U-charts (sometimes referred to as "rate" charts) deal with event counts when the area of opportunity is not constant during each period. The steps to follow for constructing a u-chart are the same as a c-chart, except that the control limits are computed for each individual quarter because the number of standard units varies.

P-chart

P-charts (sometimes referred to as "proportion" charts) are used to show the fraction nonconforming of a nonstandard sample size over a constant area of opportunity (e.g., each period of interest). The steps to follow for constructing a P-chart are the same as a c-chart, except that the control limits are computed for each time period because the sample size varies.

NP-chart

Like p-charts, np-charts are used to analyze nonconforming items over a constant area of opportunity; however, the np-chart focuses on the number of nonconforming items when the sample size is constant. The steps to follow for constructing an np-chart are the same as for a p-chart.

Use of Control Charts

Control charts serve to direct management attention toward special causes of variation in a system when they appear. Limit lines drawn on the charts provide guides for evaluation of performance. These lines (called control lines) indicate the dispersion of data on a statistical basis and indicate if an abnormal situation (e.g., the process is not in control or special causes are adversely influencing a process in control) has occurred.

In evaluating control charts, managers should look for the following indications:

• Outliers - Data that fall outside the control lines.

• Runs - Series of data points over or below the central line. A "run" of 7 consecutive points or 10 out of 11 points indicates an abnormality. Other approaches exist for identifying runs, such as detecting two of the last three data points beyond two standard deviations (2-sigma) and the more general CUSUM (cumulative sum discussed in Section 3.2.7) procedures, which involve adding up standardized deviations from the calculated mean to detect abnormalities (such as runs or trends) sooner.

• Trends - Continual rise or fall of data points. If seven data points rise or fall continuously, an abnormality is considered to exist.

• Periodicity - Data show the same pattern of change over time. Also known as a cyclical pattern.

  In summary, the two main uses for control charts are: (1) to monitor whether the system is stable and under control (to warn of changes), and (2) to substantiate results from changes introduced into the system (to confirm results).

In constructing control charts, individual or groups of data points may appear near or beyond the calculated control limit lines. Since these data appear to indicate that a system is or may not be in control (e.g., stable), additional evaluation may be needed to ascertain if the data in question are the result of common cause or special cause variation. If the data are clearly influenced by a one-time aberration (e.g., special cause), there could be a basis for excluding the number or estimating what the actual value should have been for the purpose of determining actual system control limits.

Treatment of "Rare" Events

For trending PI data using c- and u-charts, the average used to calculate control limits should be equal to or greater than five. Where the limited nature of the data does not support the use of control charts, the use of more sensitive trend tests may provide a better indication of actual trends. An example would be multinomial likelihood ratio tests that involve comparing the likelihood of postulated rates of data change (e.g., constant, increasing, or decreasing) assuming the data are generated by a multinomial distribution.

Scaling of Control Charts

The following general criteria should be applied to the depiction of trend data on control charts: (1) the scale should be set so that the chart can be quickly understood, and (2) the data together with the limit lines should span at least half of the vertical axis.

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