Nonlinear transformations

Nonlinear transformations arise when the values are replaced by a nonlinear function of the original measurements, such as their logarithm or inverse. They have a more fundamental effect on the shape of a distribution than linear transformations.

The most commonly used nonlinear transformation is:

new value   = log10 (old value)

Natural logarithms (base e) have a similar effect on the distribution of values but base-10 logarithms are easier to interpret so we use them here.

Properties of logarithms

Consider four values 1, 10, 100 and 1000. The first two values are much closer to each other than the last two values. However their logarithms are 0, 1, 2 and 3, so their logarithms are evenly spaced out.

Effect on the shape of a distribution

A logarithmic transformation selectively spreads out low values in a distribution and compresses high values. It is therefore useful before analysing skew data with a long tail towards the high values. It will spread out a dense cluster of low values and may detect clustering or outliers that would not be visible in graphical displays of the original data.