Is the reciprocal of a positively skewed distribution also positively skewed?

@RISK (risk analysis using Monte Carlo, software for Excel) can be a simple yet effective tool to explore statistical concepts and properties of distributions.  For example, one interesting question is whether the reciprocal of a positively skewed distribution is positively or negatively skewed.

One’s first thought may be that such a reciprocal is negatively skewed. Of course, when reflecting on such an issue for the lognormal distribution it becomes clear that this is not the case. Since the lognormal distribution is the result of multiplying many independent random processes, the reciprocal of such a distribution is the result of multiplying the reciprocals of these individual distributions. Therefore the reciprocal of a lognormal distribution is itself lognormal, and hence always positively skewed.

Turning to triangular distributions the situation is not so clear. @RISK can be used to sample a distribution and calculate its reciprocal. The @RISK Statistics functions can be used to compute the moments of the input distributions (mean, stddev, skewness using e.g. RiskTheoSkewness etc) and the statistics for the reciprocals are available after the simulation (using RiskSkewness etc).

The left graph shows a range of triangular distributions which are either symmetric or positively skewed (as model inputs), and the right hand graph shows the (simulated) reciprocals. It is interesting to note that the reciprocal of the symmetric distribution is positively skewed, whereas as the parameters of the distribution are adjusted, the reciprocal may either be positively or negatively skewed, but with a general prevalence for positive skew.

These properties could of course be explored further through mathematical manipulations, many of which are not trivial. However, @RISK provides an easy and intuitive way to explore such issues for the best possible decision making under uncertainty.

Dr. Michael Rees
Director of Training and Consulting

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