Johnson's SU-distribution¶

An unbounded four-parameter distribution obtained by applying an inverse-hyperbolic-sine transformation to a standard normal variable.

Version: v2 · 2026-09-19 · History
Domain-specific #: 5144
Domain group: Formal Sciences
Origin domain: Experimental Design & Statistics
Subdomain: Probability Distributions → Experimental Design & Statistics
Aliases: Johnson SU distribution

Core Idea¶

Location xi and positive scale lambda transform x, while shape parameters gamma and positive delta make gamma plus delta asinh of the standardized value normally distributed, allowing broad skewness and kurtosis. A normal variate is shifted and rescaled in transformed space, then a hyperbolic-sine inverse map produces real-valued observations with adjustable asymmetry and tail behavior. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

No faithful explanation at this level. Any picture simple enough for a five-year-old turns this into 'a bendy, lopsided hill' that can be drawn freely. It loses that the shape is a rule for how likely different numbers are. It also loses that the shape comes from one specific transformation that turns the values into a normal bell curve.

The Stretchy Bell Curve

When scientists collect many numbers, they often draw how common each value is, and a lot of data makes a bell-shaped curve. But real data can lean to one side or have more extreme values than a bell curve allows. Johnson's SU-distribution is a family of shapes made by starting from a bell curve and running it through a special stretching formula. Four settings control where it sits, how wide it is, how much it leans, and how heavy its far ends are. Its values can be any number, positive or negative.

Asinh-Transformed Normal Family

Johnson's SU-distribution is a probability distribution built by transforming a normal (bell-curve) variable. It has four parameters: location ξ, scale λ > 0, and shape parameters γ and δ > 0. A value x belongs to the distribution if γ + δ·asinh((x − ξ)/λ) is standard normal, where asinh is the inverse hyperbolic sine. Going the other way, you take a normal value, shift and rescale it, and apply the hyperbolic sine to get x. Because of this mapping, the distribution can produce a wide range of skewness (lopsidedness) and kurtosis (tail heaviness). Values can be any real number, unlike some related distributions that are bounded or one-sided.

Johnson's SU-distribution is a member of Johnson's translation system, defined by a normalizing transformation. With location ξ, scale λ > 0, and shape parameters γ and δ > 0, the variable X has an SU distribution if Z = γ + δ·asinh((X − ξ)/λ) is standard normal. Equivalently, X = ξ + λ·sinh((Z − γ)/δ), so a normal variate is shifted and rescaled in transformed space and then mapped through sinh, yielding support on the entire real line. The parameter γ chiefly governs asymmetry and δ governs tail behavior, giving coverage of a broad region of skewness–kurtosis combinations. Because the transformation is explicit, the density and CDF follow directly from the normal ones via change of variables. A proper use states all four parameters with their sign constraints, the location-scale convention, the density and CDF, which moments exist, and how the parameters were estimated or fitted. It is distinct from other Johnson families (such as bounded SB) and from generic skewed distributions that lack this asinh-to-normal structure.

Scope of Application¶

Johnson's SU-distribution belongs to probability distributions and is useful where the analyst can specify the typed probability distributions carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the four parameters and sign constraints, support on the real line, normalizing transformation, density and CDF, location-scale convention, moment existence and estimation or fitting method are explicit. The scope is broad within that domain but bounded by the need for the four parameters and sign constraints, support on the real line, normalizing transformation, density and CDF, location-scale convention, moment existence and estimation or fitting method are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the four parameters and sign constraints, support on the real line, normalizing transformation, density and CDF, location-scale convention, moment existence and estimation or fitting method are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Johnson's SU-distribution can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Johnson's SU-distribution. Johnson's SU-distribution compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed probability distributions carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the four parameters and sign constraints, support on the real line, normalizing transformation, density and CDF, location-scale convention, moment existence and estimation or fitting method are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of probability distributions because they reuse the typed probability distributions carrier, including its objects, relations, parameters, conventions, evidence, boundary cases, and comparison targets, A normal variate is shifted and rescaled in transformed space, then a hyperbolic-sine inverse map produces real-valued observations with adjustable asymmetry and tail behavior., and type the carrier, state every parameter and convention in the definition, test that the four parameters and sign constraints, support on the real line, normalizing transformation, density and CDF, location-scale convention, moment existence and estimation or fitting method are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Johnson's SU-distribution Domain-specific

Parents (1) — outgoing typed relations

Johnson's SU-distribution is a kind of Probability Prime

It remains its own entry because its identity is fixed by the four parameters and sign constraints, support on the real line, normalizing transformation, density and CDF, location-scale convention, moment existence and estimation or fitting method.

Hierarchy path (1) — routes to 1 parentless root

Johnson's SU-distribution → is a kind of Probability → is a kind of Measure → presupposes Set and Membership

Neighborhood in Abstraction Space¶

Johnson's SU-distribution sits in a crowded region of the domain-specific corpus (21^st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Probability Distributions & Transforms (22 abstractions)

Nearest neighbors

Modified half-normal distribution — 0.91
Inverse distribution — 0.90
Hyperbolic distribution — 0.90
Truncated normal distribution — 0.89
Exchangeable random variables — 0.89

Computed from structural-signature embeddings · 2026-10-25