Johnson Distributions
What is the Johnson family of distributions?
The Johnson family is a group of continuous probability distributions that can model many different shapes of data. It includes distributions for values that are unbounded (U), bounded on both sides (B), or bounded on one side (L). Each Johnson distribution is created by transforming a normal distribution and uses parameters that control its location, spread, symmetry, and tail behavior.
What are the Johnson distributions?
The family includes three distributions. The best choice depends on the range of values the data can take:
- Johnson SU: For data with no fixed lower or upper limit, such as measurements that could extend in either direction. It has four parameters.
- Johnson SB: For data with both a lower limit and an upper limit, such as a proportion between 0 and 1. It has four parameters.
- Johnson SL: For data with a limit on only one side, such as positive measurements with no fixed maximum. It has three parameters.
When should I consider using a Johnson distribution?
Consider the Johnson family when a familiar distribution does not match the shape of your data. It is especially useful when the data are skewed, have unusually light or heavy tails, or are restricted by one or two boundaries. The family can represent many combinations of mean, variance, skewness, and kurtosis. Because the method transforms the data to a normal distribution, it can also support analyses that rely on normality, such as calculating the process capability index Cpk.
What are the characteristics of the Johnson distributions?
All three Johnson distributions are related to the normal distribution through a mathematical transformation. Their parameters describe where the distribution is centered, how widely values are spread, how asymmetric it is, and how much probability lies in the tails. In the table, $\theta$ is a location parameter, $\sigma$ is a scale parameter, and $\gamma$ and $\delta$ are shape parameters. The transformation differs for each member of the family because each one supports a different range of possible values.
| Model parameters | |
| Johnson SU |
$\theta$, location $\sigma$, scale $\gamma$, shape, skewness $\delta$, shape, tail weight |
| Johnson SB |
$\theta$, location $\sigma$, scale $\gamma$, shape, skewness $\delta$, shape, tail weight |
| Johnson SL |
$\sigma$, scale $\gamma$, shape, skewness $\delta$, shape, tail weight |
Let Z be a standard normal random variable. The transformed normal is $Z = \gamma + \delta f(Y)$.
| Transformation | |
| Johnson SU | $ \begin{aligned} f(Y) &= \log\!\left(Y + \sqrt{1 + Y^2}\right) = \sinh^{-1} Y \\ Y &= \frac{X-\theta}{\sigma}, \quad -\infty < X < \infty \end{aligned} $ |
| Johnson SB | $ \begin{aligned} f(Y) &= \log\!\left(\frac{Y}{1-Y}\right) \\ Y &= \frac{X-\theta}{\sigma}, \quad \theta < X < \theta + \sigma \end{aligned} $ |
| Johnson SL |
$ \begin{aligned} \sigma &= \pm 1 \\ f(Y) &= \log(Y) \end{aligned} $ $ Y=\frac{X-\theta}{\sigma}, \begin{aligned}[t] &X>\theta,\text{ if }\sigma=1\\ &X<\theta,\text{ if }\sigma=-1 \end{aligned} $ |