Cumulative Sum (Cusum) charts display cumulative sums of subgroup or individual measurements from a target value. Cusum charts are graphical and analytical tools for deciding whether a process is in a state of statistical control and for detecting a shift in the process mean.
JMP cusum charts can be one-sided, which detect a shift in one direction from a specified target mean, or two-sided to detect a shift in either direction. Both charts can be specified in terms of geometric parameters (h and k shown in Illustration of a V-Mask for a Two-Sided Cusum Chart); two-sided charts allow specification in terms of error probabilities α and β.
To interpret a two-sided Cusum chart, you compare the points with limits that compose a V-mask. A V-mask is formed by plotting V-shaped limits. The origin of a V-mask is the most recently plotted point, and the arms extended backward on the x-axis, as in Illustration of a V-Mask for a Two-Sided Cusum Chart. As data are collected, the cumulative sum sequence is updated and the origin is relocated at the newest point.
Shifts in the process mean are visually easy to detect on a cusum chart because they produce a change in the slope of the plotted points. The point where the slope changes is the point where the shift occurs. A condition is out-of-control if one or more of the points previously plotted crosses the upper or lower arm of the V-mask. Points crossing the lower arm signal an increasing process mean, and points crossing the upper arm signal a downward shift.
|
•
|
A Shewhart control chart plots points based on information from a single subgroup sample. In cusum charts, each point is based on information from all samples taken up to and including the current subgroup.
|
|
•
|
On a Shewhart control chart, horizontal control limits define whether a point signals an out-of-control condition. On a cusum chart, the limits can be either in the form of a V-mask or a horizontal decision interval.
|
|
•
|
The control limits on a Shewhart control chart are commonly specified as 3σ limits. On a cusum chart, the limits are determined from average run length, from error probabilities, or from an economic design.
|
A cusum chart is more efficient for detecting small shifts in the process mean. Lucas (1976) comments that a V-mask detects a 1σ shift about four times as fast as a Shewhart control chart.
When you choose Analyze > Quality And Process > Control Chart > Cusum, the Control Chart launch dialog appears, including appropriate options and specifications as shown here.
See Parameters in Statistical Control Charts for a description of K Sigma. The following items pertain only to cusum charts:
Requests a two-sided cusum scheme when checked. If it is not checked, a one-sided scheme is used and no V-mask appears. If an H value is specified, a decision interval is displayed.
Specifies that the cumulative sums be computed without standardizing the subgroup means or individual values so that the vertical axis of the cusum chart is scaled in the same units as the data.
H is the vertical distance h between the origin for the V-mask and the upper or lower arm of the V-mask for a two-sided scheme. You also enter a value for the increase in the lower V-mask per unit change on the subgroup axis (Illustration of a V-Mask for a Two-Sided Cusum Chart). For a one-sided scheme, H is the decision interval. Choose H as a multiple of the standard error.
Appends the panel shown here to the Control Charts launch dialog, which lets you enter the process variable specifications.
is the target mean (goal) for the process or population. The target mean must be scaled in the same units as the data.
specifies the absolute value of the smallest shift to be detected as a multiple of the process standard deviation or of the standard error. This depends on whether the shift is viewed as a shift in the population mean or as a shift in the sampling distribution of the subgroup mean, respectively. Delta is an alternative to the Shift option (described next). The relationship between Shift and Delta is given by
where δ represents Delta, Δ represents the shift, σ represents the process standard deviation, and n is the (common) subgroup sample size.
is the minimum value that you want to detect on either side of the target mean. You enter the shift value in the same units as the data, and you interpret it as a shift in the mean of the sampling distribution of the subgroup mean. You can choose either Shift or Delta.
specifies a known standard deviation, σ0, for the process standard deviation, σ. By default, the Control Chart platform estimates sigma from the data.
specifies an initial value for the cumulative sum, S0, for a one-sided cusum scheme (S0 is usually zero). Enter Head Start as a multiple of standard error.
displays the JMP color palette for you to select a color for the connect line when the Connect Points option is in effect.
shows or hides the V-mask based on the parameters (statistics) specified in the Cusum Control Charts launch window.
displays a Parameters table (Show Parameters) that summarizes the Cusum charting parameters.
