Methods and formulas for estimating sigma for Xbar-S Chart

The process standard deviation is also called sigma, or σ. If you enter a historical value for sigma, then Minitab uses the historical value. If you do not enter a value, Minitab uses one of the following methods to estimate sigma from the data.

Sbar method

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(ni), then Sbar is calculated as follows:

When the subgroup size is constant, Sbar is:

Notation

TermDescription
c4 (ni)value of the unbiasing constant c4 that corresponds to the value that is specified in parentheses.
Sistandard deviation of subgroup i
mnumber of subgroups

Pooled standard deviation method

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

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

Notation

TermDescription
xijjth observation in the ith subgroup
mean of subgroup i
ninumber of observations in subgroup i
μvmean of the subgroup variances
c4(·)value of the unbiasing constant c4 that corresponds to the value that is specified in parentheses.
ddegrees of freedom for Sp, given by the following formula:

Unbiasing constants c4() and c5()

c4()

c5()

Notation

TermDescription
Γ()gamma function
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