WOLFRAM|DEMONSTRATIONS PROJECT

Choosing a Data Transformation with the Box-Whisker Plot

999

data distribution

normal

exponential

lognormal

inverse Gaussian

Weibull

random seed

212

This Demonstration shows the effect of a power data transformation,

(λ)

=

 λ y -1λ	λ≠0
log(y)	λ=0

on data,

, from simulated samples of size

n=50

n=999

from normal, exponential, lognormal, inverse Gaussian, or Weibull distributions for

λ∈(-2,2)

In practice, a suitable power transformation can be selected by examining the effect of the transformation using a box-and-whisker plot. The simplest power transformation which makes the data approximately symmetric is selected. With actual data, often

λ=-1,0,0.5,1

corresponding to reciprocal, log, square root, or no transformation.

Two skewness statistics—the usual Pearson skewness,

, and the Bowley skewness,

—are displayed for comparison with the plot.

Another method for choosing

treats

as a parameter and makes the assumption that for some value of

, the data is normally distributed. Under this assumption, the likelihood function

ℒ(λ)

may be obtained and it may be numerically maximized to obtain the maximum likelihood estimate for



=argmaxℒ(λ)

. A range of plausible values for

is given by all

for which

R(λ)>1%

, where

R(λ)=ℒ(λ)ℒ





Try experimenting with different sample sizes

and different distributions.

In actual applications, real data (not simulated data) would be used. Using a suitable power transformation often simplifies the statistical analysis.