Methods and formulas for overall capability measures in Normal Capability Analysis

Pp

Notation

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

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

PPL

Notation

TermDescription
Process mean (estimated from the sample data or a historical value)
LSLLower specification limit
TolerMultiplier of the sigma tolerance (Minitab uses 6 as the default value)
Overall standard deviation

PPU

Notation

TermDescription
USLUpper specification limit
Process mean (estimated from the sample data or a historical value)
TolerMultiplier of the sigma tolerance (Minitab uses 6 as the default value)
Overall standard deviation

Ppk

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

Cpm is available only when a target is specified. Minitab calculates Cpm based on known values of LSL, USL, and T.

Known value Cpm
LSL and USL only *
LSL, USL, and T = m
LSL, USL, and T ≠ m
USL and T only
LSL and T only
Either LSL or USL only *

Notation

TermDescription
*Missing value
USLUpper specification limit
LSLLower specification limit
mMidpoint between USL and LSL
TTarget value
Xijjth observation in the ith subgroup
niNumber of observations in the ith subgroup
TolerMultiplier of the sigma tolerance (Minitab uses 6 as the default value)

Cpm confidence interval and bound

Minitab calculates the (1 – α) 100% two-sided confidence interval for Cpm as follows:

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

Notation

TermDescription
α quantile of the chi-square distribution with ν degrees of freedom
Degrees 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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