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EWMACHART Statement |
Here denotes the gamma function, and denotes the i^{ th} subgroup mean. A subgroup standard deviation s_{i} is included in the calculation only if . If the observations are normally distributed, then the expected value of s_{i} is .Thus, is the unweighted average of N unbiased estimates of . This method is described in the ASTM Manual on Presentation of Data and Control Chart Analysis (1976).
A subgroup standard deviation s_{i} is included in the calculation only if , and N is the number of subgroups for which .The MVLUE assigns greater weight to estimates of from subgroups with larger sample sizes, and it is intended for situations where the subgroup sample sizes vary. If the subgroup sample sizes are constant, the MVLUE reduces to the default estimate.
The weights are the degrees of freedom n_{i}-1. A subgroup standard deviation s_{i} is included in the calculation only if ,and N is the number of subgroups for which .
If the unknown standard deviation is constant across subgroups, the root-mean-square estimate is more efficient than the minimum variance linear unbiased estimate. However, in process control applications it is generally not assumed that is constant, and if varies across subgroups, the root-mean-square estimate tends to be more inflated than the MVLUE.
where N is the number of observations, and x_{1},x_{2}, ... ,x_{N} are the individual measurements. This formula is given by Wetherill (1977), who states that the estimate of the variance is biased if the measurements are autocorrelated.
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