CUSUM Control Chart: Cumulative Sums for Fast Shift Detection

A CUSUM control chart (cumulative sum chart) monitors a process by accumulating deviations from a target value, so that small but persistent shifts add up into a clear alarm long before a conventional control chart reacts. Where a Shewhart chart judges each sample in isolation, CUSUM has memory. That makes it the fastest standard tool for catching shifts of roughly 0.5 to 1.5 standard deviations, exactly the slow drifts from tool wear, temperature creep, or material variation that quietly eat margin.

Why small shifts are the expensive ones

A Shewhart chart with 3 sigma limits spots large, sudden jumps quickly but reacts slowly to small ones. If the process mean drifts by one standard deviation, the average run length (ARL) to detection is about 44 samples. Sampling hourly, that is nearly two days of off-target production inflating your scrap rate without a single point ever breaching a limit.

A properly tuned CUSUM detects that same shift in about 10 samples at a comparable false alarm rate: four times faster. That advantage is why the chart belongs in any mature statistical process control program.

How the tabular CUSUM works: the k and h parameters

The tabular CUSUM tracks two running sums for each observation x against the target mu0:

• Upper sum: C+ = max(0, x - (mu0 + k) + previous C+), which catches upward shifts
• Lower sum: C- = max(0, (mu0 - k) - x + previous C-), which catches downward shifts

Two parameters govern everything:

• k, the reference value (allowance): a dead band the chart ignores, set to half the shift you want to detect (k = 0.5 sigma to catch a 1 sigma shift). Deviations smaller than k drain the sums back toward zero; larger ones build them up.
• h, the decision interval: the alarm threshold, typically 4 or 5 sigma. With k = 0.5 sigma and h = 5 sigma, the in-control ARL is about 465 samples, comparable to a Shewhart chart, while a 1 sigma shift signals in about 10 samples.

A lower h detects faster but false-alarms more often. After a reset, starting the sums at h/2 (a fast initial response, or FIR) re-alarms quickly if the problem persists.

Worked example: catching a small drift in fill weight

A bottling line fills to a target of mu0 = 500 g with sigma = 1 g. Quality wants to catch a 1 g upward drift, so k = 0.5 g and h = 5 g. Shewhart limits would sit at 497 g and 503 g. After sample 5, a filler valve begins to stick and the true mean rises by about 1.5 g.

1. Samples 1 to 5 (in control): 500.2, 499.4, 500.4, 499.8, 500.1 g. All sit below the 500.5 g threshold (mu0 + k), so C+ stays at zero.
2. Sample 6: 501.5 g. Excess over 500.5 is 1.0, so C+ = 1.0.
3. Sample 7: 501.2 g. Excess is 0.7, so C+ = 1.7.
4. Sample 8: 501.8 g. Excess is 1.3, so C+ = 3.0.
5. Sample 9: 501.4 g. Excess is 0.9, so C+ = 3.9.
6. Sample 10: 501.7 g. Excess is 1.2, so C+ = 5.1, which exceeds h = 5. Alarm.

The chart signaled just five samples after the shift began, yet not one individual reading came near the 503 g Shewhart limit; a Shewhart chart would have kept waiting while every bottle gave away product. CUSUM even estimates the new mean: mu0 + k + C+/N, where N is the samples since the sum left zero. Here that is 500 + 0.5 + 5.1/5 = 501.52 g, telling the technician how far to adjust.

CUSUM vs Shewhart: when to use each

• Shewhart charts excel during process start-up, for large special causes, and for operator intuition; the Nelson rules extend their sensitivity.
• CUSUM excels on mature, stable processes threatened by slow drift: tool wear, heat exchanger fouling, reagent aging, seasonal temperature effects.
• CUSUM amplifies small deviations, so measurement noise can masquerade as drift. Run a gauge R&R study before trusting the chart.
• It assumes a stable process with known sigma. Establish control and confirm process capability first. And since CUSUM is slower than Shewhart on very large shifts, many plants run both in parallel.

Implementing CUSUM on the shop floor

1. Estimate mu0 and sigma from at least 20 to 30 in-control subgroups; borrowing the specification target instead of real process data is a classic failure mode.
2. Pick the shift that matters economically and set k to half of it.
3. Choose h for the false alarm rate you can live with: 4 sigma reacts faster, 5 sigma cries wolf less often.
4. Automate the arithmetic. Tabular CUSUM is trivial for software and error prone by hand.
5. Write the reaction into your control plan: verify the measurement, find and fix the assignable cause, then reset both sums to zero (or to an FIR headstart) so the chart keeps driving action instead of becoming wallpaper.

Where Fabrico fits

A CUSUM chart is only as fast as the data feeding it. Fabrico is the real-time data foundation: it captures production and machine data as it happens, including through computer vision on machines with no PLC, and turns it into live OEE and production monitoring, so drift shows up within the shift, not in next week's report. When a signal is confirmed, the loop closes in the same system: raise a work order, log the root cause against the asset, and schedule the follow-up preventive task in Fabrico's field-ready CMMS. And because Fabrico is EU-built with EU data residency, your process data stays inside the EU.

Frequently Asked Questions

What do the k and h parameters mean in a CUSUM chart?

k (the reference value) is the tolerance band the chart ignores, normally half the shift you want to detect; h (the decision interval) is the alarm threshold on the cumulative sums, normally 4 to 5 standard deviations.

How does a CUSUM chart compare to an EWMA chart?

Both have memory and offer similar small-shift performance. EWMA weights recent points exponentially and is more robust to non-normal data; tabular CUSUM is tuned to one specific shift size and gives a direct estimate of the new mean after a signal.

What should I do after a CUSUM alarm?

Verify the measurement first, then find and correct the assignable cause and document the action. Reset C+ and C- to zero, or to an h/2 headstart for fast confirmation that the fix worked, and continue charting.

Want the live machine data that makes fast shift detection possible? Book a Fabrico demo and see real-time OEE monitoring and a field-ready CMMS working as one system.