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PROC CAPABILITY and General Statements |

You can use the NORMALTEST option in the PROC CAPABILITY statement
to request several tests of the hypothesis that
the analysis variable values are a random sample from a normal
distribution.
These tests, which are summarized
in the table
labeled *Tests for Normality*,
include the following:

- Shapiro-Wilk test
- Kolmogorov-Smirnov test
- Anderson-Darling test
- Cramr-von Mises test

Tests for normality are particularly important
in process capability analysis because the
commonly used capability indices are difficult
to interpret unless the data are at least approximately
normally distributed.
Furthermore, the confidence limits for capability
indices displayed in the table labeled
*Process Capability Indices*
require the assumption of normality.
Consequently,
the tests of normality are always
computed when you specify the SPEC statement,
and a note is added to the
table when the hypothesis of normality
is rejected.
You can specify the particular test and
the significance level
with the CHECKINDICES option.

Small values of *W*
lead to rejection of the null hypothesis.
The method for computing the
*p*-value (the probability of obtaining a *W*
statistic less than or equal to the observed value)
depends on *n*.
For *n*=3, the probability distribution
of *W* is known and is used to determine the *p*-value.
For *n*>4, a normalizing transformation
is computed:

The empirical distribution function is defined for a set
of *n* independent observations *X _{1}*, ... ,

Note that *F*_{n}(*x*) is a step function that takes a
step of height [1/*n*] at each observation.
This function estimates the distribution function
*F*(*x*). At any value *x*, *F*_{n}(*x*) is the proportion
of observations less than or equal to *x*, while *F*(*x*)
is the probability of an observation less than or equal
to *x*. EDF statistics measure the discrepancy between
*F*_{n}(*x*) and *F*(*x*).

The EDF tests
make
use of the probability integral transformation *U*=*F*(*X*).
If *F*(*X*) is the distribution function of *X*, the random
variable *U* is uniformly distributed between 0 and 1.
Given *n* observations *X _{(1)}*, ... ,

The Kolmogorov-Smirnov statistic is computed as the
maximum of *D ^{+}* and

PROC CAPABILITY uses a modified Komogorov *D*
statistic to test the data against
a normal distribution with mean and variance equal to
the sample mean and variance.

The Anderson-Darling statistic (*A ^{2}*) is defined as

The Anderson-Darling statistic is computed as

The Cramr-von Mises statistic is computed as

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