This function computes the biweight scale, a robust measure of scale or dispersion for a numeric vector. The biweight scale is less sensitive to outliers than the sample standard deviation.
Arguments
- x
A numeric vector.
- loc
The location about which the scale is computed (default: median of
x).- c
A numeric value specifying the tuning constant for the biweight estimator (
c = 9by default).- reduced
A logical value specifying whether the sample size, n, should be reduced to the number of non-rejected values. If
TRUE, n is reduced to the number of observations that pass a rejection criteria (\(|u_i| < 1\)). IfFALSE(default), n is equal to the length ofx(the input data).- drop.na
A logical value indicating whether to remove missing values (
NA) from the calculations. IfTRUE, missing values will be removed. IfFALSE(the default), the result isNAwhenxcontains missing values.
Details
The biweight scale is the square root of the biweight midvariance: $$\zeta = \frac{\sqrt{n \sum_{|u_i|<1} (x_i - M)^2 (1 - u_i^2)^4}} {\left| \sum_{|u_i|<1} (1 - u_i^2)(1 - 5u_i^2) \right|}, \quad u_i = \frac{x_i - M}{c \cdot \mathrm{MAD}}$$ where \(M\) is the location (the median by default) and MAD is the (unscaled) median absolute deviation. For normally distributed data the biweight scale is a consistent estimator of the standard deviation.
References
Mosteller, F., and Tukey, J. W. (1977). Data Analysis and Regression: A Second Course in Statistics. Addison-Wesley, pp. 203-209.
Beers, T.C., Flynn, K., Gebhardt, K., (1990). Measures of location and scale for velocities in clusters of galaxies - A robust approach. The Astronomical Journal, 100:32-46.