The Weibull distribution is a continuous probability distribution that can fit an extensive range of distribution shapes. Like the normal distribution, the Weibull distribution describes the probabilities associated with continuous data. However, unlike the normal distribution, it can also model skewed data. In fact, its extreme flexibility allows it to model both left- and right-skewed data.
This distribution is an unusually versatile probability distribution because it can fit a variety of shapes. It can even approximate the normal distribution and other distributions. Because of its flexibility, analysts use it in a broad range of settings, such as quality control, capability analysis, medical studies, and engineering. It’s frequently used in life data, reliability analysis, and warranty analysis to assess time to failure for systems and parts.
The distribution is named after Swedish mathematician Waloddi Weibull, who presented it to the American Society of Mechanical Engineers (ASME) in 1951. However, Weibull didn’t discovered this distribution. Indeed, other mathematicians had been using this probability distribution for decades. One early use for it was modeling particle sizes in 1933.
There are two versions of this distribution. The three-parameter Weibull distribution, unsurprisingly, has three parameters, shape, scale, and threshold. When analysts set the threshold parameter to zero, it is known as the two-parameter Weibull distribution.
Analysts use the Weibull distribution frequently because it is so adaptable to varying conditions. The nature of the distribution changes significantly based on the values of the parameters. In fact, with certain parameter values, the Weibull distribution is equivalent to other probability distributions. Let’s investigate how changing the parameters affects it!
Related post: Understanding Probability Distributions
Weibull Threshold Parameter (γ)
The threshold parameter defines the lowest possible value in a Weibull distribution. Some analysts refer to this parameter as the location. All values must be greater than the threshold. Consequently, negative threshold values let the distribution handle both negative and positive values. Zero allows it to contain only positive values. A two-parameter Weibull distribution simply has the threshold set to zero. Statisticians denote the threshold parameter with γ.
When you hold the shape and scale parameters constant, the threshold shifts the distribution left and right. In the probability distribution plot below, I show the threshold’s effect in action!
Weibull Shape Parameter (β, k)
Unsurprisingly, the shape parameter describes the shape of your data’s distribution. Statisticians also refer to it as the Weibull slope because its value equals the slope of the line on a probability plot. Statisticians denote the shape parameter using either beta (β) or k.
Keep in mind the following four key ranges of shape values for the Weibull distribution. In these probability distribution plots, I hold the scale and threshold parameters constant to highlight the impact of changing the shape.
Shape < 1: Steadily decreasing values
When the shape parameter equals 1, the Weibull distribution is equivalent to a two-parameter exponential distribution.
Related post: Using the Exponential Distribution
Shape between 1 and 2.6: Right-skewed
A Weibull distribution with a shape value of 2 is a Rayleigh distribution, which is equivalent to a Chi-square distribution with two degrees of freedom.
Shape near 3: Approximates a normal distribution.
Related post: Normal Distribution
Shape > 3.7: Left-skewed
Weibull Shapes and Failure Rates
Additionally, if you’re studying failure rates, shape values provide critical information about how the failure rate changes over time. When the shape is:
- < 1, the failure rate decreases over time (e.g., infant mortality failures).
- = 1, the failure rate is constant over time.
- > 1, the failure rate increases over time (e.g., wear-out failures).
Weibull Scale Parameter (η, λ)
The scale parameter represents the variability present in the distribution. Changing the scale parameter affects how far the probability distribution stretches out. As you increase the scale, the distribution stretches further right, and the height decreases. Decreasing the scale shrinks the distribution to the left and increases its peak, as shown below. Statisticians denote the scale parameter using either eta (η) or lambda (λ).
The value of the scale parameter equals the 63.2 percentile in the distribution. 63.2% of the values in the distribution are less than the scale value.
Even though the Weibull distribution fits many shapes, it’s not always the best choice. Learn how to identify the probability distribution that best fits your data.