How is the standard deviation calculated for the different moving average control charts in JMP®?

[#1]

EWMA and UWMA Charts

The estimate for σ is

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

and

Here

denotes the gamma function and

denotes the i ^th subgroup mean. A subgroup standard deviation s_i is included in the claculation only if n_i≥2. If the observations are normally distributed, then the expected value of s_i is c₄(n_i)σ. Thus,

is the unweighted average of N unbiased estimates of σ.

When each subgroup sample contains a single observation (n_i=1), the process standard deviation σ is estimated as

where N is the number of observations and x_i,x₂,...,x_n are the individual measurements.

[Previously JMP Note 35442]