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Mean Deviation

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The mean deviation (also called the mean absolute deviation) is the mean of the absolute deviations of a set of data about the data's mean. For a sample size N, the mean deviation is defined by

MD=1/Nsum_(i=1)^N|x_i-x^_|,
(1)

where x^_ is the mean of the distribution. The mean deviation of a list of numbers is implemented in the Wolfram Language as MeanDeviation[data].

The mean deviation for a discrete distribution P_i defined for i=1, 2, ..., N is given by

MD=sum_(i=1)^NP_i|x_i-x^_|.
(2)

Mean deviation is an important descriptive statistic that is not frequently encountered in mathematical statistics. This is essentially because while mean deviation has a natural intuitive definition as the "mean deviation from the mean," the introduction of the absolute value makes analytical calculations using this statistic much more complicated than the standard deviation

sigma=sqrt(1/Nsum_(i=1)^N(x_i-x^_)^2).
(3)

As a result, least squares fitting and other standard statistical techniques rely on minimizing the sum of square residuals instead of the sum of absolute residuals.

For example, consider the discrete uniform distribution consisting of N possible outcomes with P_i=1/N for i=1, 2, ..., N. The mean is given by

x^_=sum_(i=1)^NiP_i=1/Nsum_(i=1)^Ni=1/2(N+1).
(4)

The variance (and therefore its square root, namely the standard deviation) is also straightforward to obtain as

sigma^2=sum_(i=1)^N(i-x^_)^2P_i=1/(12)(N-1)(N+1).
(5)

On the other hand, the mean deviation is given by

MD=sum_(i=1)^N|i-x^_|P_i=1/Nsum_(i=1)^N|i-x^_|.
(6)

This can be obtained in closed form, but is much more unwieldy since it requires breaking up the summand into two pieces and treating the cases of N even and odd separately.

For a random variable X satisfying E|X|<infty, let F be its cumulative distribution function and mu its mean. The mean absolute deviation about a real number a can be written without an absolute value inside the integral as

H(a)=E|X-a|=mu-a+2int_(-infty)^aF(x)dx
(7)

(Pinsky 2025, p. 3). In particular, H(mu) is the mean deviation defined above. If Q is the quantile function, then

H(Q(p))=mu-2int_0^pQ(u)du+(2p-1)Q(p)
(8)

for 0<p<1 (Pinsky and Kundu 2026, pp. 8-9). The choice p=1/2 gives the mean absolute deviation about a median.

The following table summarizes the mean absolute deviations for some named continuous distributions, where B(z;a,b) is an incomplete beta function, B(a,b) is a beta function, Gamma(z) is a gamma function, Gamma(a,z) is the upper incomplete gamma function, gamma is the Euler-Mascheroni constant, G_(2,1)^(0,2)(a,b|m) is a Meijer G-function, Ei(z) is the exponential integral function, erf(z) is erf, and erfc(z) is erfc.

distributionM.D.
beta distribution(2alphaB(alpha/(alpha+beta);alpha,beta)-2(alpha+beta)B(alpha/(alpha+beta);alpha+1,beta))/((alpha+beta)B(alpha,beta))
chi-squared distribution((2e)^(-nu/2)nu^(nu/2+1)(nu+2))/(Gamma(2+1/2nu))
exponential distribution2/(elambda)
F-distribution(4mz^(n/2)(1-z)^(m/2-1))/(n(m-2)B(n/2,m/2))
gamma distribution(2e^(-alpha)alpha^alphabeta)/(Gamma(alpha))
generalized Pareto distribution2sigma(1-xi)^(1/xi-2)
Gumbel distributionbeta[G_(2,1)^(2,0)(e,gamma^(-1)|1,1; 0)-Ei(sinhgamma-coshgamma)]
half-normal distribution2/theta[e^(-1/pi)+erf(1/(sqrt(pi)))]
Laplace distributionbeta
logistic distribution2betaln2
log normal distribution2e^(mu+sigma^2/2)erf(sigma/(2sqrt(2)))
Maxwell distribution4e^(-4/pi)sqrt(2/pi)sigma[1+e^(4/pi)erf(2/(sqrt(pi)))]
normal distributionsqrt(2/pi)sigma
Pareto distribution(2k)/(alpha-1)((alpha-1)/alpha)^(alpha-1)
Rayleigh distributionsqrt(2pi)erfc((sqrt(pi))/2)
Student's t-distribution(2sqrt(nu))/((nu-1)B(1/2nu,1/2))
Student's t-distribution(2sqrt(nu)sigma)/((nu-1)B(1/2nu,1/2))
triangular distribution1/6(b-a)
triangular distribution{(2(b+c-2a)^3)/(81(a-b)(a-c)) for a+b<2c; (2(a+c-2b)^3)/(81(a-b)(b-c)) for a+b>2c
uniform distribution1/4(b-a)
Weibull distribution(2beta)/alphaGamma(1/alpha,Gamma(1+1/alpha)^alpha)

For the F-distribution, n>0 and m>2 are the numerator and denominator degrees of freedom, and z=n/(m+n-2). For the generalized Pareto distribution, sigma>0 and xi<1 are the scale parameter and shape parameter. At xi=0, the tabulated expression is interpreted as its limit 2sigma/e (Pinsky 2025, pp. 7 and 10). For the two Student's t-distribution rows, nu>1 is the number of degrees of freedom. The first row is the standard case and the second has scale parameter sigma>0. The mean deviation is independent of the location parameter. For the Weibull distribution, alpha>0 is the shape parameter and beta>0 is the scale parameter (Pinsky 2025, p. 11; Zhang et al. 2026, p. 3).

