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

## Cp

### Notation

TermDescription
USLUpper specification limit
LSLLower specification limit
TolerMultiplier of the sigma tolerance (Minitab uses 6 as the default value)
Between/within standard deviation

## Cp confidence interval for between/within capability analysis

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

where ν is approximated by

and

### Notation

TermDescription
χ2α,νThe α percentile of the chi-square distribution with ν degrees of freedom
αAlpha for the confidence level
νDegrees of freedom
Yij The jth observation in the ith subgroup
The ith subgroup average
The average across all subgroups
I The number of subgroups
ni The ith subgroup size

## CPL

### Notation

TermDescription
Average of the observations
LSLLower specification limit
TolerMultiplier of the sigma tolerance (Minitab uses 6 as the default value)
Between/within standard deviation

## CPU

### Notation

TermDescription
USLUpper specification limit
Average of the observations
TolerMultiplier of the sigma tolerance (Minitab uses 6 as the default value)
Between/within standard deviation

## Cpk confidence interval bounds for between/within capability

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

###### Note

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.

### Notation

TermDescription
NThe total number of observations
αAlpha for the confidence level
vThe 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/2The 1-α/2 percentile from the standard normal distribution

## CCpk

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

Known value CCpk
USL only
LSL only
LSL and USL

where:

### Notation

TermDescription
Estimator for the population mean
USLUpper specification limit
LSLLower specification limit
Between/within standard deviation
Average of the observations
TolerMultiplier of the sigma tolerance (Minitab uses 6 as the default value)
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