Formalized by John Tukey, the Tukey lambda distribution is a continuous, symmetric probability distribution defined in terms of its quantile function.

It is typically used to identify an appropriate distribution (see the comments below) and not used in statistical models directly.

For the standard form of the Tukey lambda distribution, the quantile function,

Differently from the central moments, L-moments can be expressed in a closed form.

For example, The most common use of this distribution is to generate a Tukey lambda PPCC plot of a data set.

Based on the value for λ with best correlation, as shown on the PPCC plot, an appropriate model for the data is suggested.

Similarly, an optimal curve-fit value of λ greater than 0.14 suggests a distribution with exceptionally thin tails (based on the point of view that the normal distribution itself is thin-tailed to begin with; the exponential distribution is often chosen as the exemplar of tails intermediate between fat and thin).

Except for values of λ approaching 0 and those below, all the PDF functions discussed have finite support, between   ⁠−1  /|λ|⁠   and   ⁠+1  / |λ| ⁠ .

[4] This article incorporates public domain material from the National Institute of Standards and Technology