# meanAD: The Mean Absolute Deviation In MKmisc: Miscellaneous Functions from M. Kohl

## Description

Computes (standardized) mean absolute deviation.

## Usage

 `1` ```meanAD(x, na.rm = FALSE, constant = sqrt(pi/2)) ```

## Arguments

 `x` a numeric vector. `na.rm` logical. Should missing values be removed? `constant` standardizing contant; see details below.

## Details

The mean absolute deviation is a consistent estimator of sqrt(2/pi)sigma for the standard deviation of a normal distribution. Under minor deviations of the normal distributions its asymptotic variance is smaller than that of the sample standard deviation (Tukey (1960)).

It works well under the assumption of symmetric, where mean and median coincide. Under the normal distribution it's about 18% more efficient (asymptotic relative efficiency) than the median absolute deviation (`(1/qnorm(0.75))/sqrt(pi/2)`) and about 12% less efficient than the sample standard deviation (Tukey (1960)).

## Author(s)

Matthias Kohl Matthias.Kohl@stamats.de

## References

Tukey, J. W. (1960). A survey of sampling from contaminated distribution. In Olink, I., editor, Contributions to Probablity and Statistics. Essays in Honor of H. Hotelling., pages 448-485. Stanford University Press.

`sd`, `mad`, `sIQR`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35``` ```## right skewed data ## mean absolute deviation meanAD(rivers) ## standardized IQR sIQR(rivers) ## median absolute deviation mad(rivers) ## sample standard deviation sd(rivers) ## for normal data x <- rnorm(100) sd(x) sIQR(x) mad(x) meanAD(x) ## Asymptotic relative efficiency for Tukey's symmetric gross-error model ## (1-eps)*Norm(mean, sd = sigma) + eps*Norm(mean, sd = 3*sigma) eps <- seq(from = 0, to = 1, by = 0.001) ARE <- function(eps){ 0.25*((3*(1+80*eps))/((1+8*eps)^2)-1)/(pi*(1+8*eps)/(2*(1+2*eps)^2)-1) } plot(eps, ARE(eps), type = "l", xlab = "Proportion of gross-errors", ylab = "Asymptotic relative efficiency", main = "ARE of mean absolute deviation w.r.t. sample standard deviation") abline(h = 1.0, col = "red") text(x = 0.5, y = 1.5, "Mean absolute deviation is better", col = "red", cex = 1, font = 1) ## lower bound of interval uniroot(function(x){ ARE(x)-1 }, interval = c(0, 0.002)) ## upper bound of interval uniroot(function(x){ ARE(x)-1 }, interval = c(0.5, 0.55)) ## worst case optimize(ARE, interval = c(0,1), maximum = TRUE) ```