Home / Statistical Tools / Control Charts / CUSUM / CUSUM How-To
CUSUM How-To¶
This walkthrough builds a complete CUSUM chart from thirty hourly readings. Four things about this chart are unlike the Shewhart charts, and the data is chosen so that all four are visible in one run:
- Your measurements are not on the chart. What is plotted is two running sums of how far they have drifted from a center value.
- There is no band that widens. The two limits are straight horizontal lines. This is the opposite of the EWMA chart, which readers often expect it to resemble.
- The lower series is drawn negated, so a plotted value of -1.9 means the lower sum has reached 1.9.
- Once a sum crosses its limit it stays across, because Reset accumulation after a signal is off until you turn it on. One drift therefore produces a run of signalled points, not one.
The data is on this page rather than in a file to download. Press Copy for Excel, then paste it into a blank worksheet.
The data¶
Thirty consecutive hours from a coating line, one thickness reading per hour, in microns. From reading 23 onward the coating runs thicker and stays there.
| Coating Thickness |
|---|
| 23.69 |
| 24.76 |
| 26.18 |
| 23.22 |
| 25.70 |
| 24.36 |
| 26.00 |
| 24.67 |
| 24.85 |
| 25.42 |
| 24.42 |
| 24.96 |
| 24.79 |
| 26.36 |
| 25.28 |
| 23.35 |
| 25.30 |
| 25.18 |
| 24.96 |
| 26.27 |
| 25.95 |
| 23.48 |
| 28.42 |
| 26.66 |
| 26.66 |
| 26.57 |
| 26.93 |
| 27.83 |
| 25.87 |
| 26.90 |
The first twenty-two readings average 24.96. The last eight average about 27.0. No single reading looks impossible: 26.36 appears at hour 14 while the line is behaving normally, and several of the later readings are no larger than that.
Do not leave a gap in this column
A CUSUM chart refuses to run if any point has no usable value, and stops with the message "Time-weighted charts require a value at every time step - fill or remove the missing value(s)." The Shewhart charts plot a gap as a gap; this one will not start. Delete a row rather than blanking a cell.
Steps¶
-
Put the data in Excel
Press Copy for Excel above the table. In Excel, open a blank worksheet, click cell A1, and press Ctrl+V. The header lands in A1 and the thirty readings in A2:A31.
-
Start the chart
From the Excel ribbon: QXL Stat Tools Tab > Control Charts > Time Weighted Charts > CUSUM.
-
Give it the cells
In the Data Selection window set Selected Range: to A1:A31. Leave Data in Columns selected and First Row/Column is Header checked. Press Next >.
-
Check the measurement column and leave the subgroup size at 1
The CUSUM Chart dialog opens on the Data tab. Check Coating Thickness. Leave the Subgroups group on Constant size: with
n = 1, which is one reading per point.With
n = 1the estimators offered on the next tab change to the individuals set: Average moving range (selected when the group appears), Median moving range, and Square root of MSSD. Leave the default. -
Look at the CUSUM Parameters group, and change nothing in it
On the Limits & Estimation tab, the CUSUM Parameters group holds every setting that is specific to this chart. This group exists on no other chart. Read it before running, because these five values are what the rest of this page describes:
Control Leave it at Allowance (k): 0.5 Decision interval (h): 4.0 Target (optional): blank Use fast initial response (FIR) unchecked Reset accumulation after a signal unchecked Headstart: shows 2.0 but is greyed out, because it is editable only while FIR is checked.
Note what is not on this tab: there is no Number of standard deviations (k): box. This chart never reads that setting, and its place is taken by the decision interval. On a CUSUM chart,
kalways means the allowance. -
Finish
Press Finish. The chart is written to a new worksheet named CUSUM and the Control Chart analysis pane opens on the right.
What you should see¶
One chart titled CUSUM Chart, with Subgroup along the bottom and Cumulative Sum up the side. Its legend names five series: C+, C-, UCL, LCL, and Center. There is one chart, not two stacked panels, and both sums share the center line.
Four numbers define everything on it:
| Quantity | Value | Where it comes from |
|---|---|---|
| Center line | 0 | Fixed. Both series share it, and it is not the center of the accumulation. |
| Center of the accumulation | 25.50 | The mean of all thirty readings, because Target (optional): was left blank. |
| Allowance | 0.56 | Allowance (k): 0.5 times the estimated standard deviation of 1.12, which came from an average moving range of 1.27. |
| Decision interval | 4.49 | Decision interval (h): 4.0 times that same 1.12. It is drawn as UCL at +4.49 and LCL at -4.49. |
The limits are two straight horizontal lines. They sit at +4.49 and -4.49 from the first point to the last and never move, because the decision interval is a fixed multiple of a standard deviation that does not change while the subgroup size is constant. Nothing about this chart funnels.
