var-methods: Variance and Standard Deviation

In actuar: Actuarial Functions and Heavy Tailed Distributions

Variance and Standard Deviation

Description

Generic functions for the variance and standard deviation, and methods for individual and grouped data.

The default methods for individual data are the functions from the stats package.

Usage

var(x, ...)

## Default S3 method:
var(x, y = NULL, na.rm = FALSE, use, ...)

## S3 method for class 'grouped.data'
var(x, ...)

sd(x, ...)

## Default S3 method:
sd(x, na.rm = FALSE, ...)

## S3 method for class 'grouped.data'
sd(x, ...)

Arguments

x	a vector or matrix of individual data, or an object of class "grouped data".
y	see stats::var.
na.rm	see stats::var.
use	see stats::var.
...	further arguments passed to or from other methods.

Details

This page documents variance and standard deviation computations for grouped data. For individual data, see var and sd from the stats package.

For grouped data with group boundaries c_0, c_1, \dots, c_r and group frequencies n_1, \dots, n_r, var computes the sample variance

\frac{1}{n - 1} \sum_{j = 1}^r n_j (a_j - m_1)^2,

where a_j = (c_{j - 1} + c_j)/2 is the midpoint of the jth interval, m_1 is the sample mean (or sample first moment) of the data, and n = \sum_{j = 1}^r n_j. The sample sample standard deviation is the square root of the sample variance.

The sample variance for grouped data differs from the variance computed from the empirical raw moments with emm in two aspects. First, it takes into account the degrees of freedom. Second, it applies Sheppard's correction factor to compensate for the overestimation of the true variation in the data. For groups of equal width k, Sheppard's correction factor is equal to -k^2/12.

Value

A named vector of variances or standard deviations.

Author(s)

Vincent Goulet [e-mail]. Variance and standard deviation methods for grouped data contributed by Walter Garcia-Fontes [e-mail].

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (1998), Loss Models, From Data to Decisions, Wiley.

Heumann, C., Schomaker, M., Shalabh (2016), Introduction to Statistics and Data Analysis, Springer.

See Also

grouped.data to create grouped data objects; mean.grouped.data for the mean and emm for higher moments.

Examples

data(gdental)
var(gdental)
sd(gdental)

## Illustration of Sheppard's correction factor
cj <- c(0, 2, 4, 6, 8)
nj <- c(1, 5,  3,  2)
gd <- grouped.data(Group = cj, Frequency = nj)
(sum(nj) - 1)/sum(nj) * var(gd)
(emm(gd, 2) - emm(gd)^2) - 4/12

actuar documentation built on Nov. 8, 2023, 9:06 a.m.