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The Weibull equation describes how long something lasts before an event such as a failure or a death. It is a two-parameter (sometimes three-parameter) probability distribution defined by a shape and a scale. This page gives the PDF and CDF, a fully worked numeric example, the notation differences between textbooks and software, and short Excel, R and Python recipes.
Weibull equation and probability distribution: the definition
The Weibull probability distribution is a continuous distribution for non-negative values, usually time to an event. Its shape parameter controls whether the event rate falls, stays flat or rises over time; its scale parameter stretches the distribution along the time axis. The NIST/SEMATECH e-Handbook of Statistical Methods (section 1.3.6.6.8, checked 2026-11-07) gives the standard reference form, and the formulas below follow it.
The Weibull PDF and CDF formulas, with every symbol
For the two-parameter form, with shape k > 0 and scale λ > 0, for x ≥ 0:
- PDF: f(x) = (k/λ) · (x/λ)^(k−1) · exp(−(x/λ)^k)
- CDF: F(x) = 1 − exp(−(x/λ)^k)
- Survival function: S(x) = 1 − F(x) = exp(−(x/λ)^k)
- Hazard function: h(x) = (k/λ) · (x/λ)^(k−1)
- Mean: λ · Γ(1 + 1/k); variance: λ² · [Γ(1 + 2/k) − Γ(1 + 1/k)²]
- Median: λ · (ln 2)^(1/k), which follows from setting F(x) = 0.5.
Here x is the elapsed time (or other non-negative quantity), exp is the exponential function, ln is the natural logarithm, and Γ is the gamma function.
What are the three parameters of the Weibull distribution?
The three parameters are shape, scale and location (threshold). The location parameter shifts the whole distribution to the right so that nothing can happen before it; setting it to zero gives the common two-parameter Weibull. The NIST handbook writes the three-parameter PDF as f(x) = (γ/α) · ((x−μ)/α)^(γ−1) · exp(−((x−μ)/α)^γ) for x ≥ μ, with γ the shape, μ the location and α the scale. When μ = 0 and α = 1 it is the standard Weibull, whose CDF is 1 − exp(−x^γ).
Notation crosswalk: the same parameters under different letters
| Role | NIST handbook | Common textbook/Wikipedia | ReliaSoft Weibull++ |
|---|---|---|---|
| Shape | γ | k | β |
| Scale | α | λ | η (characteristic life) |
| Location (failure-free time) | μ | θ | γ |
The letter γ means the shape in NIST but the location in ReliaSoft (sources: NIST handbook section 1.3.6.6.8, Wikipedia “Weibull distribution”, ReliaSoft Weibull++ help, all checked 2026-11-07). Check which convention your software uses before copying a number into a report.
Worked example with the arithmetic shown
Suppose time to failure of a component is Weibull with shape k = 1.5 and scale λ = 1,000 hours. This is an illustrative calculation, not data from a study.
- Probability of failure by 500 hours. (500/1000)^1.5 = 0.5^1.5 = 0.35355. F(500) = 1 − exp(−0.35355) = 1 − 0.70219 = 0.2978, so about 29.8% fail by 500 hours and 70.2% survive.
- Median. 1000 · (ln 2)^(1/1.5) = 1000 · 0.6931^0.6667 = 1000 · 0.78322 = 783.2 hours.
- Mean. 1000 · Γ(1 + 1/1.5) = 1000 · Γ(1.6667) = 1000 · 0.90275 = 902.7 hours. The mean exceeds the median because the distribution is right-skewed at this shape.
- Standard deviation. Variance = 1000² · [Γ(1 + 2/1.5) − Γ(1.6667)²] = 10⁶ · (1.19064 − 0.81495) = 375,690, so the standard deviation is 612.9 hours.
- 90th percentile. Set F(x) = 0.9: x = 1000 · (−ln 0.1)^(1/1.5) = 1000 · 2.3026^0.6667 = 1,743.7 hours.
- Hazard at 500 hours. h(500) = (1.5/1000) · 0.5^0.5 = 0.0015 · 0.70711 = 0.001061 per hour. The PDF there is h(500) × S(500) = 0.001061 × 0.70219 = 0.000745.
A check on any Weibull: at x = λ, F = 1 − e^(−1) = 0.632 regardless of the shape. ReliaSoft describes the scale η as the characteristic life for this reason.
