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
$u_{i}$	number of nonconformities per unit in the ith subgroup. In general, $u_ i = c_ i/n_ i$ .
$c_{i}$	total number of nonconformities in the ith subgroup
$n_{i}$	number of inspection units in the ith subgroup
$\bar{u}$	average number of nonconformities per unit taken across subgroups. The quantity $\bar{u}$ is computed as a weighted average: $\bar{u} = \frac{n_{1}u_{1} + \cdots + n_{N}u_{N}}{n_{1} + \cdots + n_{N}} = \frac{c_{1} + \cdots + c_{N}}{n_{1} + \cdots + n_{N}}$
N	number of subgroups
$\chi ^{2}_{\nu }$	has a central $\chi ^{2}$ distribution with $\nu$ degrees of freedom

Plotted Points

Each point on a u chart indicates the number of nonconformities per unit ( $u_{i}$ ) in a subgroup. For example, Figure 17.94 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 ith subgroup is denoted by $n_ i$ , which is the subgroup sample size.

Figure 17.94: Terminology for c Charts and u Charts

The number of nonconformities in the ith subgroup is denoted by $c_{i}$ . The number of nonconformities per unit in the ith subgroup is denoted by $u_{i}=c_{i}/n_{i}$ . In Figure 17.94, the number of defective welds per unit in the third subgroup is $u_{3}=2/2.5=0.8$ .

A u chart plots the quantity $u_{i}$ for the ith subgroup. A c chart plots the quantity $c_{i}$ for the ith subgroup (see CCHART Statement: SHEWHART Procedure). An advantage of a u chart is that the value of the central line at the ith subgroup does not depend on $n_{i}$ . This is not the case for a c chart, and consequently, a u chart is often preferred when the number of units $n_{i}$ is not constant across subgroups.

Central Line

On a u chart, the central line indicates an estimate of u, which is computed as $\bar{u}$ by default. If you specify a known value ( $u_{0}$ ) for u, the central line indicates the value of $u_{0}$ .

Control Limits

You can compute the limits in the following ways:

as a specified multiple (k) of the standard error of $u_{i}$ above and below the central line. The default limits are computed with k = 3 (these are referred to as $3\sigma$ limits).
as probability limits defined in terms of $\alpha$ , a specified probability that $u_{i}$ exceeds the limits

The lower and upper control limits, LCLU and UCLU, respectively, are given by

$\begin{eqnarray*} \mbox{LCLU} & = & \mbox{max}\left(\bar{u} - k\sqrt {\bar{u}/n_ i} \; ,0 \right) \\ \mbox{UCLU} & = & \bar{u} + k\sqrt {\bar{u}/n_ i} \end{eqnarray*}$

The limits vary with $n_{i}$ .

The upper probability limit UCLU for $u_{i}$ can be determined using the fact that

$\begin{array}{ll} P\{ u_{i} > \mbox{UCLU}\} & = 1 - P\{ u_{i} \leq \mbox{UCLU} \} \\ & = 1 - P\{ c_{i} \leq n_{i}\mbox{UCLU} \} \\ & = 1 - P\{ \chi ^{2}_{2(n_{i} \times {\mbox{\scriptsize UCLU}}+1)} \geq 2n_{i}\bar{u}\} \end{array}$

The limit UCLU is then calculated by setting

$1 - P\{ \chi ^{2}_{2(n_{i} \times {\mbox{\scriptsize UCLU}}+1)} \geq 2n_{i}\bar{u}\} = \alpha /2$

and solving for UCLU.

Likewise, the lower probability limit LCLU for $u_{i}$ can be determined using the fact that

$\begin{array}{ll} P\{ u_{i} < \mbox{LCLU}\} & = P\{ c_{i} < n_{i}\mbox{LCLU} \} \\ & = P\{ \chi ^{2}_{2(n_ i \times {\mbox{\scriptsize LCLU}}+1)} > 2n_{i}\bar{u}\} \end{array}$

The limit LCLU is then calculated by setting

$P\{ \chi ^{2}_{2(n_ i \times {\mbox{\scriptsize LCLU}}+1)} > 2n_{i}\bar{u}\} = \alpha /2$

and solving for LCLU. For more information, refer to Johnson, Kotz, and Kemp (1992). This assumes that the process is in statistical control and that $c_{i}$ has a Poisson distribution. Note that the probability limits vary with $n_{i}$ and are asymmetric around the central line. If a standard value $u_0$ is available for u, replace $\bar{u}$ with $u_0$ 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 $\alpha$ with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set.
Specify a constant nominal sample size $n_ i \equiv n$ for the control limits with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.
Specify $u_0$ with the U0= option or with the variable _U_ in a LIMITS= data set.

UCHART Statement: SHEWHART Procedure

Constructing Charts for Nonconformities per Unit (u Charts)

Plotted Points

Central Line

Control Limits