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Johnson SB distribution

From phenomenon to model

N. L. Johnson introduced his curve systems in 1949 as transformations of a normal variable. SB means bounded, contrasting with unbounded SU.

What to look at

The support extends from xi to xi+lambda, with lambda > 0. gamma and delta control the normalizing transformation; the bounds are explicit model components.

Useful relationships

The logistic transformation links SB to normal and Beta-like bounded models. Johnson SU uses a hyperbolic-sine transformation and spans the real line; both families normalize non-normal data.

Names you may also encounter

It is also called the Johnson bounded system or Johnson SB family.

When it makes sense

SB can outperform Beta when transformation to normality gives the needed flexibility. Bounds xi and xi+lambda must be known or estimated from enough information.

What not to assume

If sample endpoints are censoring, do not treat them as physical bounds without modelling censoring; SB can absorb the artefact.

Straightening a bounded variable until it is Normal

Johnson SB applies a log-odds transformation to relative position inside an interval and requires the result to be Normal. The letters SB denote the bounded system: support lies between xi and xi+lambda.

In 1949 Johnson designed a transformation system to cover regions of skewness and kurtosis left open by classical families. SB helps when bounds are real and the interior shape is complex. Its transformation also supplies a diagnostic: transformed data should look Normal. If bounds are estimated, they may move too close to observed extremes and make log odds fragile. Known bounds substantially strengthen interpretation.

Decision guide

A good candidate when: data are bounded and skewed, and a logistic transformation can make them approximately Normal.

Compare it with: Beta and Kumaraswamy. Johnson SB is more flexible, but estimated endpoints can dominate the transformation; validate their stability out of sample.

References

Johnson SB Distribution: equations and calculator

Distribution defintion

X∼JohnsonSB(ξ,λ,γ,δ)X\sim\mathrm{JohnsonSB}\left(\xi,\lambda,\gamma,\delta\right )

Distribution domain

x∈(ξ,ξ+λ)x\in\left(\xi,\xi+\lambda\right)

Parameters domain and parameters constraints

ξ∈R,λ∈R+,γ∈R,δ∈R+\xi\in\mathbb{R},\lambda\in\mathbb{R}^{+},\gamma\in\mathbb{R},\delta\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=Φ(γ+δln⁡z(x)1−z(x))F_{X}\left(x\right)=\Phi\left(\gamma+\delta\ln\frac{z(x)}{1-z(x)}\right)

Probability density function

fX(x)=δλ2πz(1−z(x))exp⁡[−12(γ+δln⁡z(x)1−z(x))2]f_{X}\left(x\right)=\frac{\delta}{\lambda\sqrt{2\pi}z(1-z(x))}\exp\left[-\frac{1}{2}\left(\gamma+\delta\ln\frac{z(x)}{1-z(x)}\right)^2\right]

Percent point function/Sample

FX−1(u)=λexp⁡(Φ−1(u)−γδ)1+exp⁡(Φ−1(u)−γδ)+ξF^{-1}_{X}\left(u\right)=\frac{\lambda\exp\left(\frac{\Phi^{-1}(u)-\gamma}{\delta}\right)}{1+\exp\left(\frac{\Phi^{-1}(u)-\gamma}{\delta}\right)}+\xi

Non-central parametric moments

μk′=E[Xk]=∫ξξ+λxkfX(x)dx\mu'_{k}=E[X^k]=\int_{\xi}^{\xi+\lambda }x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′\mathrm{Mean}(X)=\mu'_{1}

Parametric variance

Variance(X)=μ2′−μ1′2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}

Parametric median

Median(X)=λexp⁡(Φ−1(1/2)−γδ)1+exp⁡(Φ−1(1/2)−γδ)+ξ\mathrm{Median}(X)=\frac{\lambda\exp\left(\frac{\Phi^{-1}\left(1/2\right)-\gamma}{\delta}\right)}{1+\exp\left(\frac{\Phi^{-1}\left(1/2\right)-\gamma}{\delta}\right)}+\xi

Parametric mode

Mode(X)=arg⁡max⁡xfX(x)\mathrm{Mode}(X)=\arg\max_{x}f_{X}\left(x\right)

Additional information and definitions

ξ:Location parameter\xi:\text{Location parameter}
λ:Scale parameter\lambda:\text{Scale parameter}
z(x)=(x−ξ)/λz\left(x\right)=\left(x-\xi\right)/\lambda
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Φ(x):CDF normal standard distribution\Phi\left(x\right):\text{CDF normal standard distribution}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}