(Weighted) Variance and Standard Deviation

varX() computes the variance of x, allowing the definition of weights (unlike base R's var() function). Using the estimator ml returns the uncorrected sample variance (which is a biased estimator for the sample variance).
sdX yields the standard deviation following the same logic.

Usage

sdX(x, estimator = c("unbiased", "ml"), weights = NULL, na.rm = FALSE, ...)

varX(x, ...)

# Default S3 method
varX(x, estimator = c("unbiased", "ml"), weights = NULL, na.rm = FALSE, ...)

# S3 method for class 'Freq'
varX(x, breaks, estimator = c("unbiased", "ml"), ...)

Arguments

x

a numeric vector, matrix, or data frame

estimator

determines the estimator type; if "unbiased" (the default) then the usual unbiased estimate (using \(n - 1\) as denominator) is returned, if "ml" then it is the maximum likelihood estimate for a Gaussian distribution (denominator \(n\)).

weights

non-negative numeric vector of weights the same length as x, interpreted as frequency (replication) weights. Observations with larger weights contribute more strongly to the empirical distribution. Weights are supported for vector input only.

na.rm

logical. Should missing values be removed?

...

further arguments passed to or from other methods

breaks

breaks for calculating the variance for classified data as composed by freq()

Value

varX()

a numeric scalar for vector input or a covariance matrix for a matrix or data frame

sdX()

a numeric scalar containing the standard deviation

Details

Using estimator "unbiased" the denominator \(n - 1\) is used (known as "Bessel's correction") which gives an unbiased estimator of the (co)variance for i.i.d. observations.
"ml" yields the biased version using the denominator \(n\). With frequency weights \(n\) is the sum of the weights.

These functions return NA() when there is only one observation and NA when x has length zero.

**Note:** Analytic (precision) weights are not supported. For likelihood-based weighted variance estimation, see stats::cov.wt().

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

See also

lumen::varCI() for confidence intervals, lumen::varTest() for tests and base R's implementations var(), sd(), cov()

Other dispersion: coefVar(), iqrX(), madX(), meanAD(), meanSE(), rangeX()

Examples

varX(1:10)                 # 9.166667
#> [1] 9.166667
sdX(1:10)
#> [1] 3.02765

# frequency weights replicate the observations, so the result is the
# variance of the expanded vector c(1, 2,2, 3,3,3, 4,4,4,4, 5,5,5,5,5)
varX(1:5, weights=1:5)     # 1.666667
#> [1] 1.666667
varX(rep(1:5, times=1:5))  # 1.666667
#> [1] 1.666667

# weighted Variance
set.seed(45)
(z <- as.numeric(names(w <- table(x <- sample(-10:20, size=50, replace=TRUE)))))
#>  [1] -9 -8 -7 -6 -5 -4 -3 -2  0  3  4  5  6  7  8  9 10 12 13 15 16 17 18 19 20
varX(z, weights=w)
#> [1] 86.53429
sdX(z, weights=w)
#> [1] 9.302381

# check!
all.equal(varX(x), varX(z, weights=w))
#> [1] TRUE


# Variance for frequency tables
varX(freq(as.table(c(6,16,24,25,17))),
          breaks=c(0, 10, 20, 30, 40, 50))
#> [1] 140.3213