A simple transformation for sets of range sizes

Transformation of data to normality may be illuminating and useful statistically. There are two standard families of transformations, power transformations for positive numbers, bounded at the left, and folded transformations for proportions, bounded both at the left and the right. It has been shown that there is no one satisfactory power transformation for range size data. However, such measures are limited to the right as well as the left, and we consider applying folded transformations to them. Seven data sets of range sizes recorded by 10 km squares are studied. Six are British (native and introduced plants, mammals, dragonflies and two breeding bird surveys) the seventh is of Swiss breeding birds. Using these we show that the right hand limit of the distribution can be estimated and the best folded transformation found. In all cases the right hand limit is larger than the range size of the most widespread species and smaller than the notional scope of the survey. In all cases the logit or flog, the logarithmic folded transformation, is satisfactory: in five cases it is the best. It is well known that abundance is approximately (though not exactly) log-normally distributed. The relationship of that to our discovery that range size data are approximately logit-normal is discussed. There is no fully satisfactory explanation for either observation at present.