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CUSUM¶
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QXL Stat Tools Tab > Control Charts > Time Weighted Charts > CUSUM
A CUSUM chart does not plot your measurements. It plots two running sums of how far those measurements have drifted from a center value, one sum that accumulates upward drift and one that accumulates downward drift. Both sums are held at zero until the drift is large enough to overcome an allowance, so a process sitting on center leaves both of them flat along the zero line, and a small shift that persists makes one of them climb.
The two sums are drawn as one chart, not as two stacked panels. The chart is titled CUSUM Chart and carries two series named C+ and C-, sharing a center line at zero. The upper sum rides toward a limit above the center, the lower sum toward the mirror image of it below.
The dialog is titled CUSUM Chart and the worksheet it creates is named CUSUM. Every option in that dialog is described on the Options page.
When to use it¶
- Your measurements are continuous, and you have either one measurement per point in time or several. Unlike the subgroup charts, this chart accepts a subgroup size of 1, so it serves individual readings and subgroups from the same dialog.
- You want a chart whose signal depends on the whole history of the deviations rather than on where the latest point landed. A Shewhart chart judges each point on its own; each CUSUM point carries everything that came before it.
- You want the sensitivity of the chart set by two numbers you choose, the allowance and the decision interval, rather than by a number of standard deviations. The sigma multiplier box is not on this dialog at all.
If you want the plotted value to stay in the units of your measurement, use an Xbar-R, Xbar-S, or I-MR chart. The other time weighted chart, EWMA, also carries history forward, but it plots a smoothed measurement against limits that widen over the first few points, while CUSUM plots an accumulated deviation against a limit that does not widen.
How it works¶
Quantum XL first reduces your data to one value per plotted point: the observation itself when the subgroup size is 1, and the subgroup mean when the data are subgrouped.
Each of those values is compared with a center. The center is the Target (optional): value if you entered one, and the mean of the section if you left it blank. On a chart split into phases, a target you entered applies to every phase, while a blank target lets each phase center on its own mean.
The deviation from that center is then reduced by an allowance before anything is accumulated, and the running total is never allowed to fall below zero. That is what keeps ordinary variation from building up: only deviations bigger than the allowance push a sum away from zero, and a run of deviations in the other direction pulls it straight back to zero rather than into negative territory. The allowance is the value in Allowance (k): multiplied by the standard deviation of the plotted value at that point.
A point signals when its accumulated sum has grown past the decision interval, which is the value in Decision interval (h): multiplied by that same per-point standard deviation. Because both are multiples of the same quantity, raising the allowance makes the chart slower to react and raising the decision interval makes it harder to trigger.
The lower sum is drawn negated. The accumulation itself is a non-negative magnitude, exactly like the upper one, but it is plotted as a negative number so that downward drift appears below the center line where a reader expects it. What you see below the axis is minus the lower sum, and the lower limit is the negative of the decision interval.
The limits do not widen. The decision interval is a fixed multiple of the per-point standard deviation, so on a chart with a constant subgroup size the two limits are straight horizontal lines from the first point to the last. They step only where the subgroup size changes, because the per-point standard deviation divides by the square root of the subgroup size. This is the main difference from an EWMA chart, whose limits start narrow and widen point by point toward an asymptote. Readers group the two charts together and assume they behave alike; they do not.
Two settings change how the accumulation starts. Use fast initial response (FIR) seeds each sum with a head start instead of zero, so a process that is already off center is caught sooner. Reset accumulation after a signal starts the sum again after a point has signalled, instead of leaving it above the decision interval where every following point signals too.
Two more settings change what the limits describe rather than how they are computed. Estimate Limits From computes the center and the standard deviation from the first few points only. Split Control Limits divides the chart into phases, and the accumulation restarts at every phase boundary.
The exact formulas are on the Math Details page.
Output¶
The analysis writes one worksheet per chart, named CUSUM. Under GroupBy the name becomes CUSUM - {GroupName}. On the worksheet, from the top:
- A User Input block in the first two columns, holding the analysis title CUSUM Chart, the measurement columns, the X-Label column, the split column and the GroupBy columns where you used them, and whether the calculation type was Excel or GroupBy. Modify Existing Analysis reads it back.
- One chart titled CUSUM Chart, with the X axis titled Subgroup and the Y axis titled Cumulative Sum. Its legend sits on the right and names the series that were drawn: C+, C-, UCL, LCL, and Center. Reference lines and phase dividers appear here too if you asked for them, and points that signalled are drawn in a different color.
- A summary table to the right of the chart. Its Summary block has one section per sum, headed Upper CUSUM (C+) and Lower CUSUM (C-), and each section reports the center, the one limit that applies to that side, the count of out-of-control points, and the count of points you marked as outliers. A transformation adds rows naming the type and its parameters; a split chart adds a per-phase breakdown.
- No capability block and no histogram. This chart has no Capability tab, and the capability calculation and the histogram are switched off for it, so nothing you enter anywhere can add that block to a CUSUM worksheet.
When the analysis finishes, the Control Chart task pane opens on the right. See Options for what it can and cannot change on this chart.