For the purposes of data transformation, the degree of skewness of a skewed distribution can be classified as moderate, high or extreme. Depending upon the degree of skewness and whether the direction of skewness is positive or negative, a different approach to transformation is often required. As a short-cut, uni-modal distributions can be roughly classified into the following transformation categories:
This article explores the transformation of a positively skewed distribution with a high degree of skewness. The following example takes medical device sales in thousands for a sample of 2000 diverse companies. The histogram below indicates that the original data could be classified as “high(er)” positive skewed.
The skew is in fact quite pronounced - the maximum value on the x axis extends beyond 250 (the frequency of sales volumes beyond 60 are so sparse as to make the extent of the right tail imperceptible) - it is however the highly leptokurtic distribution that that lends this variable to be better classified as high rather than extreme. It is in fact log-normal - convenient for the present demonstration. From inspection it appears that the log transformation will be the best fit in terms of normalising the distribution.
Starting with a more conservative option, the square root transformation, a major improvement in the distribution is achieved already. The extreme observations contained in the right tail are now more visible. The right tail has been pulled in considerably and a left tail has been introduced. The kurtosis of the distribution has reduced by more than two thirds.
A natural log transformation proves to be an incremental improvement yielding the following results:
This is quite a good outcome - the right tail has been reduced considerably while the left tail has extended along the number line to create symmetry. The distribution now roughly approximates a normal distribution. An outlier has emerged at around -4.25, while extreme values of the right tail have been eliminated. The kurtosis has again reduced considerably.
Taking things a step further and apply a log to base 10 transformation yields the following:
In this case the right tail has been pulled in even further and the left tail extended less than the previous example. Symmetry has improved and the extreme value in the left tail has been bought closer in to around -2. The log to base ten transformation has provided an ideal result - successfully transforming the log normally distributed sales data to normal.
In order to illustrate what happens when a transformation that is too extreme for the data is chosen, an inverse transformation has been applied to the original sales data below.
Here we can see that the right tail of the distribution has been brought in quite considerably to the extent of increasing the kurtosis. Extreme values have been pulled in slightly but still extend sparsely out towards 100. The results of this transformation are far from desirable overall.
Some thing to note is that in this case the log transformation has caused data that was previously greater than zero to now be located on both sides of the number line. Depending upon the context, data containing zero may become problematic when interpreting or calculating the confidence intervals of un-back-transformed data. As log(1)=0, any data containing values <=1 can be made >0 by adding a constant to the original data so that the minimum raw value becomes >1 . Reporting un-back-transformed data can be fraught at the best of times so back-transformation of transformed data is recommended. Further information on back-transformation can be found here.
Adding a constant to data is not without it’s impact on the transformation. As the below example illustrates the effectiveness of the log transformation on the above sales data is effectively diminished in this case by the addition of a constant to the original data.
Depending on the subsequent intentions for analysis this may be the preferred outcome for your data - it is certainly an adequate improvement and has rendered the data approximately normal for most parametric testing purposes.
Taking the transformation a step further and applying the inverse transformation to the sales + constant data, again, leads to a less optimal result for this particular set of data – indicating that the skewness of the original data is not quite extreme enough to benefit from the inverse transformation.
It is interesting to note that the peak of the distribution has been reduced whereas an increase in leptokurtosis occurred for the inverse transformation of the raw distribution. This serves to illustrate how a small alteration in the data can completely change the outcome of a data transformation without necessarily changing the shape of the original distribution.
There are many varieties of distribution, the below diagram depicting only the most frequently observed. If common data transformations have not adequately ameliorated your skewness, it may be more reasonable to select a non-parametric hypothesis test that is based on an alternate distribution.