CUSUM Chart
A cumulative sum control chart that plots the running total of deviations from a target value over time — designed to detect small, sustained shifts in a process mean earlier than standard control charts.
// 01 — The chart
What it looks like
A CUSUM chart tracking cumulative deviations from a target fill weight. The rising curve signals a sustained upward shift. An alarm triggers when the cumulative sum exceeds the decision interval H.
// 02 — Definition
What is a CUSUM chart?
A CUSUM (Cumulative Sum) chart is a sequential analysis technique used in statistical process control (SPC). Instead of plotting individual measurements like a standard control chart, it plots the cumulative sum of deviations from a target value over time.
At each time point, the chart adds the difference between the current observation and the target to a running total. When the process is on target, these deviations cancel out and the cumulative sum stays near zero. When a sustained shift occurs — even a small one — the cumulative sum drifts steadily upward or downward, creating a visible slope change.
The key advantage of CUSUM over standard Shewhart control charts is its sensitivity to small, persistent shifts. A Shewhart chart waits for a single point to exceed control limits, which may take many samples for a small shift. A CUSUM chart accumulates evidence over time, detecting a 0.5–1.5 standard deviation shift far more quickly.
Origin: The CUSUM chart was introduced by E. S. Page in 1954 in his paper “Continuous Inspection Schemes.” It was developed at the University of Cambridge as a more efficient alternative to Shewhart charts for detecting small process shifts in industrial quality control.
// 03 — Anatomy
Parts of a CUSUM chart
// 04 — Usage
When to use it — and when not to
- You need to detect small, sustained shifts (0.5–1.5σ) in a process mean
- Standard Shewhart control charts are too slow to catch gradual drifts
- Your process has low variability and even small mean shifts matter (e.g., pharmaceutical dosing)
- You want to monitor a continuous process with sequential observations
- Early detection of calibration drift or tool wear is critical
- You have a well-defined target value and known in-control standard deviation
- You need to detect large, sudden shifts — a Shewhart chart is simpler and equally fast for shifts > 2σ
- Your data is not approximately normal or has heavy-tailed distributions
- The process has frequent, known assignable causes — the CUSUM accumulates past signals
- You need to monitor process variability (spread) rather than the mean
- Your audience is unfamiliar with SPC — CUSUM charts are harder to interpret than standard control charts
- The target value is not well-established or keeps changing
// 05 — Reading guide
How to read a CUSUM chart
Follow these steps whenever you encounter a CUSUM chart in the wild.
Identify the target and decision interval
The chart should show the target value (zero line) and the decision interval H (upper and lower threshold). These define what “in control” and “out of control” mean for this process.
Check whether the cumulative sum stays near zero
When the process is on target, the CUSUM line fluctuates randomly around zero. A flat, meandering path means no detectable shift.
Look for a sustained slope
A consistent upward or downward slope indicates a persistent shift in the process mean. The steeper the slope, the larger the shift. An upward slope means the mean has increased above the target; downward means it has decreased.
Find the alarm point
The point where the CUSUM first crosses the decision interval H is the alarm. This is where the chart signals that a statistically significant shift has been detected and investigation is needed.
Estimate when the shift started
Look backward from the alarm point to where the slope change began. The CUSUM’s slope change point often indicates when the process actually shifted, which can be well before the alarm was triggered.
// 06 — Pitfalls
Common mistakes
× Not resetting after an alarm
After a CUSUM signals an out-of-control condition, the cumulative sum should be reset (or partially reset) once the assignable cause is fixed. Failing to reset means the chart carries forward old evidence, making future signals unreliable.
× Using wrong reference value (k)
The reference value k (also called the allowance or slack) determines the shift size the CUSUM is tuned to detect. Setting k too large misses small shifts; too small causes excessive false alarms. A common choice is k = δ/2 where δ is the shift to detect in standard deviation units.
