7.
Product and Process Comparisons
7.2.

Comparisons based on data from one process


Questions answered in this section

For a single process, the current state of the process can be
compared with a nominal or hypothesized state. This section
outlines techniques for answering the following questions from
data gathered from a single process:
 Do the observations come from a particular
distribution?
 ChiSquare GoodnessofFit test
for a continuous or discrete distribution
 Kolmogorov Smirnov test for a
continuous distribution
 AndersonDarling and ShapiroWilk tests for a
continuous distribution
 Are the data consistent with the assumed
process mean?
 Confidence interval
approach
 Sample sizes required
 Are the data consistent with a nominal
standard deviation?
 Confidence interval
approach
 Sample sizes required
 Does the proportion of defectives meet
requirements?
 Confidence intervals
 Sample sizes required
 Does the defect density meet requirements?
 What intervals contain a fixed
percentage of the data?
 Approximate intervals that
contain most of the population values
 Percentiles
 Tolerance intervals
 Tolerance intervals based on the
smallest and largest observations

General forms of testing

These questions are addressed either by an hypothesis
test or by a confidence interval.

Parametric vs. nonparametric testing

All hypothesistesting procedures can be broadly described as
either parametric or nonparametric/distributionfree. Parametric
test procedures are those that:
 Involve hypothesis testing of specified parameters (such
as "the population mean=50 grams"...).
 Require a stringent set of assumptions about the underlying
sampling distributions.

When to use nonparametric methods?

When do we require nonparametric or distributionfree methods?
Here are a few circumstances that may be candidates:
 The measurements are only categorical; i.e., they are
nominally scaled, or ordinally (in ranks) scaled.
 The assumptions underlying the use of parametric methods
cannot be met.
 The situation at hand requires an investigation of such
features as randomness, independence, symmetry, or goodness
of fit rather than the testing of hypotheses about
specific values of particular population parameters.

Difference between nonparametric and distributionfree

Some authors distinguish between nonparametric and
distributionfree procedures.
Distributionfree test procedures are broadly defined as:
 Those whose test statistic does not depend on the form of
the underlying population distribution from which the sample
data were drawn, or
 Those for which the data are nominally or ordinally scaled.
Nonparametric test procedures are defined as those that are
not concerned with the parameters of a distribution.

Advantages of nonparametric methods.

Distributionfree or nonparametric methods have several advantages,
or benefits:
 They may be used on all types of datacategorical data, which
are nominally scaled or are in rank form, called ordinally
scaled, as well as interval or ratioscaled data.
 For small sample sizes they are easy to apply.
 They make fewer and less stringent assumptions than their
parametric counterparts.
 Depending on the particular procedure they may be
almost as powerful as the corresponding parametric
procedure when the assumptions of the latter are met, and
when this is not the case, they are generally more
powerful.

Disadvantages of nonparametric methods

Of course there are also disadvantages:
 If the assumptions of the parametric methods can be met, it
is generally more efficient to use them.
 For large sample sizes, data manipulations tend to become
more laborious, unless computer software is available.
 Often special tables of critical values are needed for the
test statistic, and these values cannot always be generated
by computer software. On the other hand, the critical values
for the parametric tests are readily available
and generally easy to incorporate in computer programs.
