# Methods and formulas for G Chart

Select the method or formula of your choice.

## Plotted points

If your data are recorded as the date of each event, each plotted point, xi, represents the number of days between successive events. If your data are recorded as the number of opportunities between events, each plotted point represents the number of opportunities between successive events.

## Center line and control limits

### Center line (CL)

The center line is the 50th percentile of the distribution. The center line equals G2 – 1.

###### Note

1 is subtracted because Minitab uses the "number until" definition of the geometric distribution in its calculations but plots the "number between" values on the G chart..

G2 equals INVCDF (0.5) for a geometric distribution with parameter p.

Minitab gives 2 values, G2a and G2b (G2a = G2b – 1), with 2 probabilities p2a and p2b (p2a < p2b). Using simple linear interpolation, G2 = G2a + (0.5 – p2a) / (p2b – p2a).

### Lower control limit (LCL)

LCL = G1 – 1

G1 equals INVCDF (0.00135) for a geometric distribution with parameter p.

Minitab gives 2 values, G1a and G1b (G1a = G1b – 1), with 2 probabilities p1a and p1b (p1a < p1b). Using simple linear interpolation, G1 = G1a + (.00135 – p1a) / (p1b – p1a).

### Upper control limit (UCL)

UCL = G3 – 1

G3 equals INVCDF (0.99865) for a geometric distribution with parameter p.

Minitab gives 2 values, G3a and G3b (G3a = G3b – 1), with 2 probabilities p3a and p3b (p3a < p3b). Using simple linear interpolation, we get G3 = G3a + (0.99865 – p3a) / (p3b – p3a).

### Notation

TermDescription
Nnumber of data values used in the calculations (If data are dates, subtract 1 because Minitab plots the differences.) average of the plotted points
p

## Tests for special causes, including Benneyan test

### Tests 1−4

Test 1 is based on the geometric distribution. Tests 2, 3 and 4 are identical to the tests used in the attribute charts.

If K = 3, the G1 and G3 values used for the control limits define Test 1 failures. If K is less than or greater than 3, plotted points below G1' fail Test 1 and plotted points above G3' fail Test 1.
• G1 = INVCDF (0.00135) for a geometric distribution with parameter p
• G3 = INVCDF (0.99865) for a geometric distribution with parameter p; average of the plotted points
• G1' = INVCDF (p1') for a geometric distribution with parameter p
• G3' = INVCDF (p2') for a geometric distribution with parameter p
• p1' = CDF (–K) for a normal distribution with mean = 0 and standard deviation = 1
• p2' = CDF (K) for a normal distribution with mean = 0 and standard deviation = 1

### Benneyan test

The Benneyan test counts the number of consecutive plotted points equal to the lower control limit using the following formula to generate a signal:

Minitab rounds cp up to the next integer and uses that value as the number of consecutive points equal to the lower control limit that are required to produce a signal.

See Benneyan1 for more information on the Benneyan test.

### Notation

TermDescription
CDF()CDF for a normal distribution with mean 0, standard deviation 1
kparameter for Test 1. The default is 3.
1 J. C. Benneyan (2001). "Performance of Number-Between g-Type Statistical Control Charts for Monitoring Adverse Events", Health Care Management Science, 4, 319−336.
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