Calculating minimum and maximum possible variances from n-tile grouped data

Abstract

Summary social science data are frequently encountered for which the raw data are unavailable and for which the standard deviations are unreported. This often precludes further analysis or even meaningful use of the summary data. At times, however, the published data include the means for different quartile, quintile, decile, or, in general, n-tile groups. This is particularly the case for income distribution figures, which will often report the average income of the bottom 20% of income units, of the second 20%, etc. We demonstrate here that when such submeans are known it is possible to calculate minimum and maximum possible values for the standard deviation or variance, which will then permit tests of statistical significance to be employed.