× Ignoring the ARL trade-off
The Average Run Length (ARL) describes how many observations it takes on average to signal. Decreasing H catches shifts faster but increases false alarm rate. Always design k and H together to balance in-control ARL vs. out-of-control ARL.
× Applying CUSUM to non-stationary data
CUSUM assumes the in-control process is stable. If the baseline naturally drifts (seasonality, trend), the chart will accumulate deviations and produce false signals. Detrend or difference the data first.
× Plotting raw values instead of cumulative sums
A CUSUM chart must plot the running total of (xᵢ − target), not the raw measurements. Plotting raw values with CUSUM limits is a meaningless hybrid that detects nothing correctly.
// 07 — In the wild
Real-world examples
Pharmaceutical manufacturing
CUSUM charts monitor tablet weight, coating thickness, and active ingredient concentration. Even a 0.5% drift in dosage can affect efficacy, making CUSUM’s sensitivity to small shifts essential.
Healthcare surveillance
Public health agencies use CUSUM to detect disease outbreaks. A cumulative excess in case counts above expected levels triggers an alert earlier than waiting for a single large spike.
Semiconductor fabrication
Wafer thickness, etch depth, and deposition rates are monitored with CUSUM charts. Process drift of fractions of a nanometer can cause chip failures, so early detection is critical.
// 08 — Quick reference
Key facts
| Also known as | Cumulative Sum chart, Page’s CUSUM |
| Invented | 1954 by E. S. Page |
| Category | Statistical / Process control |
| Difficulty | Advanced |
| Input data | Sequential measurements from a process |
| Key parameters | Target (μ₀), reference value (k), decision interval (H) |
| Best for detecting | Small sustained shifts (0.5–1.5σ) |
| Typical ARL₀ | 370–500 observations (in-control) |
// 09 — Variations
Common variations
Two-sided (tabular) CUSUM
Tracks both upward (C+) and downward (C−) cumulative sums simultaneously. The most common implementation, detecting shifts in either direction.
V-mask CUSUM
The original graphical method where a V-shaped mask is overlaid on the cumulative sum plot. If past points fall outside the mask arms, a shift is detected. Largely replaced by the tabular method.
Fast Initial Response (FIR) CUSUM
Starts the cumulative sum at a non-zero “head start” value (typically H/2) to improve sensitivity at the beginning of monitoring when an initial out-of-control state is possible.
Self-starting CUSUM
Estimates the in-control mean and standard deviation from the data as observations arrive, rather than requiring pre-specified parameters. Useful when historical data is limited.
// 10 — FAQs
Frequently asked questions
What is a cusum chart?+
A CUSUM (Cumulative Sum) chart is a sequential analysis technique used in statistical process control (SPC). Instead of plotting individual measurements like a standard control chart, it plots the cumulative sum of deviations from a target value over time.
When should you use a cusum chart?+
Use a cusum chart when you need to detect small, sustained shifts (0.5–1.5σ) in a process mean. It also works well when standard Shewhart control charts are too slow to catch gradual drifts, and when your process has low variability and even small mean shifts matter (e.g., pharmaceutical dosing).
When should you avoid a cusum chart?+
Avoid a cusum chart when you need to detect large, sudden shifts — a Shewhart chart is simpler and equally fast for shifts > 2σ. It is also a poor fit when your data is not approximately normal or has heavy-tailed distributions, or when the process has frequent, known assignable causes — the CUSUM accumulates past signals.
Is a cusum chart suitable for dashboards?+
Yes — a cusum chart can work well in dashboards as long as the panel is large enough for readers to perceive the encoded values, has a clear title, and includes the legend or axis labels needed to interpret it.
What category of chart is a cusum chart?+
CUSUM Chart belongs to the Statistical family of charts. Charts in that family are designed to answer the same kind of question, so they often work as alternatives when one doesn't quite fit your data.
How do you read a cusum chart?+
Start with the axis labels and legend, then look at the overall shape before zooming into individual marks. Compare prominent features against the rest of the data, and verify any conclusion against the underlying numbers when precision matters.