Gini {DescTools} | R Documentation |

## Gini Coefficient

### Description

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

### Usage

```
Gini(x, weights = NULL, unbiased = TRUE,
conf.level = NA, R = 10000, type = "bca", na.rm = FALSE)
```

### Arguments

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

`weights` |
a numerical vector of weights the same length as |

`unbiased` |
logical. In order for G to be an unbiased estimate of the true population value,
calculated gini is multiplied by |

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

`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. |

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

`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: https://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 |

and
if a `conf.level`

is provided, a named numeric vector with 3 elements:

`gini` |
Gini coefficient |

`lwr.ci` |
lower bound of the confidence interval |

`upr.ci` |
upper bound of the confidence interval |

### Author(s)

Andri Signorell <andri@signorell.net>

### 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

```
# 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(x=fl, weights=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), weights=c(2,3), unbiased=FALSE)
# perfect balance
Gini(c(10, 10, 10))
```

*DescTools*version 0.99.55 Index]