# dmandelk: Mandel's k statistic. In metRology: Support for Metrological Applications

## Description

Density, distribution function, quantile function and random generation for Mandel's k statistic, a measure of relative precision compared to a common variance.

## Usage

 1 2 3 4 dmandelk(x, g, n, log = FALSE) pmandelk(q, g, n, lower.tail = TRUE, log.p = FALSE) qmandelk(p, g, n, lower.tail = TRUE, log.p = FALSE) rmandelk(B, g, n) 

## Arguments

 x, q vector of quantiles. p vector of probabilities. g number of groups for which k is calculated. n number of observations in each group of data for which k is calculated. B Number of observations. If 'length(B) > 1', the length is taken to be the number required. lower.tail logical; if TRUE (default), probabilities are P[X <= x]; otherwise, P[X > x]. log, log.p logical; if TRUE, probabilities p are given as log(p).

## Details

Mandel's k for one of a set of g standard deviations s is calculated as

k=\frac{s_{ij}^2}{∑_{i=1}^p{s_{ij}^2/p}}

Since the numerator is chi-squared(n-1), or Gamma((n-1)/2, 2), and the denominator can be written as the sum of the same quantity and a pooled variance with distribution Gamma((g-1)*(n-1)/2, 2), k is distributed as Beta((n-1)/2, (g-1)(n-1)/2). Quantiles, probabilities, density and random numbers can therefore be generated from the Beta distribution. For example, qmandelk is calculated as sqrt( g * qbeta( (n-1)/2, (g-1)*(n-1)/2)).

## Value

dmandelh returns the density at x, pmandelh the cumulative probability, qmandelh the quantiles for probability p and rmandelh returns B random values drawn from the distribution.

Vector values of x, p, q and g are permitted, in which case the functions return vectors.

## Warning

Note that rmandelk uses B and not n (as do most R random number functions) for number of random draws; this is because n is conventionally used for the number of replicates per group. Be careful when using named parameters!

## Author(s)

S. L. R. Ellison, s.ellison@lgc.co.uk

## References

None.

pmandelh
 1 2 3 4 5 6 7  #Generate the 95% and 99% quantiles for comparison with tables in #ISO 5725:1996 Part 2: round(qmandelk(0.95, g=3:30, n=3), 2) #95% upper tail round(qmandelk(0.99, g=3:30, n=3), 2) #99% upper tail