To see an example of a two-sided cusum chart, open the Oil1 Cusum.jmp file from the Quality Control sample data folder. A machine fills 8-ounce cans of two-cycle engine oil additive. The filling process is believed to be in statistical control. The process is set so that the average weight of a filled can, μ0, is 8.10 ounces. Previous analysis shows that the standard deviation of fill weights, σ0, is 0.05 ounces.
Subgroup samples of four cans are selected and weighed every hour for twelve hours. Each observation in the Oil1 Cusum.jmp data table contains one value of weight along with its associated value of hour. The observations are sorted so that the values of hour are in increasing order. The Control Chart platform assumes that the data are sorted in increasing order.
A two-sided cusum chart is used to detect shifts of at least one standard deviation in either direction from the target mean of 8.10 ounces.
|
1.
|
Choose the Analyze > Quality And Process > Control Chart > CUSUM command.
|
|
2.
|
Click the Two Sided check box if it is not already checked.
|
|
3.
|
|
4.
|
|
5.
|
Click the H radio button and enter 2 into the text box.
|
|
6.
|
|
7.
|
Set Target to the average weight of 8.1.
|
|
8.
|
Enter a Delta value of 1.
|
|
9.
|
Set Sigma to the standard deviation of 0.05.
|
Control Chart(Sample Size( :hour), H(2), Chart Col( :weight, CUSUM(Two sided(1), Target(8.1), Delta(1), Sigma(0.05))));
Cusum Chart for Oil1 Cusum.jmp Data
You can interpret the chart by comparing the points with the V-mask whose right edge is centered at the most recent point (hour=12). Because none of the points cross the arms of the V-mask, there is no evidence that a shift in the process has occurred.
A shift or out-of-control condition is signaled at a time t if one or more of the points plotted up to the time t cross an arm of the V-mask. An upward shift is signaled by points crossing the lower arm, and a downward shift is signaled by points crossing the upper arm. The time at which the shift occurred corresponds to the time at which a distinct change is observed in the slope of the plotted points.
|
2.
|
Open the Oil2 Cusum.jmp sample data table.
|
|
3.
|
Copy rows 49 through 60. Paste them into the end of the Oil1 Cusum.jmp sample data table.
|
Notice that the graph in the CUSUM of weight report updates automatically. See Updated Cusum Chart for the Oil1 Cusum.jmp Data.
Updated Cusum Chart for the Oil1 Cusum.jmp Data
You can move the origin of the V-mask by using the grabber tool to click a point. The center line and V-mask adjust to reflect the process condition at that point.
Consider the data used in Example 1. Two-Sided Cusum Chart with V-mask, where the machine fills 8-ounce cans of engine oil. Consider also that the manufacturer is now concerned about significant over-filling in order to cut costs, and not so concerned about under-filling. A one-sided Cusum Chart can be used to identify data approaching or exceeding the side of interest. Anything 0.25 ounces beyond the mean of 8.1 is considered a problem. To do this example,
|
•
|
Open the Oil1 Cusum.jmp sample data table.
|
|
•
|
Choose the Analyze > Quality And Process > Control Chart > CUSUM command.
|
|
•
|
Deselect the Two Sided check box.
|
|
•
|
|
•
|
|
•
|
Click the H radio button and enter 0.25 into the text box.
|
|
•
|
|
•
|
Set Target to the average weight of 8.1.
|
|
•
|
Enter a Delta value of 1.
|
|
•
|
Set Sigma to the standard deviation 0.05.
|
Control Chart(Sample Size( :hour), H(0.25), Show Limits Legend(0), Chart Col( :weight, CUSUM(Two Sided(0), Target(8.1), Delta(1), Sigma(0.05))));
The resulting output should look like the picture in One-Sided Cusum Chart for the Oil1 Cusum.jmp Data.
One-Sided Cusum Chart for the Oil1 Cusum.jmp Data
Notice that the decision interval or horizontal line is set at the H-value entered (0.25). Also note that no V-mask appears with One-Sided Cusum charts.
The Show Parameters option in the Cusum chart popup menu shows the Parameters report in Show Parameters. The parameters report summarizes the charting parameters from the Known Statistics for CUSUM Chart tab on the Control Chart launch dialog. An additional chart option, Show ARL, adds the average run length (ARL) information to the report. The average run length is the expected number of samples taken before an out-of-control condition is signaled:
|
•
|
|
•
|
