Methods and formulas for overall capability measures in Nonnormal Capability Sixpack

In This Topic

Pp
Ppk
PPM Total for expected "overall" performance
Z.Bench Z.LSL Z.USL

Pp

Pp is calculated using the parameter values that you specify or is estimated using maximum likelihood method for a distribution used in the analysis. Minitab calculates the Pp statistic using either the Z-Score method (default) or the ISO method, depending on the settings that are selected in File > Options > Control Charts and Quality Tools > Capability Analysis. The following formulas are based on the default tolerance of K = 6, which is 6 standard deviations wide (3 standard deviations on each side of the process mean).

Z-Score method

Notation

Term	Description
Zlsl	Φ^–1(p₁)
Zusl	Φ^–1(p₂)
Φ^–1(p)	p * 100^th percentile of a standard normal distribution
p₁	Prob (X ≤ LSL)
p₂	Prob (X ≤ USL)
X	X follows the distribution used in the analysis

ISO method

Notation

Term	Description
USL	Upper specification limit
LSL	Lower specification limit
X_0.99865	The 99.865^th percentile for the specified distribution
X_0.00135	The 0.135^th percentile for the specified distribution

Ppk

PPM Total for expected "overall" performance

The expected number of parts per million that are outside of the specification limits is calculated as follows:

PPM Total (Exp. Overall) = PPM < LSL (Exp. Overall) + PPM > USL(Exp. Overall)

PPM Total (Exp. Overall) = [1,000,000 * F(LSL)]+ [1,000,000 * (1 – F(USL))]

Notation

Term	Description
F(X)	The cumulative distribution function (CDF) of the distribution used in the analysis based on estimated or specified parameters
LSL	Lower specification limit
USL	Upper specification limit
PPM	Parts per million

Z.Bench Z.LSL Z.USL

Benchmark Z statistics for overall capability are calculated by finding the Z value using the standard normal (0,1) distribution for the corresponding statistics.

Note

PPL and PPU are based on the method (ISO or Minitab) used in the analysis. For more information, go to Methods and formulas for overall capability measures in Nonnormal Capability Analysis. The formulas are based on the default tolerance of K = 6, which is 6 standard deviations wide (3 standard deviations on each side of the process mean).

Notation

Term	Description
P₁	Probability (Observations < LSL) based on the nonnormal distribution used in the analysis and the parameters that you specify (or the parameters that are estimated from the data)
P₂	Probability (Observations > USL) based on the nonnormal distribution used in the analysis and the parameters that you specify (or the parameters that are estimated from the data)
Φ (X)	Cumulative distribution function (CDF) of a standard normal distribution
Φ^–1 (X)	Inverse CDF of a standard normal distribution