Both sums stay near zero while the line is behaving. Over the first twenty-two points C+ never rises above 0.30, because no run of readings beats the allowance for long enough to accumulate anything.
Then C+ climbs, point by point:
| Plotted point | Reading | C+ | Signalled? |
|---|---|---|---|
| 22 | 23.48 | 0.00 | |
| 23 | 28.42 | 2.36 | |
| 24 | 26.66 | 2.96 | |
| 25 | 26.66 | 3.56 | |
| 26 | 26.57 | 4.07 | |
| 27 | 26.93 | 4.94 | yes |
| 28 | 27.83 | 6.71 | yes |
| 29 | 25.87 | 6.52 | yes |
| 30 | 26.90 | 7.36 | yes |
Four points signal: 27, 28, 29 and 30. Point 27 is the first, reaching 4.94 against the decision interval of 4.49.
Read that table alongside the readings and the accumulation is easy to follow. Each point adds its own deviation from 25.50 and then gives back the allowance of 0.56. Point 25 reads 26.66, which is 1.16 above center, so it adds 1.16 minus 0.56, that is 0.60, to the 2.96 already there and reaches 3.56. Point 29 reads 25.87, only 0.37 above center, which is less than the allowance, so the sum goes down slightly, from 6.71 to 6.52. That is the allowance working as intended: a deviation smaller than it reduces the total rather than adding to it.
Why four points signal and not one
Reset accumulation after a signal is unchecked by default, so nothing restarts when point 27 crosses. The sum simply stays above the decision interval, and every point after it is above the line as well. Four signalled points here mean one event, not four.
This is worth knowing before you read a real chart, because a long run of signalled points at the right-hand end of a CUSUM chart usually means a single shift that has not been corrected, not a process failing repeatedly. The first section of Things to try next turns the option on so you can see the difference.
Why C- carries a small standing value
C- is not pinned at zero over the early stretch. It wanders up to 1.9 and back, and the reason is the blank target.
With the target blank the center of the accumulation is the mean of all thirty readings, 25.50, and that mean already contains the drift. It sits 0.54 above the average of the twenty-two settled readings, 24.96. So while the line was behaving, its readings sat a little below the center they are being measured against, and the lower sum picks that up. It never gets near the decision interval, peaking at 1.9 out of 4.49, so nothing signals.
Entering a target removes the effect entirely, and the next section does that.
Only one test runs on this chart
A point signals when its accumulation passes the decision interval, and that is the only test. The run, trend, and zone rules are not offered here and no zone boundaries are produced. The test is applied as each sum is computed, so it happens whatever the Rule 1 checkbox is set to. See Out-of-Control tests.
The summary table to the right of the chart reports the two sums separately, under Upper CUSUM (C+) and Lower CUSUM (C-), each with the one limit that applies to its side.
Things to try next¶
- Turn on Reset accumulation after a signal. Run it again with that box checked. Only point 27 signals. The sum restarts at the next point, dropping from 4.94 to 1.77, then 1.58, then 2.42, so the run of four signalled points becomes one. Nothing is removed from the chart: point 27 is still drawn at the 4.94 it reached.
- Enter a target of 25.0. Put 25.0 in Target (optional): and the center of the accumulation stops being the drifting mean. The chart now signals from point 25 rather than 27, two hours earlier, and C+ finishes at 11.35 instead of 7.36. It also removes the standing value on C- described above. This one box is the difference between asking "has the line moved away from its own average" and "has the line moved away from 25 microns".
- Turn on Use fast initial response (FIR). With FIR checked, Headstart: becomes editable at 2.0 and each sum starts from 2.0 standard deviations instead of zero. On this data that puts C- at 3.49 at the very first point, which is most of the way to the 4.49 limit, because the first reading of 23.69 is well below the center. A head start makes an early signal much easier to produce, which is the point of it, and also makes an early false one easier.
- Change the two parameters and watch them trade off. Raise Allowance (k): to 1.0 and more of each deviation is absorbed, so the sum climbs more slowly and the signal arrives later. Lower Decision interval (h): to 3.0 and the limit moves in to 3.37, so the existing accumulation crosses it sooner. Both boxes are in standard deviations, not microns.
- Split at the change. Add a split at point 23. The accumulation restarts at the phase boundary and each phase gets its own center and its own decision interval, so drift built up at the end of one phase never carries into the next.