# Gini

##### Gini Coefficient

Compute the Gini coefficient, the most commonly used measure of inequality.

- Keywords
- univar

##### Usage

```
Gini(x, n = rep(1, length(x)), unbiased = TRUE,
conf.level = NA, R = 1000, type = "bca", na.rm = FALSE)
```

##### Arguments

- x
a vector containing at least non-negative elements. The result will be

`NA`

, if x contains negative elements.- n
a vector of frequencies (weights), must be same length as x.

- unbiased
logical. In order for G to be an unbiased estimate of the true population value, calculated gini is multiplied by \(n/(n-1)\). Default is TRUE. (See Dixon, 1987)

- conf.level
confidence level for the confidence interval, restricted to lie between 0 and 1. If set to

`TRUE`

the bootstrap confidence intervals are calculated. If set to`NA`

(default) no confidence intervals are returned.- R
number of bootstrap replicates. Usually this will be a single positive integer. For importance resampling, some resamples may use one set of weights and others use a different set of weights. In this case R would be a vector of integers where each component gives the number of resamples from each of the rows of weights. This is ignored if no confidence intervals are to be calculated.

- type
character string representing the type of interval required. The value should be one out of the c(

`"norm"`

,`"basic"`

,`"stud"`

,`"perc"`

or`"bca"`

). This argument is ignored if no confidence intervals are to be calculated.- na.rm
logical. Should missing values be removed? Defaults to FALSE.

##### Details

The range of the Gini coefficient goes from 0 (no concentration) to \(\sqrt(\frac{n-1}{n})\) (maximal concentration). The bias corrected Gini coefficient goes from 0 to 1.
The small sample variance properties of the Gini coefficient are not known, and large sample approximations to the variance of the coefficient are poor (Mills and Zandvakili, 1997; Glasser, 1962; Dixon et al., 1987),
therefore confidence intervals are calculated via bootstrap re-sampling methods (Efron and Tibshirani, 1997).
Two types of bootstrap confidence intervals are commonly used, these are
percentile and bias-corrected (Mills and Zandvakili, 1997; Dixon et al., 1987; Efron and Tibshirani, 1997).
The bias-corrected intervals are most appropriate for most applications. This is set as default for the `type`

argument (`"bca"`

).
Dixon (1987) describes a refinement of the bias-corrected method known as 'accelerated' -
this produces values very closed to conventional bias corrected intervals.
(Iain Buchan (2002) *Calculating the Gini coefficient of inequality*, see: http://www.statsdirect.com/help/default.htm#nonparametric_methods/gini.htm)

##### Value

If `conf.level`

is set to `NA`

then the result will be

a single numeric value

Gini coefficient

lower bound of the confidence interval

upper bound of the confidence interval

##### References

Cowell, F. A. (2000) Measurement of Inequality in Atkinson, A. B. / Bourguignon, F. (Eds): *Handbook of Income Distribution*. Amsterdam.

Cowell, F. A. (1995) *Measuring Inequality* Harvester Wheatshef: Prentice Hall.

Marshall, Olkin (1979) *Inequalities: Theory of Majorization and Its
Applications*. New York: Academic Press.

Glasser C. (1962) Variance formulas for the mean difference and coefficient of concentration.
*Journal of the American Statistical Association* 57:648-654.

Mills JA, Zandvakili A. (1997). Statistical inference via bootstrapping for measures of inequality.
*Journal of Applied Econometrics* 12:133-150.

Dixon, PM, Weiner J., Mitchell-Olds T, Woodley R. (1987) Boot-strapping the Gini coefficient of inequality.
*Ecology* 68:1548-1551.

Efron B, Tibshirani R. (1997) Improvements on cross-validation:
The bootstrap method. *Journal of the American Statistical Association* 92:548-560.

##### See Also

See `Herfindahl`

, `Rosenbluth`

for concentration measures,
`Lc`

for the Lorenz curve
`ineq()`

in the package ineq contains additional inequality measures

##### Examples

```
# NOT RUN {
# generate vector (of incomes)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
# compute Gini coefficient
Gini(x)
# working with weights
fl <- c(2.5, 7.5, 15, 35, 75, 150) # midpoints of classes
n <- c(25, 13, 10, 5, 5, 2) # frequencies
# with confidence intervals
Gini(fl, n, conf.level=0.95, unbiased=FALSE)
# some special cases
x <- c(10, 10, 0, 0, 0)
plot(Lc(x))
Gini(x, unbiased=FALSE)
# the same with weights
Gini(x=c(10, 0), n=c(2,3), unbiased=FALSE)
# perfect balance
Gini(c(10, 10, 10))
# }
```

*Documentation reproduced from package DescTools, version 0.99.36, License: GPL (>= 2)*