Methods and formulas for G Chart

Select the method or formula of your choice.

In This Topic

Plotted points
Center line and control limits
Tests for special causes, including Benneyan test

Plotted points

If your data are recorded as the date of each event, each plotted point, x_i, 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 50^th 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, G2_a and G2_b (G2_a = G2_b – 1), with 2 probabilities p2_a and p2_b (p2_a < p2_b). Using simple linear interpolation, G2 = G2_a + (0.5 – p2_a) / (p2_b – p2_a).

Lower control limit (LCL)

LCL = G1 – 1

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

Minitab gives 2 values, G1_a and G1_b (G1_a = G1_b – 1), with 2 probabilities p1_a and p1_b (p1_a < p1_b). Using simple linear interpolation, G1 = G1_a + (.00135 – p1_a) / (p1_b – p1_a).

Upper control limit (UCL)

UCL = G3 – 1

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

Minitab gives 2 values, G3_a and G3_b (G3_a = G3_b – 1), with 2 probabilities p3_a and p3_b (p3_a < p3_b). Using simple linear interpolation, we get G3 = G3_a + (0.99865 – p3_a) / (p3_b – p3_a).

Notation

Term	Description
N	number 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

Term	Description
CDF()	CDF for a normal distribution with mean 0, standard deviation 1
k	parameter 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.