Concentration of a total across units

Usage

gini(x, na.rm = TRUE)

lorenz(x, na.rm = TRUE)

top_share(x, fractions = c(0.01, 0.05, 0.1, 0.25), na.rm = TRUE)

Arguments

x: Non-negative values, one per unit.
na.rm: Whether to drop missing values. They are dropped either way; this argument exists so the call reads the same as base R's.
fractions: Fractions of the units, largest first, to report the share of.

Value

gini() a single number in [0, 1], NA when the total is zero. lorenz() a data frame of cumulative population and value shares, including the origin. top_share() a data frame of the requested fractions, the share each holds, and how many units that was.

Details

Gini is the mean absolute difference between pairs of units over twice the mean, which is also twice the area between the Lorenz curve and the diagonal. Zero is a perfectly even spread; the maximum for n units is 1 - 1/n, not 1, so a Gini near 1 requires many units as well as an uneven spread.

References

Hedderich, J. and Sachs, L. (2020). Applied Statistics: Methods Using R. Springer-Verlag, Berlin Heidelberg. Section 3.14, p. 117, gives the construction used here: the units are placed at equal intervals on the horizontal axis and the cumulated, ascendingly ordered shares of the total on the vertical one, so that the curve is the diagonal exactly when p percent of the units account for p percent of the total, and sags further the greater the concentration.

Examples

# Ten units holding one each: no concentration.
gini(rep(1, 10))
#> [1] 0

# One unit holding everything: the maximum for ten units.
gini(c(rep(0, 9), 1))
#> [1] 0.9
1 - 1 / 10
#> [1] 0.9

placements <- c(rep(1, 1200), rep(3, 430), rep(8, 110), rep(20, 38))
gini(placements)
#> [1] 0.461225

# What the most frequent few account for.
top_share(placements, c(0.01, 0.05, 0.1))
#>   fraction units      share
#> 1     0.01    18 0.08716707
#> 2     0.05    89 0.28280872
#> 3     0.10   178 0.41888620

head(lorenz(placements))
#>     population        value
#> 1 0.0000000000 0.0000000000
#> 2 0.0005624297 0.0002421308
#> 3 0.0011248594 0.0004842615
#> 4 0.0016872891 0.0007263923
#> 5 0.0022497188 0.0009685230
#> 6 0.0028121485 0.0012106538