CUSUM ARL

CUSUM ARL (LET)

Let Subcommand

Compute the average run length (ARL) for the cumulative sum control chart.

The average run length for the cumulative sum control chart is the average number of observations that are entered before the system is declared out of control.

This command assumes the following parameters have been defined:

• S0 = start-up value for the cumulative sum. This is usually zero. However, it can be set to a positive initial value for a fast initial response (FIR) cusum chart.
• H = defines the value which signals that the cusum is "out of control". A value of 5 is a common choice.
• K = the value of k is set to one half of the smallest shift in location (in standard deviation units) that you want to detect. A common choice is a 1-sigma shift, that is k = 0.5.

The cusum arl is defined relative to DELTA. DELTA is the difference between the target value of the process and the true value of the process. DELTA is usually defined to be a sequence of values. For example,

LET DELTA = SEQUENCE 0 0.01 0.5

LET <y> = CUSUM ARL <delta>             <SUBSET/EXCEPT/FOR qualification>
where <delta> is a number, parameter, or variable that contains the delta values;
            <y> is a parameter or variable (depending on <delta>) where the cusum arl is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used for a 2-sided cusum chart.

LET <y> = ONE-SIDED CUSUM ARL <delta>             <SUBSET/EXCEPT/FOR qualification>
where <delta> is a number, parameter, or variable that contains the delta values;
            <y> is a parameter or variable (depending on <delta>) where the cusum arl is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used for a 1-sided cusum chart.

LET Y = CUSUM ARL DELTA
LET Y = ONE-SIDED CUSUM ARL DELTA

The LET command is used to specify the necessary parameters before entering the CUSUM ARL command. See the Program example below. Note that these parameters names do not have synonyms (i.e., you have to define K, H, and S0).

Dataplot provides macros for generating ARL plots for the XBAR, C, and P control charts. In addition, the OC curve is closely related to the ARL. The OC curve is a plot of the Type II, or Beta, error. The Beta error is the probability of incorrectly accepting the hypothesis of statistical control. That is, the OC curve plots the probability of the Type II error while the ARL plots the average number of runs until a Type II error occurs. Dataplot provides macros for generating the OC curves for the XBAR, C, and P control charts.

Enter the following commands for details on how to use these macros:

LIST ARL_XBAR.DP
LIST ARL_C.DP
LIST ARL_P.DP
LIST OC_XBAR.DP
LIST OC_C.DP
LIST OC_P.DP

Dataplot uses Applied Statistics algorithm 258 to compute the cusum arl. Specifically, it uses the ARL1 and ARL2 routines.

None

None

CONTROL CHART	= Generate a control chart.
PLOT	= Generate a data or function plot.

Algorithm 258, Applied Statistics, Vol. 39, No. 3, 1990.

Quality Control

1999/5

LET K = 0.5
LET H = 5
LET S0 = 0
LET DELTA = SEQUENCE 0 0.01 1.0
LET Y = CUSUM ARL DELTA
Y1LABEL AVERAGE RUN LENGTH
X1LABEL DELTA
PLOT Y DELTA