Constructing Charts for Numbers of Nonconformities (c Charts)
The following notation is used in this section:
|u||expected number of nonconformities
per unit produced by the process|
|ui||number of nonconformities
per unit in the i th subgroup|
|ci||total number of nonconformities in the
i th subgroup|
|ni||number of inspection units in the i th subgroup.
Typically, ni = 1 and ui=ci for c charts.
In general, ui=ci/ni.|
|average number of nonconformities per unit taken
across subgroups. The quantity is computed
as a weighted average:
|N||number of subgroups|
|has a central distribution with degrees of freedom|
Each point on a c chart represents the
total number of nonconformities (ci) in a subgroup.
For example, Figure 33.10 displays three sections of pipeline
that are inspected for defective welds (indicated by an
X). Each section represents a subgroup composed
of a number of inspection units, which are 1000-foot-long
sections. The number of units in the i th
subgroup is denoted by ni, which is
the subgroup sample size. The value of ni can be fractional;
Figure 33.10 shows n3=2.5 units in the third subgroup.
Figure 33.10: Terminology for c Charts and u Charts
The number of nonconformities in the i th
subgroup is denoted by ci. The number of nonconformities
per unit in the i th subgroup is
denoted by ui=ci/ni. In Figure 33.10, the number
of welds per inspection unit in the third subgroup is u3=2/2.5=0.8.
A u chart created with the UCHART statement plots the quantity
ui for the i th subgroup (see Chapter 41).
An advantage of a u chart is that the value of the central
line at the i th subgroup does not depend
on ni. This is not the case for a c chart, and consequently,
a u chart is often preferred when the number of units ni
is not constant across subgroups.
On a c chart, the central line indicates an estimate
for niu, which is computed as .If you specify a known value (u0) for u,
the central line indicates the value of niu0.
Note that the central line varies with subgroup sample size ni.
When ni=1 for all subgroups, the central line has the constant
You can compute the limits in the following ways:
- as a specified multiple (k) of the standard error of ci
above and below the central line.
The default limits are computed
with k=3 (these are referred to as limits).
- as probability limits defined in terms of , a specified probability that ci
exceeds the limits
The lower and upper control limits, LCLC and UCLC respectively, are
The upper and lower control limits vary with the number of inspection
units per subgroup ni. If ni=1 for all subgroups, the control limits
have constant values.
An upper probability limit UCLC for ci can be determined using the
The upper probability limit UCLC is then calculated by setting
and solving for UCLC.
A similar approach is used to calculate the lower probability limit
LCLC, using the fact that
The lower probability limit LCLC is then calculated by setting
and solving for LCLC. This assumes that the process is in
statistical control and that ci has a Poisson distribution.
For more information, refer to Johnson, Kotz, and Kemp (1992).
Note that the probability limits vary with the number of inspection
units per subgroup (ni) and are asymmetric about the central line.
If a standard value u0 is available
for u, replace with u0 in the formulas for the
You can specify parameters for the limits as follows:
- Specify k with the SIGMAS= option
or with the variable _SIGMAS_ in a LIMITS=
- Specify with the ALPHA= option
or with the variable _ALPHA_ in a LIMITS= data set.
- Specify a constant nominal sample size for the
control limits with the LIMITN= option or with the
variable _LIMITN_ in a LIMITS= data set.
- Specify u0 with the U0= option
or with the variable _U_ in a LIMITS= data set.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.