Constructing Charts for Nonconformities per Unit (u Charts)

UCHART Statement

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 i^th subgroup. In general, u_i = c_i/n_i.

c_i total number of nonconformities in the i^th subgroup

n_i number of inspection units in the i^th subgroup

$\bar{u}$ average number of nonconformities per unit taken across subgroups. The quantity $\bar{u}$ is computed as a weighted average:
$\bar{u} = \frac{n_{1}u_{1} + ... + n_{N}u_{N}} {n_{1} + ... + n_{N}} = \frac{c_{1} + ... + c_{N}} {n_{1} + ... + n_{N}}$

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 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 n_i, 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 c_i. The number of nonconformities per unit in the i^th subgroup is denoted by u_i=c_i/n_i. 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 u_i for the i^th subgroup. A c chart plots the quantity c_i 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 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 (u₀) for u, the central line indicates the value of u₀.

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

${LCLU} & = & {max}(\bar{u} - k\sqrt{\bar{u}/n_i} \; ,0 ) \ {UCLU} & = & \bar{u} + k\sqrt{\bar{u}/n_i}$

The limits vary with n_i.

The upper probability limit UCLU for u_i can be determined using the fact that

$P\{u_{i} \gt {UCLU}\} & = 1 - P\{u_{i} \leq {UCLU} \} \ & = 1 - P\{c_{i} \leq n_... ...\} \ & = 1 - P\{\chi^2_{2(n_{i}(\!{{\scriptsize UCLU}}+1))} \geq 2n_{i}\bar{u}\}$

The limit UCLU is then calculated by setting

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

and solving for UCLU.

Likewise, the lower probability limit LCLC for u_i can be determined using the fact that

$P\{u_{i} \lt {LCLC}\} & = P\{c_{i} \lt n_{i}{LCLU} \} \ & = P\{\chi^2_{2(n_i(\!{{\scriptsize LCLC}}+1)} \gt 2n_{i}\bar{u}\}$

The limit LCLC is then calculated by setting

$P\{\chi^2_{2(n_i(\!{{\scriptsize LCLC}}+1)} \gt 2n_{i}\bar{u}\} = \alpha/2$

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 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 u₀ is available for u, replace $\bar{u}$ 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 $\alpha$ 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 u₀ with the U0= option or with the variable _U_ in a LIMITS= data set.

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