For the Pareto distribution in the table, k>0 is the minimum value and alpha>1 is the shape parameter. Its mean is mu=alphak/(alpha-1), so splitting the integral at mu gives (Pinsky 2025, p. 9)

MD=2int_k^mu(mu-x)(alphak^alpha)/(x^(alpha+1))dx=(2k)/(alpha-1)((alpha-1)/alpha)^(alpha-1).
(9)

For 0<alpha<=1, the mean is infinite and the mean deviation about a finite mean is not defined.

The following table summarizes the mean absolute deviations for some named discrete distributions, where mu=zeta(rho)/zeta(rho+1).

distributionM.D.
Bernoulli distribution2p(1-p)
binomial distribution2(1-p)^(N-|_Np_|)p^(|_Np_|+1)(|_Np_|+1)(N; |_Np_|+1)
discrete uniform distribution{1/4N for N even; ((N-1)(N+1))/(4N) for N odd
geometric distribution2(1-p)^(|_1/p_|)|_1/p_|
log-series distribution2mu[p^j+(B(p;j+1,0))/(ln(1-p))]
negative binomial distribution2r(j+r; r)(1-p)^(j+1)p^(r-1)
Poisson distribution(2e^(-nu)nu^(|_nu_|+1))/(|_nu_|!)
Zipf distribution(2[zeta(rho+1)zeta(rho,|_mu_|+1)-zeta(rho)zeta(rho+1,|_mu_|+1)])/(zeta^2(rho+1))

For the log-series distribution, 0<p<1, mu=-p/[(1-p)ln(1-p)], and j=|_mu_|, where |_x_| is the floor function. The probability mass function is P(X=k)=-p^k/[kln(1-p)] for k=1, 2, .... Summing its truncated first moment gives

sum_(k=1)^jkP(X=k)=mu(1-p^j),
(10)

so the tabulated result follows from MD=2[muF(j)-sum_(k=1)^(j)kP(X=k)], where F is the cumulative distribution function. For the negative binomial distribution, r>0, 0<p<1, and j=|_r(1-p)/p_|, with probability mass function P(X=k)=(k+r-1; k)p^r(1-p)^k for k=0, 1, ... (Pinsky 2025, p. 12).

Mean absolute deviations about a median must be distinguished from those about the mean. For the Gumbel distribution, Fréchet distribution, and Weibull distribution, respectively, the deviations about a median are

D_(Gumbel)=beta[gamma+ln(ln2)-2Ei(-ln2)]
(11)
D_(Frechet)=beta[Gamma(1-1/alpha)-2Gamma(1-1/alpha,ln2)]
(12)
D_(Weibull)=beta[2Gamma(1+1/alpha,ln2)-Gamma(1+1/alpha)],
(13)

where beta>0 is the scale parameter. The shape parameter satisfies alpha>1 for the Fréchet distribution and alpha>0 for the Weibull distribution (Pinsky and Kundu 2026, p. 12; Zhang et al. 2026, p. 3).

See also

Absolute Deviation, QuantileFunction, Standard Deviation, Variance

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References

Havil, J. "Ways of Means." §13.1 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 119-121, 2003.Kenney, J. F. and Keeping, E. S. "Mean Absolute Deviation." §6.4 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 76-77 1962.Pinsky, E. "Computation and Interpretation of Mean Absolute Deviations by Cumulative Distribution Functions." Front. Appl. Math. Stat. 11, Art. 1487331, 2025. https://doi.org/10.3389/fams.2025.1487331.Pinsky, E. and Kundu, T. "Skewness and Kurtosis for Heavy-Tailed Distributions: A Comparison of Mean Absolute Deviation and L-Moment Approaches for Pareto and Extreme-Value Families." Mathematics 14, Art. 3656, 2026. https://doi.org/10.3390/math14203656.Zhang, W.; Wang, Z.; and Pinsky, E. "Mean Absolute Deviations for the Weibull Distribution: Applications in Survival Analysis and Insurance Claims." Statistics in Transition new series 27, 1-14, 2026. https://doi.org/10.59139/stattrans-2026-013.

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Mean Deviation

Cite this as:

Weisstein, Eric W. "Mean Deviation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MeanDeviation.html

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