The Weibull distribution calculator above evaluates the two- and three-parameter Weibull in both directions. Give it a shape, a scale and a value, and it returns the probability density, the cumulative probability, the reliability, the instantaneous hazard rate, the mean, median, mode and standard deviation, the probability of landing in an interval, and any percentile you ask for. It is the workhorse distribution of reliability engineering, survival analysis and wind resource assessment, and it earns that position by being flexible enough to describe wear-in, random failure and wear-out with a single equation.
Arb Digital publishes this page with one convention stated loudly at the top, because that convention is where nearly every Weibull error comes from. This calculator uses the shape and scale parameterisation: k for shape, λ for scale, with λ measured in the same units as x. It does not use the rate form, where the second parameter is the reciprocal of the scale. Feeding a rate of 0.001 into a field expecting a scale of 1,000 does not produce a slightly wrong answer, it produces a distribution six orders of magnitude off.
What This Weibull Distribution Calculator Does
Forward, it takes a value of x and returns probabilities. Backward, it takes a probability and returns the value — the quantile, or inverse CDF, which reliability practice calls the B-life. The default percentile of 0.10 returns the B10 life, the age by which one in ten of the population has failed, which is the figure bearing manufacturers and many warranty schemes are actually specified against.
It also reports the hazard rate separately from the density, because the two are routinely confused and they answer different questions. The density is the unconditional chance of failing near x, computed across the whole original population. The hazard rate is the chance of failing near x given survival to x, computed across the survivors only. For a wear-out distribution the density eventually falls while the hazard rate keeps climbing, which sounds contradictory until you notice there are hardly any survivors left to fail.
The three-bar split shows the probability mass below x, between x and x₂, and above x₂, so an interval question can be read without any subtraction. The third parameter, the location or threshold θ, shifts the whole distribution to the right and encodes a guaranteed failure-free period. Leave it at zero unless you have a physical reason for it.
How to Use It
- Enter the shape parameter k. This is the number that determines the character of the distribution. Values below 1 mean a falling hazard rate, exactly 1 means constant, above 1 means rising.
- Enter the scale parameter λ in the same units as x. Check your source is quoting a scale and not a rate; if it gives a rate, take the reciprocal before entering it.
- Set the location θ to zero unless you are fitting a three-parameter Weibull with a known failure-free period.
- Enter x and x₂. x drives the headline probability, reliability, density and hazard. x₂ only sets the upper end of the interval.
- Choose a percentile between 0 and 1 to get the corresponding B-life, then read the summary line for the mean, median, mode and standard deviation.
The Formula and How It Is Calculated
Write z = (x − θ) ÷ λ. Then, for x greater than or equal to θ:
PDF: f(x) = (k ÷ λ) · zk−1 · e−zk
CDF: F(x) = 1 − e−zk
Reliability: R(x) = e−zk
Hazard: h(x) = (k ÷ λ) · zk−1
Quantile: xp = θ + λ · (−ln(1 − p))1÷k
The mean is θ + λΓ(1 + 1÷k) and the variance is λ²[Γ(1 + 2÷k) − Γ(1 + 1÷k)²], where Γ is the gamma function, evaluated here with a Lanczos approximation accurate to about fifteen significant figures across the range this tool uses. The median is θ + λ(ln 2)1÷k. The mode is θ + λ((k−1)÷k)1÷k when k is greater than 1; when k is less than 1 the density is unbounded as x approaches θ and there is no interior mode. These are the same expressions given in the NIST/SEMATECH e-Handbook entry for the Weibull distribution, which is the reference this page follows.
Worked check, using the defaults: with k = 2, λ = 1,000 and θ = 0, at x = 500 we have z = 0.5, so zk = 0.25, R = e−0.25 = 0.7788 and F = 0.2212. The density is (2÷1000)(0.5)(0.7788) = 0.00077880 and the hazard is (2÷1000)(0.5) = 0.001 exactly. The mean is 1,000Γ(1.5) = 886.23 and the B10 life is 1,000(−ln 0.9)0.5 = 324.59. Those are the figures the calculator prints on load.
Why λ Is the 63.2nd Percentile
Set x = λ with θ = 0 and z becomes 1, so zk is 1 regardless of the shape, and F(λ) = 1 − e−1 = 0.6321. That is not a coincidence or an approximation. Whatever the shape parameter, 63.2% of the population has failed by the scale parameter. Reliability engineers use this as a sanity check: if a fitted λ sits somewhere other than around the 63rd percentile of your data, something has gone wrong in the fit or in the parameterisation you assumed.
It is also the cleanest way to explain the scale to a non-specialist. λ is not a mean and it is not a median. It is the characteristic life, and it always lands at the same cumulative probability.
The Exponential Is the Special Case at k = 1
Put k = 1 into the formulas above and zk collapses to z, the hazard becomes 1÷λ for every x, and the CDF becomes 1 − e−x÷λ. That is precisely the exponential distribution with mean λ, equivalently rate 1÷λ. The Weibull is the exponential generalised so that the hazard rate may rise or fall with age instead of staying flat, and if you set k to 1 in this calculator every figure will agree with our exponential distribution calculator exactly, provided you remember that page offers a rate parameterisation as well as a mean.
This relationship is why the Weibull is worth reaching for at all. The exponential assumes the memoryless property: a component that has run for a year is exactly as likely to fail in the next hour as a brand-new one. That is true of radioactive decay and roughly true of some electronic failure modes. It is plainly false for anything that wears, corrodes, fatigues or beds in. The shape parameter is the dial that lets you say so.
