Chapter 7

Process Capability

Introduction

• A “capable” process is one for which the distributions of the process characteristics do lie almost entirely within the engineering tolerances or customer’s needs.

• Process capability indices– Simple– Careful in use and interpretation

• Two phases in a process capability study– Determining how to data are to be collected, and then

collecting the data – Selecting 1 or more indices and performing the computations

7.1 Data Acquisition for Capability Indices

• Data must come from an in-control process• The sample must be representative of the population• The sample size must be large enough

– To assess the extent of the non-normality– To allow a non-normal distribution to be fit to the data

• Process capability indices <> Process Performance indices

7.2 Process Capability Indices

• Should be easy to compute• Should not be undermined by slight-to-moderate

departures from normality

(7.1)

(7.2)

7.3 Estimating the Parameters in Process Capability Indices

7.4 Distributional Assumption for Capability Indices

• It is assumed that the observations have come from a normal distribution.

• A normal distribution is also assumed when the capability indices are used.

7.5 Confidence Intervals for Process Capability Indices

• Unless the sample size was large, it is desirable to also report a confidence interval for the index.

• Lower confidence bound is more appropriate than a 2-sided confidence interval.

• It is assumed that individual observations are used in computing the parameter estimates (7.5.1~7.5.4)

(7.3)

7.5.5 Confidence Intervals Computed Using Data in Subgroups

7.5.6 Nonparametric Capability Indices and Confidence Limits

• Some quality characteristics such as diameter, roundness, mold dimensions, and customer waiting time will be non-normal, and flatness, runout, and % contamination will have skewed distributions.

• Process capability indices are not robust to non-normality in the individual observations.

• 4 approaches for non-normal distributions:– Robust capability index– Fit a distribution to a set of data and use percentiles in an index– Transform the data to approximate normal– Resample from the n sampled observations

7.5.6.1 Robust Capability Indices

7.5.6.2 Capability Indices Based on Fitted Distributions

7.5.6.3 Data Transformation

• Data can be transformed so the transformed data will be approximately normally distributed.

• Lognormal data

7.5.6.4 Capability Indices Computed Using Resampling Methods

• Resampling methods have been used to approximate sampling distributions when no assumption is made of the distribution of the random variable.

• Bootstrapping is one type of resampling.– Naïve bootstrap: keep the original sample size, resample with

replacement• The standard bootstrap methods can not be relied on to

produce a lower confidence limit for a capability index.

7.6 Asymmetric Bilateral Tolerances

(7.4)

7.6.1 Example

A B CUSL 62 62 62LSL 50 50 50 50 56 62 1 1 1T 59 59 59

Cp 2 2 2

Cpk 0 2 0

Cpk’ 0 0 0

Cpm 0.2209 0.6325 0.6325

Cpmk 0 0.6325 0% def. 0.500000 0.000000 0.500000

7.7 Capability Indices that are a Function of % Non-conforming

7.11 Process Capability Indices vs. Process Performance Indices

7.13 Software for Process Capability Indices

• Minitab: