Table 1

Chart formulae and settings used in chart comparisons

Shewhart p-chartg-chartEWMACUSUM
Chart statistic Embedded Image
where pj is the proportion in the jth reporting period, nj is the volume and xij is the ith observation in the jth period
Embedded Image
where sk is the number of observations between events, ik is the sequence number of the kth event
Embedded Image
where si is the ith value of the trace, xi is the ith observation and λ is the weighting parameter
Embedded Image
where
Embedded Image
Embedded Image
where si is the ith value of the trace, xi is the ith observation, p is the baseline event rate and OR is the OR corresponding to the minimum shift size the chart should detect
Target line Baseline event rate, pBaseline mean number between events, Embedded Image or (1−p)/pBaseline event rate, pNo target line. Trace moves between zero and the control limit
Control limits Embedded Image
where L is the number of SEs (sigma) from the target line (see main text for runs rules)
Embedded Image Embedded Image Absolute value for limit
Chart settings pn per periodOnly limits are alteredλ=0.01 OR=2 for an increase and 0.5 for a decrease
<0.02
0.02 to <0.1
0.1 to 0.5
200
100
 50
In-control ARLs* For L=3 with runs rules, n as above:To detect a decrease:For λ=0.01:For OR=2:
pIn-control ARLpLIn-control ARLpLIn-control ARLpLimitIn-control ARL
0.01
0.10
0.50
14 400
 9 800
 8 750
0.01
0.10
0.50
4.0
6.0
8.0
14 800
10 500
 8 200
0.01
0.10
0.50
4.3
3.3
2.9
15 250
10 050
 8 460
0.01
0.10
0.50
3.5
3.5
6.0
14 290
 9 300
 8 900
  • *In-control ARLs are given for specified values of control limits. Control limits were more finely adjusted to achieve in-control ARLs for comparisons in table 2.

  • ARL, average run length; CUSUM, cumulative sum; EWMA, exponentially weighted moving average.