Reading the Shape Parameter as Physics
Because k controls the hazard rate rather than just the spread, a fitted shape parameter carries a claim about failure mechanism. A k well below 1 gives a hazard rate that falls steeply with age, which is the signature of infant mortality: manufacturing defects and installation errors surfacing early, after which survivors are more reliable than new units. The engineering response is burn-in and better incoming inspection, not longer warranties.
A k near 1 means a flat hazard rate and no useful age signal. Failures arrive at random and replacing a component on a schedule buys you nothing, because the replacement is no less likely to fail than the part it replaced. Preventive replacement intervals only make sense when the hazard is rising.
A k comfortably above 1 means wear-out, and the higher it goes the tighter the failures cluster around a characteristic life. Values in the range of three to four produce a roughly symmetric distribution that looks much like a normal curve, which is one reason a Weibull fit sometimes wins over a normal fit on the same data without changing any conclusion. NIST’s reliability chapter on Weibull lifetime models sets out the extreme-value argument for why a weakest-link failure process produces this distribution in the first place.
What the Calculator Cannot Tell You
Every figure on this page is conditional on the parameters you typed. The Weibull is a model, and fitting one to a data set is a separate exercise with its own uncertainty. A shape parameter estimated from twelve failures has a confidence interval wide enough to span infant mortality and wear-out simultaneously, and a point estimate reported without that interval is close to meaningless.
Censoring matters just as much. Reliability data almost always includes units that were still running when the study ended, and treating those as failures or dropping them entirely both bias the fit — in opposite directions. A proper fit uses maximum likelihood with the censoring structure represented explicitly, and that is the point at which a general-purpose calculator hands over to statistical software.
Finally, a Weibull fitted to one operating environment does not transfer to another. Temperature, duty cycle, vibration and load all move the parameters, which is the whole subject of accelerated life testing. If your source data came from a bench test, the numbers here describe the bench.
Where the Weibull Turns Up Outside Reliability
Wind engineering uses it to describe the distribution of wind speeds at a site, with shape parameters typically clustering around 2 — the special case known as the Rayleigh distribution, which is the Weibull with k fixed at 2. Hydrology uses it for flood and drought extremes. Materials science uses it for the strength of brittle solids, where the weakest-link argument applies literally: a ceramic or glass specimen fails at its worst flaw, and the distribution of worst flaws across specimens is Weibull.
Survival analysis in medicine uses it as a parametric alternative to Cox regression when a specific hazard shape is defensible. In each of these fields the same two parameters mean the same two things, which is why getting the parameterisation right transfers across all of them. If you are working with count data instead of times, our Poisson distribution calculator and geometric distribution calculator cover the discrete equivalents, and our normal distribution calculator handles the symmetric case a high-shape Weibull starts to resemble.
Arb Digital builds free calculators and reference pages that state their assumptions instead of hiding them. Browse the library, or tell us what your readers keep getting wrong.
Browse Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Entering a rate where a scale is expected. Some texts and some software parameterise the Weibull by 1÷λ. Check which one your source uses before typing anything.
- Reading the scale as an average life. λ is the 63.2nd percentile, not the mean. The mean is λΓ(1 + 1÷k) and only equals λ at k = 1 by coincidence of the gamma function’s value there.
- Confusing the density with the hazard rate. The density is measured across the original population, the hazard across survivors only. They diverge sharply at high x.
- Fitting a three-parameter Weibull without a physical reason. A free threshold will always improve the fit and will frequently be fitting noise.
- Quoting a shape parameter without its confidence interval. On small samples that interval commonly spans values implying opposite failure mechanisms.
Related Free Tools From Arb Digital
Set k to 1 and compare with the exponential distribution calculator, work with counts instead of times using the Poisson distribution calculator or the geometric distribution calculator, handle skewed positive data with the lognormal distribution calculator, and cover the symmetric case with the normal distribution calculator or the beta distribution calculator. The full free online tools hub lists everything else.
Frequently Asked Questions
The shape and scale form. The second parameter is the scale, in the same units as x, not a rate. If your source quotes a rate, take its reciprocal before entering it here, otherwise the answer will be wrong by a large factor rather than a small one.
It sets how the hazard rate changes with age. Below 1 the hazard falls, which suggests infant mortality. At exactly 1 it is constant. Above 1 it rises, which is the signature of wear-out, and higher values cluster failures more tightly.
Yes. Setting the shape parameter to 1 collapses the Weibull to the exponential with mean equal to the scale parameter, and a constant hazard rate of one divided by the scale. Every figure will then match an exponential calculator exactly.
Because at x equal to the scale, the exponent in the reliability function becomes 1 for any shape, so reliability is e to the minus 1, or 36.8 percent. That leaves 63.2 percent failed, independent of the shape parameter.
The time by which 10 percent of a population has failed, which is the 10th percentile of the distribution. It is a common specification figure for bearings and mechanical components. Set the percentile field to 0.10 to read it.
The density gives the chance of failing near a value measured across the whole original population. The hazard rate gives the chance of failing near that value given survival up to it, measured across survivors only. For wear-out the density can fall while the hazard keeps rising.
Only when there is a physical reason to believe failure is impossible before a certain age, such as a required run-in period. A freely fitted threshold nearly always improves the apparent fit while modelling noise rather than mechanism.
This page evaluates a probability model from the parameters you supply. It does not fit a distribution to data, does not account for censoring, and produces no reliability, safety or engineering conclusion on its own. Any decision about component life or maintenance intervals should be taken with a qualified reliability engineer using your own validated data.
