How is the standard deviation calculated for different Shewhart control charts?

XR Charts
The estimate for σ is

where N is the number of subgroups for which n_i>2, and R_i is the sample range of the observations x_i1,...,x_ini in the i^th subgroup

XS Charts
The estimate for σ is

where N is the number of subgroups for which n_i>2, and s_i is the sample standard deviation of the i^th subgroup

and

Here Γ(·) denotes the gamma function and X-bar_i denotes the i^th subgroup mean.

IR Charts
The estimate for σ is

where R-bar is the average of the moving ranges, n is the number of consecutive individual measurements used to compute each moving range, and the unbiasing factor d₂(n) is defined so that if the observations are normally distributed, the expected value of R_i is E(R_i)=d₂(n_i)σ

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FAQ # 2076
Last Updated: 2005 May 16

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