Scottish Indices of Deprivation 2003
Appendix 5: Exponential Transformation of the Domain Indices
The precise transformation proposed is as follows. For any ward, denote its rank on the domain, scaled to the range [0,1], by R (with R=1/N for the least deprived, and R=N/N, i.e. R=1, for the most deprived, where N=1222 which is the number of wards in Scotland).
The transformed domain, X say, is X = -23*log{1 - R*[1 - exp(-100/23)]}
where log denotes natural logarithm and exp the exponential or antilog transformation, and * denotes multiplication. This formula may at first sight seem complicated, but it is very straightforwardly calculated and is in fact simpler than the commonly-used transformation to a normal curve which necessitates the use of a look-up table.
Each transformed domain has a range of 0 to 100, with a score of 100 for the most deprived ward. The chosen exponential distribution is one of an infinite number of possible such distributions. The constant (23) determines that ten percent of wards have a score higher than 50. When transformed scores from different domains are combined by averaging them, the skewness of the distribution reduces the extent to which deprivation on one domain can be cancelled by lack of deprivation on another. For example, if the transformed scores on two domains are simply averaged, with equal weights, a (hypothetical) ward that scored 100 on one domain and 0 on the other would have a combined score of 50 and would thus be ranked at the 90th percentile. (Averaging the untransformed ranks, or after transformation to a normal distribution, would result in such a ward being ranked instead at the 50th percentile: the high deprivation in one domain would have been fully cancelled by the low deprivation in the other.) Thus the extent to which deprivation in some domains can be cancelled by lack of deprivation in others is, by design, reduced.