Methods and formulas for estimating sigma for Xbar Chart

Minitab uses the range of each subgroup, , to calculate , which is an unbiased estimator of σ:

where

When the subgroup size is constant, the formula simplifies to the following:

where (Rbar) is the mean of the subgroup ranges, calculated as follows:

Notation

Without unbiasing constant

If you do not use an unbiasing constant, then the Sbar is the mean of the subgroup standard deviations:

With unbiasing constant

If you use the unbiasing constant, c₄(n_i), then Sbar is calculated as follows:

When the subgroup size is constant, Sbar is:

Notation

The pooled standard deviation (S_p) is given by the following formula:

When the subgroup size is constant, S_p can also be calculated as follows:

With unbiasing constant

By default, Minitab applies the unbiasing constant, c₄(), when you use the pooled standard deviation to estimate σ:

When the subgroup size is constant, the unbiased S_p can also be calculated as follows:

Notation

Term	Description
xij	jth observation in the ith subgroup
	mean of subgroup i
ni	number of observations in subgroup i
μv	mean of the subgroup variances
c4(·)	value of the unbiasing constant c4 that corresponds to the value that is specified in parentheses.
d	degrees of freedom for Sp, given by the following formula:

d₂(N) is the expected value of the range of N observations from a normal population with standard deviation = 1. Thus, if r is the range of a sample of N observations from a normal distribution with standard deviation = σ, then E(r) = d₂(N)σ.

d₃(N) is the standard deviation of the range of N observations from a normal population with σ = 1. Thus, if r is the range of a sample of N observations from a normal distribution with standard deviation = σ, then stdev(r) = d₃(N)σ.

Use the following table to find an unbiasing constant for a given value, N. (To determine the value of N, consult the formula for the statistic of interest.)

For values of N from 51 to 100, use the following approximation for d₂(N):

For values of N from 26 to 100, use the following approximations for d₃(N) and d₄(N):

c₄()

c₅()

Notation