Coefficient of Variation (Biased or Unbiased Estimator)

Computes the sample coefficient of variation \(\hat\kappa = s / \bar Y\) or, optionally, its first-order bias-corrected counterpart under normality. Either supply a precomputed cv or supply the raw mean and sd; with unbiased = TRUE the value is multiplied by the small-sample correction \((1 + 1/(4 N))\). The (biased) sample coefficient of variation (the default, unbiased = FALSE) is the form usually reported. To accompany it with a confidence interval, use ci_cv.

Usage

cv(cv = NULL, mean = NULL, sd = NULL, N = NULL, unbiased = FALSE)

Arguments

cv: The sample coefficient of variation, \(s/\bar Y\). Optional; if supplied, mean and sd must not be.
mean: Sample mean. Numeric scalar.
sd: Sample standard deviation, using \(N - 1\) in the variance denominator. Numeric scalar.
N: Sample size. Required when unbiased = TRUE.
unbiased: Logical. If TRUE, applies the first-order small-sample bias correction \((1 + 1/(4 N))\) to the plug-in estimator. Default FALSE.

Value

A 1-row data.frame with columns term and value. The term value is "cv" and value is either the plug-in estimator (default) or the first-order bias-corrected estimator (when unbiased = TRUE).

Details

The plug-in estimator \(\hat\kappa = s / \bar Y\) is the workhorse coefficient of variation in applied work and is the form usually reported. It is what this function returns by default (unbiased = FALSE). A point estimate is most informative when paired with a confidence interval; ci_cv computes one for \(\kappa\). Under normality, however, the plug-in estimator is biased downward, and the leading-order expansion is \(E[\hat\kappa] = \kappa (1 - 1/(4 N)) + O(N^{-2})\) (Sokal & Rohlf, 1995). The bias is negligible at \(N\) above about 100 but is non-trivial in small samples, where multiplying the plug-in value by \((1 + 1/(4 N))\) removes the leading-order term. The unbiased = TRUE option applies that correction.

For confidence intervals on \(\kappa\), the McKay (1932) noncentral t based interval is implemented in ci_cv; the corresponding asymptotic variances are in var_cv. The Vangel (1996) small-sample refinement of McKay's interval is a further option described in the literature; it matters more than the bias correction in this function when \(\kappa\) is larger than about 0.3 (Kelley, 2007).

References

Kelley, K. (2007). Sample size planning for the coefficient of variation from the accuracy in parameter estimation approach. Behavior Research Methods, 39(4), 755–766. doi:10.3758/BF03192966

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 3.)

McKay, A. T. (1932). Distribution of the coefficient of variation and the extended t distribution. Journal of the Royal Statistical Society, 95(4), 695–698.

Sokal, R. R., & Rohlf, F. J. (1995). Biometry: The principles and practice of statistics in biological research (3rd ed.). W. H. Freeman.

Vangel, M. G. (1996). Confidence intervals for a normal coefficient of variation. The American Statistician, 50(1), 21–26. doi:10.1080/00031305.1996.10473537

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Point estimate from raw mean and SD.
cv(mean = 100, sd = 15)
#>  term value
#>  cv   0.15 

# 2. Bias-corrected estimate at N = 50; the correction is small but
#    non-negligible at this sample size.
cv(mean = 100, sd = 15, N = 50, unbiased = TRUE)
#>  term value
#>  cv   0.151

# 3. Bias correction at N = 10; the correction is larger here.
cv(cv = .15, N = 10, unbiased = TRUE)
#>  term value
#>  cv   0.154