Free Weibull Distribution Calculator

The Weibull distribution is a flexible continuous probability distribution that can model a wide variety of data shapes, from exponential decay to near‑symmetry. This free online Weibull Distribution Calculator lets you compute probabilities, generate random samples, evaluate the pdf and cdf, determine quantiles, and obtain key statistical measures—all in one place. Below we cover the distribution’s parameters, core formulas, and how to use the tool.

Parameters

Two positive parameters define the distribution:

• Shape $k$ – determines the curve’s form. For $k=1$ the distribution reduces to the exponential; for $k=2$ it becomes the Rayleigh distribution; with $k \approx 3.6$ the pdf is nearly symmetric. Values between give various right‑skewed shapes.
• Scale $\lambda$ – stretches or compresses the distribution along the x‑axis. A larger $\lambda$ spreads the distribution out, while a smaller one concentrates it near zero.

Because $k$ controls the hazard rate (decreasing for $k<1$, constant for $k=1$, increasing for $k>1$), the Weibull distribution is especially popular in reliability engineering, survival analysis, hydrology, and electrical system modeling.

Probability Density Function (PDF)

For $x \ge 0$,

$$f(x) = \dfrac{k}{\lambda} \left( \dfrac{x}{\lambda} \right)^{k-1} e^{-(x/\lambda)^k}$$

Cumulative Distribution Function (CDF)

The CDF simplifies to a closed form:

$$F(x) = 1 - e^{-(x/\lambda)^k}, \quad x \ge 0$$

Hence $P(X \le x) = F(x)$ and $P(X > x) = 1 - F(x)$.

Quantile Function and Sampling

The quantile function (inverse CDF) for $0 \le p < 1$ is

$$Q(p) = \lambda \bigl[ -\ln(1-p) \bigr]^{1/k}$$

To generate a random Weibull value, draw $U \sim \text{Uniform}(0,1)$ and set

$$X = \lambda (-\ln U)^{1/k}$$

This method is used by the calculator’s sample generation mode.

Common Statistical Measures

Let $\Gamma$ denote the gamma function. The main measures are:

Measure	Formula
Mean	$\mu = \lambda \Gamma\left(1 + \frac{1}{k}\right)$
Median	$m = \lambda (\ln 2)^{1/k}$
Mode (when $k>1$)	$\displaystyle \lambda \left( \frac{k-1}{k} \right)^{1/k}$ (if $k \le 1$ the mode is 0)
Variance	$\sigma^2 = \lambda^2 \left[ \Gamma\!\left(1+\frac{2}{k}\right) - \bigl(\Gamma\!\left(1+\frac{1}{k}\right)\bigr)^2 \right]$
Standard deviation	$\sigma = \sqrt{\sigma^2}$
Skewness	$\displaystyle \frac{\Gamma\!\left(1+\frac{3}{k}\right) - 3\,\Gamma\!\left(1+\frac{1}{k}\right)\Gamma\!\left(1+\frac{2}{k}\right) + 2\!\left[\Gamma\!\left(1+\frac{1}{k}\right)\right]^3}{\bigl[ \Gamma\!\left(1+\frac{2}{k}\right) - \bigl(\Gamma\!\left(1+\frac{1}{k}\right)\bigr)^2 \bigr]^{3/2}}$

Skewness depends only on $k$; for example, at $k=2$ it is about 0.631, and near $k=3.6$ it approaches zero.

Using the Calculator

The tool offers six operating modes:

• Probabilities – obtain $P(X \le x)$ or $P(X > x)$.
• Samples – generate any desired number of random draws.
• CDF – evaluate $F(x)$ at a given $x$.
• PDF – evaluate $f(x)$ at a given $x$.
• Quantiles – find the value that corresponds to a specified probability.
• Distribution Measures – view mean, median, mode, variance, and skewness.

To get a result, select the mode, enter the Weibull parameters $\lambda$ and $k$, and fill in any additional required fields (such as sample size or argument $x$). The calculator then instantly returns the answer. With this free online Weibull Distribution Calculator you can explore the distribution’s behavior without manual computation.