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 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 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 average number of nonconformities per unit taken across subgroups. The quantity is computed as a weighted average: N number of subgroups 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.

### Central Line

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.

### Control Limits

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

The limits vary with ni.

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

The limit UCLU is then calculated by setting

and solving for UCLU.

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

The limit LCLC is then calculated by setting

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.

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