Constructing Charts for Nonconformities per Unit (u Charts)
The following notation is used in this section:
u | expected number of nonconformities
per unit produced by process |
ui | number of nonconformities
per unit in the i th subgroup.
In general, ui = ci/ni. |
ci | total number of nonconformities in the
i th subgroup |
ni | number of inspection units in the i th
subgroup |
![\bar{u}](images/ucheq5.gif) | average number of nonconformities per unit taken
across subgroups. The quantity is computed
as a weighted average:
![\bar{u} = \frac{n_{1}u_{1} + ... + n_{N}u_{N}}
{n_{1} + ... + n_{N}}
= \frac{c_{1} + ... + c_{N}}
{n_{1} + ... + n_{N}}](images/ucheq6.gif)
|
N | number of subgroups |
![\chi^2_{\nu}](images/ucheq7.gif) | has a central distribution with degrees of freedom |
Plotted Points
Each point on a u chart indicates the
number of nonconformities per unit (ui) in a subgroup.
For example, Figure 41.10 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.
Figure 41.10: 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 41.10, the number
of defective welds per unit in the third subgroup is u3=2/2.5=0.8.
A u chart plots the quantity
ui for the i th subgroup.
A c chart plots the quantity
ci for the i th subgroup
(see Chapter 33, "CCHART Statement").
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 u chart, the central line indicates an estimate of u,
which is computed as
by default.
If you specify a known value (u0) for u,
the central line indicates the value of u0.
You can compute the limits in the following ways:
- as a specified multiple (k) of the standard error of
ui 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 ui exceeds the limits
The lower and upper control limits, LCLU and UCLU, respectively, are
given by
![{LCLU} & = & {max}(\bar{u} -
k\sqrt{\bar{u}/n_i} \; ,0 ) \ {UCLU} & = & \bar{u} + k\sqrt{\bar{u}/n_i}](images/ucheq11.gif)
The limits vary with ni.
The upper probability limit UCLU for ui can be
determined using the fact that
![P\{u_{i} \gt {UCLU}\} & = 1 - P\{u_{i} \leq {UCLU} \} \ & = 1 - P\{c_{i} \leq n_...
...\} \ & = 1 - P\{\chi^2_{2(n_{i}(\!{{\scriptsize UCLU}}+1))} \geq 2n_{i}\bar{u}\}](images/ucheq12.gif)
The limit UCLU is then calculated by setting
![1 - P\{\chi^2_{2(n_{i}(\!{{\scriptsize UCLU}}+1))} \geq 2n_{i}\bar{u}\} = \alpha/2](images/ucheq13.gif)
and solving for UCLU.
Likewise, the lower probability limit LCLC for ui can be
determined using the fact that
![P\{u_{i} \lt {LCLC}\} & = P\{c_{i} \lt n_{i}{LCLU} \} \ & = P\{\chi^2_{2(n_i(\!{{\scriptsize LCLC}}+1)} \gt 2n_{i}\bar{u}\}](images/ucheq14.gif)
The limit LCLC is then calculated by setting
![P\{\chi^2_{2(n_i(\!{{\scriptsize LCLC}}+1)} \gt 2n_{i}\bar{u}\} = \alpha/2](images/ucheq15.gif)
and solving for LCLC. For more information, refer to
Johnson, Kotz, and Kemp (1992). This assumes that the process is in
statistical control and that ci has a Poisson distribution.
Note that the probability limits vary with ni
and are asymmetric around the central line.
If a standard value u0 is available for u, replace
with u0 in the formulas for the control limits.
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 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.