinks roughly as 1/√n and is not an error in either number.

• The shape parameter k. It is the single most important choice: it moves the distribution from a sharp peak at zero through the exponential to a near-normal bell, and it decides whether the hazard rate rises or falls.
• The scale parameter λ. It scales every value linearly, so a percentage answer is unaffected by it while an absolute answer is exactly proportional to it.
• Units. Weibull calculations are only meaningful with consistent units; 1000 cycles and 1000 hours describe different realities even though the arithmetic is identical.
• Sample size in sample mode. The spread of the generated values narrows as more values are drawn, so the sample mean approaches the theoretical mean.
• The probability chosen in quantile mode. Near 0 or 1 the quantile becomes extreme, and p = 0 or 1 returns the boundary of the distribution.

• Modelling time to failure for machinery, electronics, lamps, bearings and other components whose lifetimes are strictly non-negative.
• Describing wear-out, where k > 1 and the failure rate climbs with age, or infant mortality, where k < 1 and it falls.
• Setting warranty periods and design targets from a required reliability such as 90% or 95%.
• Turning a fitted survival curve into everyday quantities: expected life, medians, or the life exceeded by only 1% of units.
• Generating realistic lifetime data for simulations, schedules and spare-parts stock, where values must never go negative.

• Values cannot be negative. The distribution has no support below zero, which is why an entry below 0 is rejected rather than clamped.
• The model assumes the failure mechanism stays the same. A population where the mechanism changes partway through is not Weibull, and fitting it as one will understate the risk.
• Both parameters must be known or estimated. Plausible-looking numbers are not data, and the results are only as good as the fit behind k and λ.
• Generated samples are independent draws. Real equipment often fails in clusters, and a sample from this distribution will understate that clustering.
• The mean alone hides the tail. For heavy-tailed shapes the 10% of units lasting far beyond the mean matters more than the mean itself, so read quantiles as well.