Weibull Distribution Calculator

Background.

Things rarely fail at a steady rate. Electronics that survive their first weeks often run for years, while bearings, belts and tyres last well and then wear out in a cluster. The Weibull distribution became the workhorse of reliability engineering because a single shape parameter can describe all of those patterns, and because it is simple enough to work out by hand.

Two numbers define it. The scale λ places the failures on the time axis, and it has a tidy meaning of its own: by x = λ, 63.2% of the population has failed, whatever the shape. The shape k governs how the risk of failure changes with age. Below 1 the risk falls over time, the signature of early defects being weeded out. At exactly 1 it is constant, and Weibull reduces to the exponential distribution, in which an old part is as good as a new one. Above 1 the risk climbs with age, which is wear-out.

Engineers in reliability are not the only users. Wind-energy studies commonly describe the spread of wind speeds at a site with a Weibull curve, and the strength of brittle materials such as ceramics is summarised by a Weibull modulus, the same k under another name: the higher it is, the more consistent the pieces.

You supply k, λ and a value x, and get back the cumulative probability F(x), the share expected to have failed, or to fall at or below x, together with the probability density there. This is the two-parameter form, which assumes failures can start at zero. It does not fit k and λ to your data, allow for units still running when a test ended, or judge whether a Weibull model suits your data in the first place.

What is weibull distribution calculator?

The Weibull distribution is a continuous probability distribution for positive quantities such as time to failure, wind speed or breaking strength. In the two-parameter form, the probability that a value is at most x is F(x) = 1 − exp[−(x/λ)^k], with shape k > 0 and scale λ > 0. Its complement, 1 − F(x), is the reliability or survival function, the share still working at x. The density f(x) measures how concentrated the probability is near x, per unit of x.

How to use this calculator.

1. 01Enter the shape k from a fitted model, a supplier datasheet or a published study. Reliability software and Weibull probability plots report it, often under the symbol β.
2. 02Enter the scale λ in the same unit as x, whether hours, cycles, kilometres or metres per second. Some sources call it the characteristic life and write it as η.
3. 03Enter the value x you care about, such as a warranty period, a service interval or a design wind speed.
4. 04Read the cumulative probability as the fraction failed, or at or below x, by that point, and subtract it from 1 for the fraction still running. The density is per unit of x and is mostly useful for plotting or for working out the hazard rate.

The formula.

F(x) = 1 − exp[−(x/λ)^k] and f(x) = (k/λ)(x/λ)^(k−1) exp[−(x/λ)^k]. Dividing x by λ makes the argument dimensionless, so λ only stretches or squeezes the curve along the axis. Raising the ratio to the power k bends it: when x is below λ, a large k makes (x/λ)^k very small, so few failures happen early and the curve then rises steeply around λ. At x = λ the ratio is 1, so F = 1 − e⁻¹ ≈ 0.632 for every k. Dividing the density by the survival share gives the hazard rate, (k/λ)(x/λ)^(k−1), the instantaneous failure rate among units still running: it falls with x when k < 1, stays at 1/λ when k = 1 and rises when k > 1. Shape and scale must be positive and x cannot be negative. When k is below 1 the density grows without limit as x approaches zero, so it has no finite value at x = 0.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST/SEMATECH e-Handbook, Weibull Distribution.
2. [2]NIST/SEMATECH e-Handbook, Weibull reliability analysis.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.