Benchmark Z statistics for overall capability are calculated by finding the Z value using the standard normal (0,1) distribution for the corresponding statistics.
where:
Term | Description |
---|---|
Φ (X) | Cumulative distribution function (CDF) of a standard normal distribution |
Φ-1 (X) | Inverse CDF of a standard normal distribution |
![]() | Overall standard deviation |
where
To calculate , substitute the sample estimates for the parameters in the formula for
:
where
To calculate a one-sided upper confidence bound, change to
in the definition of U.
Term | Description |
---|---|
![]() | the estimated tail probabilities outside of the specificataion limits |
![]() | the (1 - α / 2)th percentile of the standard normal distribution |
α | the alpha for the confidence level |
![]() | the process mean (estimated from the sample date or a historical value) |
s | the sample standard deviation within subgroups |
υ | the degrees of freedom for s |
![]() | the Cumulative Distribution Function (CDF) from a standard normal distribution |
![]() | the Probability Density Function (PDF) from a standard normal distribution |
USL | the upper specification limit |
LSL | the lower specification limit |
![]() | the inverse CDF from a standard normal distribution |
The calculations for the confidence interval for Z.Bench depend on which specification limit the process has.
Minitab solves the following equation to find p1:
where
Minitab solves the following equation to find p2:
where
Term | Description |
---|---|
LSL | the lower specification limit |
USL | the upper specification limit |
α | the alpha for the confidence level |
![]() | the Cumulative Distribution Function (CDF) from a standard normal distribution |
![]() | the inverse CDF from a standard normal distribution |
![]() | the (1 - α/2)th percentile of the standard normal distribution |
N | the total number of measurements |
υ | the degrees of freedom for s |
![]() | the process mean (estimated from the sample date or a historical value) |
s | the sample overall standard deviation |
![]() | a random variable that is distributed as a non-central t distribution with ![]() |