Chapter 2: Income Standards, Inequality, and Poverty
partial mean is obtained by finding the mean of the incomes below a specific percentile cutoff. An upper partial mean is obtained by finding the mean of incomes above a specific percentile cutoff. Lower partial means are more commonly used than upper partial means. The lower partial mean of the pth percentile is the average or mean income of the bottom p percent of the population. The upper partial mean of the pth percentile, in contrast, is the average or mean income of the top (1 – p) percent of the population. We denote the lower partial mean and upper partial mean of distribution x for percentile p by WLPM(x; p) and WUPM(x; p), respectively. For example, if p = 50 percent, then the lower partial mean of the pth percentile of distribution x is denoted by WLPM(x; 50). If WLPM(x; 50) = $100 and WUPM(x; 50) = $10,000, then together they should be read as the mean income of the bottom 50 percent of the population is $100, and the mean income of the top 50 percent of the population is $10,000 (see example 2.1). Example 2.1: Consider the income vector x = ($2k, $4k, $8k, $10k). The lower partial mean of the 50th percentile of the distribution is ($2k + $4k)/2 = $3k, and that of the 75th percentile of the distribution is ($2k + $4k + $8k)/3 = $4.7k. In contrast, the upper partial mean of the 50th percentile of the distribution is ($8k + $10k)/2 = $9k and that of the 75th percentile of the distribution is $10k. The following is a graphical description of how partial means can be calculated using quantile function Qx. The vertical axis of figure 2.5 denotes income, and the horizontal axis denotes population share. There are two percentiles, p' and p", for describing the lower and upper partial means. The lower partial mean of the p' percentile population is the shaded area underneath the quantile function Qx to the left of p' divided by p'. The lower partial mean is the average income of all people in society X whose income is less than Qx(p'). Similarly, the upper partial mean of the p" percentile population is the shaded area underneath the quantile function Qx to the right of p" divided by (100 – p"). This upper partial mean is the average income of all people in society X whose income is larger than Qx(p"). Like the quantile incomes, any partial mean satisfies symmetry, normalization, population invariance, linear homogeneity, and weak monotonicity, but no partial mean satisfies monotonicity, transfer principle, and subgroup
61