This program is designed
to interpret the results of a sampling inspection, for the purpose of judging compliance
with chosen limits. It may also be
used to identify outlying values or departure from the assumed (Gaussian or Student’s-t)
distribution. Uncertainties for the
individual values may be entered, and a mean and standard deviation for the set
are calculated. The statistical test
is based on a selection of the “Consumer’s Risk” (CR) and the “Lot Tolerance Percent
Defective” (LTPD), so that the stringency of the test may be adjusted. Typical values are CR=0.1 and LTPD=10%. More stringent tests may be made by choice of smaller values of the CR and
LTPD. Confidence limits for the fitted
distribution may be plotted for identification of outlying points, values that do
not fit in the distribution.
Statistical analysis is used to convert a large amount of data into a manageable
amount of understandable information.
This process can involve a variety of techniques, the simplest being to determine
the average (or mean) value for a given
set of data.
This simple determination
is improved upon by also calculating the standard deviation of the data about the
mean, which gives an estimate of the variability of the data.
In many cases, this variability represents variations both in the characteristics
or values being measured and in measurement technique fluctuations.
The significance of
these quantities (mean and standard deviation) depends upon the distribution assumed
for the data. Sometimes there is a
theoretically known distribution for a particular measurement process, such as the
binominal, or Poisson distribution for counting radioactivity. These distributions are relatively well approximated by the Gaussian, or
normal, distribution. In fact, the
Gaussian distribution approximates the distribution of many different kinds of measurements
and for simplicity is generally assumed to be the proper distribution. The Gaussian distribution is generally seen in the form of a bell-shaped
curve, with most values occurring near the mean value and fewer and fewer values
existing at increasing distances from the mean, both greater and less than the mean.
However, it is difficult
to derive the bell-shaped curve from experimental data unless the data are specifically
selected to demonstrate the curve, and deviations from the distribution are difficult
to see. A better version is the so-called
“cumulative probability function” utilized in this program, which forms an S-shaped
curve when plotted in the usual manner.
This can be further improved by adjusting the abscissa (the “X” values in an X-Y
graph) so that the “S” curve becomes a straight line.
This is a standard statistical technique and is the basis for special graph
paper used for probability analysis of data.
The parameters of the Gaussian distribution (the mean and the standard deviation)
are determined by the usual calculational methods.
Where the data are
not well-represented by a Gaussian distribution (and this is true in most cases)
the departure is readily apparent; the data points do not lie along a straight line
representing the Gaussian distribution.
In most cases, this departure takes a single typical form. Much of the data lies along the theoretical straight line, with a few points
at either extreme lying somewhat above it.
This form can usually be interpreted as showing a large number of uncontaminated
measurements where the variability is due to random fluctuations in the measurements
themselves, with the balance being locations that harbor more or less residual contamination.
If the contaminated area is large, there will be many points departing from the
curve. In these cases, the points will not fit the theoretical straight line.
If most of the region in question is contaminated, the distribution will be dominated
by the contaminated data points, in a line of points generally sloping from the
lower left to the upper right, fitting more or less closely, a theoretical straight
line.
This program is used
to provide a sampling inspection test.
It uses a standard quality control technique called inspection by variables, in
which the distribution of the measured values is used to predict the probability
that other unmeasured values would exceed a specified limit. The standard test method requires calculating the mean and the standard deviation-(s). Then, depending the values chosen for
certain parameters that reflect the performance of the test in accepting bad lots,
or rejecting good lots, the necessary number of samples is determined and a multiplier,
k, is computed so that the inequality mean + ks < U where U is the acceptance limit, representing
an acceptable lot. The parameters used in the program to calculate
the multiplier
are the CR and LTPD This value of mean
+ ks, that is compared to the limit is what is called the Test Statistic, or Ts. The value of Ts is a point near the
upper end of the observed data distribution.
If this value is less than the acceptance limit U, the lot has passed the Sampling
Inspection by Variables Test, according to the criteria chosen for CR and LTPD.
The usual manner of
applying this inspection method is to use tables giving the value of the sample
size (N) and multiplier (k) for the selected values of CR and LTPD. The program uses the number of measured values (N) in the lot to compute
k, and this value is used to calculate mean + ks.