 Mean difference

The mean difference is a measure of statistical dispersion equal to the average absolute difference of two independent values drawn from a probability distribution. A related statistic is the relative mean difference, which is the mean difference divided by the arithmetic mean. An important relationship is that the relative mean difference is equal to twice the Gini coefficient, which is defined in terms of the Lorenz curve.
The mean difference is also known as the absolute mean difference and the Gini mean difference. The mean difference is sometimes denoted by Δ or as MD. The mean deviation is a different measure of dispersion.
Contents
Calculation
For a population of size n, with a sequence of values y_{i}, i = 1 to n:
For a discrete probability function f(y), where y_{i}, i = 1 to n, are the values with nonzero probabilities:
For a probability density function f(x):
For a cumulative distribution function F(x) with quantile function F(x):
Relative mean difference
When the probability distribution has a finite and nonzero arithmetic mean, the relative mean difference, sometimes denoted by ∇ or RMD, is defined by
The relative mean difference quantifies the mean difference in comparison to the size of the mean and is a dimensionless quantity. The relative mean difference is equal to twice the Gini coefficient which is defined in terms of the Lorenz curve. This relationship gives complementary perspectives to both the relative mean difference and the Gini coefficient, including alternative ways of calculating their values.
Properties
The mean difference is invariant to translations and negation, and varies proportionally to positive scaling. That is to say, if X is a random variable and c is a constant:
 MD(X + c) = MD(X),
 MD(X) = MD(X), and
 MD(c X) = c MD(X).
The relative mean difference is invariant to positive scaling, commutes with negation, and varies under translation in proportion to the ratio of the original and translated arithmetic means. That is to say, if X is a random variable and c is a constant:
 RMD(X + c) = RMD(X) · mean(X)/(mean(X) + c) = RMD(X) / (1+c / mean(X)) for c ≠ mean(X),
 RMD(X) = −RMD(X), and
 RMD(c X) = RMD(X) for c > 0.
If a random variable has a positive mean, then its relative mean difference will always be greater than or equal to zero. If, additionally, the random variable can only take on values that are greater than or equal to zero, then its relative mean difference will be less than 2.
Compared to standard deviation
Both the standard deviation and the mean difference measure dispersion—how spread out are the values of a population or the probabilities of a distribution. The mean difference is not defined in terms of a specific measure of central tendency, whereas the standard deviation is defined in terms of the deviation from the arithmetic mean. Because the standard deviation squares its differences, it tends to give more weight to larger differences and less weight to smaller differences compared to the mean difference. When the arithmetic mean is finite, the mean difference will also be finite, even when the standard deviation is infinite. See the examples for some specific comparisons. The recently introduced distance standard deviation plays similar role than the mean difference but the distance standard deviation works with centered distances. See also Estatistics.
Sample estimators
For a random sample S from a random variable X, consisting of n values y_{i}, the statistic
is a consistent and unbiased estimator of MD(X). The statistic:
is a consistent estimator of RMD(X), but is not, in general, unbiased.
Confidence intervals for RMD(X) can be calculated using bootstrap sampling techniques.
There does not exist, in general, an unbiased estimator for RMD(X), in part because of the difficulty of finding an unbiased estimation for multiplying by the inverse of the mean. For example, even where the sample is known to be taken from a random variable X(p) for an unknown p, and X(p)  1 has the Bernoulli distribution, so that Pr(X(p) = 1) = 1 − p and Pr(X(p) = 2) = p, then
 RMD(X(p)) = 2p(1 − p)/(1 + p).
But the expected value of any estimator R(S) of RMD(X(p)) will be of the form:^{[citation needed]}
where the r _{i} are constants. So E(R(S)) can never equal RMD(X(p)) for all p between 0 and 1.
Examples
Examples of Mean Difference and Relative Mean Difference Distribution Parameters Mean Standard Deviation Mean Difference Relative Mean Difference Continuous uniform a = 0 ; b = 1 1 / 2 = 0.5 ≈ 0.2887 1 / 3 ≈ 0.3333 2 / 3 ≈ 0.6667 Normal μ = 1 ; σ = 1 1 1 ≈ 1.1284 ≈ 1.1284 Exponential λ = 1 1 1 1 1 Pareto k > 1 ; x_{m} = 1 (for k > 2) Gamma k ; θ kθ k θ (2 − 4 I _{0.5} (k+1 , k)) † 2 − 4 I _{0.5} (k+1 , k) † Gamma k = 1 ; θ = 1 1 1 1 1 Gamma k = 2 ; θ = 1 2 ≈ 1.4142 3 / 2 = 1.5 3 / 4 = 0.75 Gamma k = 3 ; θ = 1 3 ≈ 1.7321 15 / 8 = 1.875 5 / 8 = 0.625 Gamma k = 4 ; θ = 1 4 2 35 / 16 = 2.1875 35 / 64 = 0.546875 Bernoulli 0 ≤ p ≤ 1 p 2 p (1 − p) 2 (1 − p) for p > 0 Student's t, 2 d.f. ν = 2 0 π / √2 = 2.2214 undefined  † I _{z} (x,y) is the regularized incomplete Beta function
See also
 Mean Deviation
 Estimator
 Coefficient of variation
 Lmoment
References
 Xu, Kuan (January, 2004). How Has the Literature on Gini's Index Evolved in the Past 80 Years?. Department of Economics, Dalhousie University. http://economics.dal.ca/RePEc/dal/wparch/howgini.pdf. Retrieved 20060601.
 Gini, Corrado (1912). Variabilità e Mutabilità. Bologna: Tipografia di Paolo Cuppini.
 Gini, Corrado (1921). "Measurement of Inequality and Incomes". The Economic Journal (The Economic Journal, Vol. 31, No. 121) 31 (121): 124–126. doi:10.2307/2223319. JSTOR 2223319.
 Chakravarty, S. R. (1990). Ethical Social Index Numbers. New York: SpringerVerlag.
 Mills, Jeffrey A.; Zandvakili, Sourushe (1997). "Statistical Inference via Bootstrapping for Measures of Inequality". Journal of Applied Econometrics 12 (2): 133–150. doi:10.1002/(SICI)10991255(199703)12:2<133::AIDJAE433>3.0.CO;2H.
 Lomnicki, Z. A. (1952). "The Standard Error of Gini's Mean Difference". Annals of Mathematical Statistics 23 (4): 635–637. doi:10.1214/aoms/1177729346.
 Nair, U. S. (1936). "Standard Error of Gini's Mean Difference". Biometrika 28: 428–436.
 Yitzhaki, Shlomo (2003). "Gini's Mean difference: a superior measure of variability for nonnormal distributions". Metron  International Journal of Statistics 61: 285–316. ftp://metron.sta.uniroma1.it/RePEc/articoli/20032285316.pdf.
Categories: Statistical deviation and dispersion
 Summary statistics
 Theory of probability distributions
 Scale statistics
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