How to interpret the Weibull distribution
Read the shape first, because it tells you how the hazard (instantaneous event rate) behaves over time. Then read the scale as the time by which about 63.2% of units have had the event.
| Shape (k) | Hazard over time | Typical reading |
|---|---|---|
| k < 1 | Decreasing | Early failures, “infant mortality” |
| k = 1 | Constant | Exponential distribution, memoryless events |
| k > 1 | Increasing | Wear-out or ageing |
| k = 2 | Increasing linearly | Rayleigh distribution |
The hazard pattern is from Wikipedia “Weibull distribution” and the NIST handbook (hazard = γx^(γ−1) in the standard form). In the example above k = 1.5, so the hazard rises steadily but not linearly.
When to use Weibull vs normal distribution
Use a Weibull when the variable is a positive waiting or lifetime measure and its shape is skewed; use a normal when values are roughly symmetric and can in principle fall on either side of the mean. A normal distribution assigns some probability to negative times, which a Weibull (with zero location) does not. The Weibull is skewed at most shapes, so a normal approximation is poor for strongly skewed lifetimes. Compare fits with a probability plot and a goodness-of-fit measure rather than choosing by habit. For censored lifetimes with covariates, see accelerated failure time models, where the Weibull has a special role. For the distribution of block maxima, which includes a reversed Weibull case, see the generalized extreme value distribution.
Assumptions and common mistakes
- Mixing parameter conventions. Excel, R, SciPy, NIST and ReliaSoft order and name parameters differently (see the table above and the recipes below).
- Using two parameters when there is a failure-free period. If nothing can happen before some time, add the location parameter.
- Ignoring censoring. Units that have not yet had the event still carry information. Choose an estimation method that handles censored data when you have it.
- Reporting a bare mean. Because the distribution is skewed, report the median and a percentile with the parameters.
How to report a Weibull fit in APA style
Name the model, the estimation method, the parameter estimates with their uncertainty, and the fit check. An illustrative sentence, using the example values above (not real data): “Time to failure was modelled with a two-parameter Weibull distribution fitted by maximum likelihood (shape k = 1.50, scale λ = 1,000 hours), indicating an increasing hazard; the median time to failure was 783 hours.” Add confidence intervals and the software and version used.
How can I use the Weibull distribution in Excel, R or Python?
Excel has WEIBULL.DIST(x, alpha, beta, cumulative); Microsoft’s documentation shows TRUE returns the cumulative distribution and FALSE the density, and that alpha = 1 gives the exponential case, so alpha is the shape and beta the scale (page checked 2026-11-07).
Excel
=WEIBULL.DIST(500, 1.5, 1000, TRUE) -> 0.2978 (CDF)
=WEIBULL.DIST(500, 1.5, 1000, FALSE) -> 0.000745 (PDF)R
pweibull(500, shape = 1.5, scale = 1000) # 0.2978
dweibull(500, shape = 1.5, scale = 1000) # 0.000745
qweibull(0.9, shape = 1.5, scale = 1000) # 1743.7
# fit to data x: MASS::fitdistr(x, "weibull")Python (SciPy)
from scipy.stats import weibull_min
d = weibull_min(c=1.5, scale=1000) # c = shape, loc = 0
d.cdf(500) # 0.2978
d.ppf(0.9) # 1743.7
# fit with the location fixed at 0: weibull_min.fit(data, floc=0)The SciPy name is weibull_min, with shape c, loc and scale. Percentile concepts are covered in percentile and quantile.
FAQ
What is the Weibull equation?
It is usually the CDF F(x) = 1 − exp(−(x/λ)^k) or its density f(x) = (k/λ)(x/λ)^(k−1)exp(−(x/λ)^k), with shape k and scale λ.
How do I interpret the shape parameter?
Below 1 the hazard falls, at 1 it is constant, and above 1 it rises with time.
Is the Weibull the same as the exponential?
The exponential is the special case k = 1; see the exponential distribution formula.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.8 Weibull Distribution: https://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm (checked 2026-11-07).
- ReliaSoft Weibull++ help, The Weibull Distribution: https://help.reliasoft.com/reference/life_data_analysis/lda/the_weibull_distribution.html (checked 2026-11-07).
- Microsoft Support, WEIBULL.DIST function: https://support.microsoft.com/en-us/office/weibull-dist-function-4e783c39-9325-49be-bbc9-a83ef82b45db (checked 2026-11-07).
- Wikipedia, Weibull distribution: https://en.wikipedia.org/wiki/Weibull_distribution (checked 2026-11-07, corroboration).
