The following discussion uses a product example
referred to as a widget to provide the basis for statistical principles
underlying parameter characterization (Figure B-1). The widget consists of
five components including a housing (e) and the internal parts (p), and a
determination is needed on the physical dimensions of the parts in a lot of
10,000. Instead of measuring each part, a random sample of 100 parts was
taken. Since the sample was truly random, any statistical estimates were
considered representative of the population.

There are two estimates being made - the first being
the arithmetic average or mean (µ), which is the balance point of the
measurements. The second index is the population standard deviation (), or statistical unit of measure relating to the dispersion of
the measurements around the balance point or mean. In other words,
would be the measuring device with which to describe the "scatter" present in
the data.

A histogram - similar to a bar chart - graphically displays the variation.
It shows how the values tend to group themselves together across the range of
values (Figure B-2).

The width of each bar is the same, and the height of any given bar is
representative of the number of part measurements associated with that bar's
width. If a smooth curve was drawn through the top of each bar, a shape
somewhat like a bell would be formed. The bell curve is referred to as normal
distribution. Most of the measurements cluster around the center of the bell
located on the measurement scale directly below the curve's hump, or average
(µ). The baseline tails never touch the baseline, even if they come quite
close.

Using the normal distribution, an estimate can be made of how many of the
widget's parts in the example lot would be found between 1
limits of distribution. That estimate is based on the fact that statisticians
have already determined how much area is under the normal curve at various
points on a scale that uses the standard deviation () as a
measurement basis. Since 68.26% of the area under the curve lies 1 of µ, with the example used here that would translate
to 68.26% of the parts being between 1.239 inches and 1.241 inches (Figure
B-3). Using Figure B-3, it can be seen that 99.73 of the parts fall within the
engineering specification range because the upper specification limit (USL)
and lower specification limit (LSL) are 3 from the center or nominal value of the specification. That is inversely stated as a 0.27% defect rate, or 2700 non-conforming parts per million (ppm). Although that number may seem low, and indeed has been in the past, if the standard deviation were smaller in another lot (#2) where = .0005
inch as the present market demands, 99.9999998% of the area under the curve
would be within the same engineering specification. That translates to only
.002 ppm non-conforming.

Under normal circumstances, it is preferable to compute the probability of
exceeding a given design constraint without the use of the bell curve - this
is accomplished through the use of the standard Z transform. This changes a
set of data so that the mean is always equal to zero (µ = O) and the standard
deviation is always equal to one ( = 1). The
measurement scale is unitless - the units of measurement such as inches are
eliminated. For example, to determine how many s there
from the mean (µ) to 1.242 inches, the necessary calculations are made using
the standard Z transform equation and the answer computes to Z = 2.000.

This indicates that the length measurement between µ and 1.242 inches is
2 to the right of µ, since the number is positive.

The Z transform is useful to determine how much of a distribution lies
beyond a given design specification - or in tolerance work, how many
non-conformities (defects) to expect. For example, using the widget part
example, how much of the area under the normal curve is beyond the upper
specification limit given a new population mean such that µ = 1.242 inches. By
modifying the equation used earlier determining how many s there are
from 1.242 inches to the upper specification limit, it is determined that Z =
1.000. This indicates that the upper specification limit lies to the right of
µ by 1. Using a table that has the areas calculated for corresponding Z
values, it can be determined how much of the area under the normal curve lies
beyond 1 - the answer being 15.87% of the part measurements lie to the
right of the upper specification limit.

Characterizing the quality of the example widget parts can be defined in
terms of a capability ratio or performance index. This index (C_{p})
contrasts the distribution width to the engineering specification width to
provide a number for comparison. It gives the ability to directly compare the
inherent quality of one product characteristic to another. It can also be used
to determine how close an item is to a given performance index. The lot
presented earlier that has the standard deviation of .0005-inch would have a
C_{p} = 2.0 or the specification width is twice as wide as the
distribution, or the +3 range
of the distribution only consumes 50% of the specification width.

Another capability index, C_{pk}, indicates how far µ is from the
nominal condition of the specification in light of the distribution width.
Whenever µ is the same value as the nominal specification - or target value
(most often given by the arithmetic mean) - then C_{p} =
C_{pk}. As µ moves left or right from the nominal specification,
C_{pk} gets smaller - assuming å remains constant. If a level of
capability such that C_{p} = 2.0 (+6) was
required, this would dictate a 0.002 ppm as long as µ is equal to the nominal
value or C(pk) = 2.0. Because the industrial world is always dynamic - or
changing - the average also differs. If µ was computed for yet another set of
like parts, µ would not still be 1.240. Although the estimate may be close, it
will vary to some extent, sometimes a little or sometimes as much as about
1.5 over a very large number of lots. Therefore, it is obvious that
natural sources of process variation are critical for allocating and analyzing
tolerances.

It is necessary to allow for naturally occurring sources of manufacturing
variation such as drifts in µ due to tool wear. For example, assume that µ =
1.2415 inches and = .0005
inches for the lot number 2. C_{pk} would be 1.5 with C_{p}
remaining the same at 2.0. The long-term yield expectation is no longer
99.9999998, but rather 99.99966%. This adjusted yield estimate accounts for
the change in µ and reflects the recognition of variation over time. This
provides substantial insight into the effects of manufacturing and material
variation on product yield.