Select the method or formula of your choice.

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.

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

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}).

*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}).

*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}).

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 | |

Event probability (p) |

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

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 Benneyan^{1} for more information on the Benneyan test.

Term | Description |
---|---|

CDF() | CDF for a normal distribution with mean 0, standard deviation 1 |

k | parameter for Test 1. The default is 3. |