Constructing Charts for Proportion Nonconforming (p Charts)
The following notation is used in this section:
|p||expected proportion of nonconforming items
produced by the process|
|pi||proportion of nonconforming items in the
i th subgroup|
|Xi||number of nonconforming items in the
i th subgroup|
|ni||number of items in the i th subgroup|
|average proportion of nonconforming items taken
|N||number of subgroups|
|incomplete beta function:
for 0<T<1, , and , where is the gamma function
Each point on a p chart represents the observed proportion
(pi=Xi/ni) of nonconforming items in a subgroup. For example,
suppose the second subgroup
(see Figure 38.10) contains 16 items,
of which two are nonconforming.
The point plotted for the second
subgroup is p2 = 2/16=0.125.
Figure 38.10: Proportions Versus Counts
Note that an np chart displays the number
(count) of nonconforming items Xi.
You can use the NPCHART statement to create np charts;
see Chapter 37, "NPCHART Statement."
By default, the central line on a p chart
indicates an estimate of
p that is computed as .If you specify a known value (p0) for p, the
central line indicates the value of
You can compute the limits in the following ways:
The lower and upper control limits, LCL and UCL, respectively,
are computed as
- as a specified multiple (k) of the standard error of pi
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 pi exceeds the limits
A lower probability limit for pi can be determined using the fact
Refer to Johnson, Kotz, and Kemp (1992). This assumes that the process is in
statistical control and that Xi is binomially distributed. The
lower probability limit LCL is then calculated by setting
and solving for LCL.
Similarly, the upper probability limit for pi can be determined
using the fact that
The upper probability limit UCL is then calculated by setting
and solving for UCL.
The probability limits are asymmetric around the central line.
Note that both the control limits and probability
limits vary with ni.
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 p0 with the P0= option
or with the variable _P_ in a LIMITS= data set.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.