# Methods and formulas for confidence intervals and bounds in Normal Capability Analysis for Multiple Variables

## Cp confidence interval bounds

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

where ν is calculated based on the method used to estimate σ2within:

• Pooled standard deviation: ν = Σ (ni- 1)
• Average moving range and Median moving range: ν ≈ k – Rspan + 1
• Square root of MSSD: ν = k - 1
• Rbar: ν = 0.9 k (n - 1)
• Sbar: ν = fn k (n - 1), where fn is the adjustment factor that varies with n, as shown in the following table.
n fn
2 0.88
3 0.92
4 0.94
5 0.95
6, 7 0.96
8, 9 0.97
10-17 0.98
18-64 0.99
65- 1.00

### Notation

TermDescription
χ2α,νThe α percentile of the chi-square distribution with ν degrees of freedom
αAlpha for the confidence level
νDegrees of freedom
σ2withinWithin-subgroup variance
ni The ith subgroup size
kNumber of samples
RspanLength of the moving range
nAverage sample size (Σ ni / k).

## Z.Bench (Within) confidence interval and bounds

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
νDegrees of freedom based on the method used to estimate σ2within (for information on the calculation of ν, see the topic on Cp confidence interval bounds)
γN, 1 -αGamma value based on the alpha level and number of observations (for more information see the Gamma table section)
Process mean (estimated from the sample data or a historical value)
Within-subgroup standard deviation

## Cpk confidence interval bounds

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

### Notation

TermDescription
NThe total number of observations
αAlpha for the confidence level
vThe degrees of freedom based on the method used to estimate σ2within (for information on the calculation of v, see the section on Cp confidence interval bounds)
TolerMultiplier of the sigma tolerance (Minitab uses 6 as the default value)
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

## Z.Bench (overall) confidence interval and bounds

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
νThe degrees of freedom (N – 1)
γN, 1 -αGamma value based on the alpha level and number of observations (for more information, see the Gamma table section)
Process mean (estimated from the sample data or a historical value)
Overall standard deviation

## 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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