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.
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 (see Details). Missing weights are treated like missing values inx. Weights are supported for vector input only.- na.rm
logical. Should missing values be removed? With weights, observations are dropped if
xor the weight is missing.- ...
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
when x has length zero. With weights, "one observation" means a total
weight of at most 1.
Weights are frequency (replication) weights: an observation with
weight 3 counts as three identical observations, and \(n\) is the sum
of the weights. Consequently the result depends on the scale of the
weights, not only on their ratios: equal weights reproduce the
unweighted result only if each equals 1. Weights of 1/3 for three
values add up to a single observation (result NA), weights of 4/3
to four observations.
Missing values are handled pairwise: with na.rm = TRUE, every
observation where x or its weight is NA is dropped; with
na.rm = FALSE, a missing value in either yields NA. Observations with
weight 0 are dropped in any case.
**Note:** Analytic (reliability) weights, whose scale does not
matter, are not supported. For these see stats::cov.wt()
(method = "unbiased").
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(),
meanAbsDev(),
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
# missing values in x or weights are dropped pairwise
varX(c(1, NA, 3), weights=c(1, 1, 1)) # NA
#> [1] NA
varX(c(1, NA, 3), weights=c(1, 1, 1), na.rm=TRUE) # 2
#> [1] 2
varX(c(1, 2, 3), weights=c(1, NA, 1), na.rm=TRUE) # 2
#> [1] 2
# frequency weights depend on scale ...
sdX(1:3, weights=rep(1/3, 3)) # NA, total weight 1 = one observation
#> [1] NA
sdX(1:3, weights=rep(4/3, 3)) # 0.942809, four observations
#> [1] 0.942809
# ... reliability weights do not
sqrt(cov.wt(cbind(1:3), wt=rep(1/3, 3))$cov) # 1
#> [,1]
#> [1,] 1
# 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