EFFECTS OF CLOSURE ON THE MOMENTS OF A DISTRIBUTION

When percentages are formed from uncorrelated, normally distributed parent variables the moments of the percentage distribution may differ considerably from those of the parents. Equations can be derived which enable the approximation of the moments of a percentage variable in terms of the moments of the parent distribution, the row sum statistics, and the correlation between a part of a sum and the sum (the part-whole correlation). If the part-whole correlation is negative the mean and variance of the percentage are increased (relative to the means and variances of those variables with a positive part-whole correlation) and the percentage variable will exhibit a positive skewness. If the part-whole correlation is positive the percentage variable will be negatively skewed if, and only if, the part-whole correlation is greater than the ratio of the coefficient of variation of the row sum (T ) to the coefficient of variation of the parent variable. The kurtosis of the percentage variable must be greater than that of the parent variable regardless of the sign of the part-whole correlation. It is obvious that the interpretation or explanation of the distribution of a percentage variable must include an assessment of the effects of percentage formation. However, at the present time the isolation of the percentage effect appears to be impossible unless the parent data set is available. © 1979 Plenum Publishing Corporation.