The Statit Help includes the details for the
formulas used by the quality control charts.
These formulas are useful when checking the
results or for comparison to other references
for these formulas. The formulas may be difficult
to understand and apply to real world situations.
This article provides another view into these
formulas, one that may help with your understanding.
This article discusses the c Chart. This chart
is one of the simplest attribute charts, and
this discussion provides a good basis for understanding
the other charts. This article should be used
along with the Statit Help for c Chart and for
QC Formulas.
c Chart
The c Chart is used to identify non-random
variation in the number of nonconformities or
defects in similar groups. For example: Say
you have 6 groups of 10 M&Ms® each.
In each group, some of the candy pieces are
missing the printed “m” or it is partially
missing. In a truly random process, the number
of defects in each group would be about the
same. In a nonrandom process, some groups will
have many more or many fewer defects than other
groups.
The mathematical formulas that the c Chart
uses to identify these groups are shown in the
Statit Help. Let’s focus on finding those
groups which have a number of defects much higher
than other groups. That is, let us look at the
upper control limit. You should realize that
the same arguments will apply for groups with
many fewer defects than other groups, that is,
when you look at the lower control limit.
The center line is the first line on this chart
to discuss, as the other values use the center
line value. Statit Help tells you:
That is, the center line ( CL ) is
equal to the average of the number of defects
in each group (
).
If this average were 2, we might be able to
say that the random variation in the process
that makes these candies results in an average
of 2 out of 10 with a defect in the printing
of the “m.”
To identify a process that does not have random
variation, we look for groups with the number
of defects greater than 3 standard deviations
away from the center line. That is, greater
than 3-sigma from the center line. The values
we want are above the upper control limit. Statit
Help provides this formula for the upper control
limit (UCL):
The Help actually uses “nsig”
rather than 3 in this formula, but nsig
= 3 when we are looking for 3 sigma or 2 if
we are working with 2 sigma. So 
,
which is added to the value of the CL to give
the UCL. If CL = 2, 3 sigma = 3 * 1.4142 = 4.2426,
and UCL = 6.2426.
What Does This Tell You?
With a center line value of two, the 3 sigma
value of 4.2 seems very large. One reason for
this is that the size of the group is not accounted
for in this calculation. Thus the help file
reminds you that the number of defects may be
larger than the number of pieces. If you counted
not only the missing “m” but also
imperfections in the “m,” the count
of defects could easily exceed the number of
candies in the group. (This is actually true
to the u chart as well.)
In the example above, I stated that there were
50 candies in each group. This could also have
been 100 without any affect on the center line
or 3 sigma values. But because the number of
pieces in each group, also called the area of
opportunity, is not considered when determining
the center line or 3 sigma, this must be the
same for all groups. If this rule is not followed,
it is incorrect to compare the groups using
these numbers. The c chart assumes that the
subgroups are of equal size.
Why is 3 sigma used? This assumes a Poisson
random distribution of values resulting from
a normal process. When values fall beyond 3
sigma, the values are beyond this distribution
and may indicate a process which can be controlled.
u Chart
In the example for the c Chart, the groups
all had the same number of candy pieces. If
the number of pieces differ between the groups,
the u Chart should be used. The u Chart takes
into account the different sized groups. Let’s
explore how this is done.
Consider the center line. Statit Help tells
us that the formula is:
That is, the center line is the total number
of defects divided by the total number of candy
pieces in all of the groups. Already a big difference
from the c Chart, as now the size of the groups
is involved in the calculation.
Next, let us look at the formula for the upper
control limit (UCL):
The i subscript indicates values specific for
each group. The UCL changes for each group because
the number if pieces in each group changes.
In essence, the contribution of this group to
the value of CL is accounted for in this calculation.
What Does This Tell You?
The plotted points are the number of defects
divided by the number of pieces in the group.
This provides a better comparison of the groups
as it incorporates the subgroup size information
into the calculations.. Think of this as a fair
way to compare the number of defects in a full
bag of candies to the number of defects in a
half of a bag of candies.
The UCL for groups with more pieces is closer
to the CL than it is for groups with fewer pieces.
This is expected because the estimation of the
sigma is better with larger subgroups; it does
not go as far from the CL with the larger sample
size. That is, with 5 pieces a defect has a
larger influence on what is known about the
process than one defect in 100 pieces.
All groups must be from the same process for
this to be a valid chart. The results seen for
the last point are affected by every other point,
because every group contributes to the CL and
UCL.
