Constructing Charts for Number Nonconforming (np 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
|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 an np chart represents the observed number (Xi)
of nonconforming items in a subgroup. For
example, suppose the first subgroup
(see Figure 37.9) contains
12 items, of which three are nonconforming. The point plotted for the
first subgroup is X1 = 3.
Figure 37.9: Proportions Versus Counts
Note that a p chart displays the proportion of nonconforming items
You can use the PCHART statement to create p charts;
see Chapter 38, "PCHART Statement."
By default, the central line on an np chart indicates an estimate for
nip, which is computed as .If you specify a known value (p0) for p, the
central line indicates the value of
nip0. Note that the central line varies with ni.
You can compute the limits in the following ways:
- as a specified multiple (k) of the standard error of
Xi 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 Xi
exceeds the limits
The lower and upper control limits, LCL and UCL respectively,
are computed as
A lower probability limit for Xi 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 Xi 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 about
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= data set.
- 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 the LIMITS= data set.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.