The text discusses two measures of concentration or diversity: Yule's 'characteristic' and Fisher's 'index of diversity'. Both are statistics calculated from sample data, not population constants. The index of diversity is mainly used with the logarithmic distribution and may not be suitable for all cases, especially when applied to infinite populations. Williams found a relationship between the two measures when applied to a logarithmic distribution. The text defines a measure of concentration in terms of population constants. An infinite population with Z groups is considered, where each individual belongs to one of Z groups. The measure λ, defined as the sum of the squares of the proportions of individuals in each group, represents the concentration of the classification. It ranges between 1/Z (maximum diversity) and 1 (minimum diversity). The third and fourth cumulants of the distribution of l are calculated, showing that as N increases, the distribution tends to normality except when λ = 1/Z, in which case the distribution of lNZ tends to a chi-squared distribution with Z-1 degrees of freedom. Yule's characteristic is defined as 1,000 times the sum of n(n-1)/N², differing from l, the sample estimator of λ, only in the denominator and scale factor. For a population with Z groups and frequencies πi = wi/Σw, the value of λ is obtained by averaging Σwi²/(Σwi)² over all possible sets of wi. This results in λ = (k+1)/(Zk+1). When samples are drawn from this population, l is an unbiased estimator of λ. The variance of l is given by a specific formula. The Poisson distribution is a special case of the negative binomial distribution, leading to λ = 1/Z. The logarithmic population is another extreme case, with λ = 1/(α+1), where α is Fisher's index of diversity. The text notes a discrepancy between Williams' and the current derivation of λ for the logarithmic distribution. E.H. Simpson is the author, with references to Yule, Fisher, and Williams.The text discusses two measures of concentration or diversity: Yule's 'characteristic' and Fisher's 'index of diversity'. Both are statistics calculated from sample data, not population constants. The index of diversity is mainly used with the logarithmic distribution and may not be suitable for all cases, especially when applied to infinite populations. Williams found a relationship between the two measures when applied to a logarithmic distribution. The text defines a measure of concentration in terms of population constants. An infinite population with Z groups is considered, where each individual belongs to one of Z groups. The measure λ, defined as the sum of the squares of the proportions of individuals in each group, represents the concentration of the classification. It ranges between 1/Z (maximum diversity) and 1 (minimum diversity). The third and fourth cumulants of the distribution of l are calculated, showing that as N increases, the distribution tends to normality except when λ = 1/Z, in which case the distribution of lNZ tends to a chi-squared distribution with Z-1 degrees of freedom. Yule's characteristic is defined as 1,000 times the sum of n(n-1)/N², differing from l, the sample estimator of λ, only in the denominator and scale factor. For a population with Z groups and frequencies πi = wi/Σw, the value of λ is obtained by averaging Σwi²/(Σwi)² over all possible sets of wi. This results in λ = (k+1)/(Zk+1). When samples are drawn from this population, l is an unbiased estimator of λ. The variance of l is given by a specific formula. The Poisson distribution is a special case of the negative binomial distribution, leading to λ = 1/Z. The logarithmic population is another extreme case, with λ = 1/(α+1), where α is Fisher's index of diversity. The text notes a discrepancy between Williams' and the current derivation of λ for the logarithmic distribution. E.H. Simpson is the author, with references to Yule, Fisher, and Williams.