# meanAD

##### The Mean Absolute Deviation

Computes (standardized) mean absolute deviation.

- Keywords
- robust, distribution, univar

##### Usage

`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)).

##### 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.

##### See Also

##### Examples

```
# NOT RUN {
## 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)
# }
```

*Documentation reproduced from package MKdescr, version 0.4, License: LGPL-3*