Methods and formulas for expected between/within performance in Normal Capability Analysis for Multiple Variables

Cp

Term	Description
USL	Upper specification limit
LSL	Lower specification limit
Toler	Multiplier of the sigma tolerance (Minitab uses 6 as the default value)
	Between/within standard deviation

The (1 -α) 100% confidence interval for Cp is calculated as follows:

where ν is approximated by

and

Term	Description
χ²_α_,_ν	The α percentile of the chi-square distribution with ν degrees of freedom
α	Alpha for the confidence level
ν	Degrees of freedom
Y_ij	The jth observation in the ith subgroup
	The ith subgroup average
	The average across all subgroups
I	The number of subgroups
n_i	The ith subgroup size

Term	Description
	Average of the observations
LSL	Lower specification limit
Toler	Multiplier of the sigma tolerance (Minitab uses 6 as the default value)
	Between/within standard deviation

Term	Description
USL	Upper specification limit
	Average of the observations
Toler	Multiplier of the sigma tolerance (Minitab uses 6 as the default value)
	Between/within standard deviation

The (1 -α) 100% confidence interval for Cpk is calculated as follows:

If the sigma tolerance value, say k, in the input is not the default 6, then 9 in the formula should be replaced by (k/2)**2.

Term	Description
N	The total number of observations
α	Alpha for the confidence level
v	The degrees of freedom. For information on the calculation of ν, see the topic on the Cp confidence interval for between/within capability analysis.
Z₁-_α_/2	The 1-α/2 percentile from the standard normal distribution

Minitab calculates CCpk based on known values of LSL, USL, and T.

where:

Term	Description
	Estimator for the population mean
USL	Upper specification limit
LSL	Lower specification limit
	Between/within standard deviation
	Average of the observations
Toler	Multiplier of the sigma tolerance (Minitab uses 6 as the default value)