# Methods and formulas for confidence intervals and bounds in Between/Within Capability Analysis

## 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

## Z.Bench (Between/Within)

The calculations for the confidence interval for Z.Bench depend on the known values of the specification limits.

• When both the lower and upper specifications limits are known, Minitab calculates only the lower bound of Z.Bench.

(1 -α) 100% lower bound = Φ-1 (1 - PU)

where:

###### Note

The lower bound is displayed only when 1-α is greater than or equal to 0.80.

• When only the lower specification limit is known, Minitab calculates both the lower and the upper bounds for Z.Bench.

(1 -α) 100% lower bound = Φ-1 (1 - PU)

(1 -α) 100% upper bound = Φ-1 (1 - PL)

where:

• When only the upper specification limit is known, Minitab calculates both the lower and the upper bounds for Z.Bench.

(1 -α) 100% lower bound = Φ-1 (1 - PU)

(1 -α) 100% upper bound = Φ-1 (1 - PL)

where:

### Notation

TermDescription
LSLLower specification limit
USLUpper specification limit
αAlpha for the confidence level
Φ (X) cdf of a standard normal distribution
NTotal number of observations
TermDescription
νDegrees of freedom based on the method used to estimate σ2B/W. See the formula for v used to calculate the Cp confidence interval for between/within capability analysis.
γN, 1 -αGamma value based on the alpha level and number of observations (for more information see the Gamma table section)
Average of the observations
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

## Pp confidence interval bounds

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

### Notation

TermDescription
χ2α,νThe α percentile of the chi-square distribution with ν degrees of freedom
αAlpha for the confidence level
νThe degrees of freedom (Σni– 1)
ni

The number of observations in the ith subgroup

## Ppk confidence interval bounds

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

### Notation

TermDescription
NThe total number of observations
αAlpha for the confidence level
vThe degrees of freedom (Σni – 1 or N – 1)
niThe number of observations in the ith subgroup
TolerMultiplier of the sigma tolerance (Minitab uses 6 as the default value)
Z1α/2The 1 – (α/2) percentile from the standard normal distribution

## Cpm confidence interval bound

The (1 – α) 100% lower confidence bound for Cpm is calculated as follows:

###### Note

Minitab displays only the lower bound for Cpm.

### Notation

TermDescription
νDegrees of freedom, defined as N ((1 + a2) 2 / (1 + 2a2))
a(Mean – Target)/
αAlpha for the confidence level
N Total number of observations
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