CUSUM Control Chart Calculator

What Is a CUSUM Control Chart?

A CUSUM (cumulative sum) control chart accumulates the deviation of each new measurement from a target value, rather than looking at each point in isolation like a standard Shewhart chart. This accumulation makes CUSUM far more sensitive to small, sustained shifts — a process that drifts 0.5 standard deviations off target may take dozens of points to trigger a standard chart, but a CUSUM chart can flag it within just a few points because the small deviations keep adding up.

Page's two-sided CUSUM tracks two running sums: an upper sum (C⁺) that grows when values run above target plus an allowance K, and a lower sum (C⁻) that grows when values run below target minus K. Both sums reset toward zero whenever the process is on-target, so they only accumulate during a genuine sustained shift rather than normal random noise. A signal is raised when either sum exceeds a decision interval H.

In energy management (ISO 50001), CUSUM is particularly valuable for catching a piece of equipment that is gradually drifting toward higher energy consumption — a slow compressor efficiency decline, for example, that a monthly bill comparison would not reveal for months, but that a CUSUM chart on daily or weekly consumption data can surface within days of the drift beginning.

Formula: C⁺ₙ = max(0, xₙ − (μ₀ + K) + C⁺ₙ₋₁) C⁻ₙ = max(0, (μ₀ − K) − xₙ + C⁻ₙ₋₁) Signal when C⁺ₙ > H or C⁻ₙ > H

Example Calculation

A process with target 100 produces the readings 100, 102, 98, 101, 99, 103, 108, 110, 112, 115 — the last four points show a clear upward drift. With auto-calculated K and H from the data's own standard deviation, C⁺ stays at 0 through the first five points (no sustained deviation yet), then climbs steadily to about 33 by the tenth point, crossing the decision interval and raising a signal at point 10. C⁻ stays at 0 throughout since there is no downward drift. The result correctly identifies a positive (upward) shift.

When to Use This Calculator

• An energy manager monitoring daily or weekly facility energy consumption against a baseline to catch gradual efficiency degradation early (ISO 50001 EnPI monitoring)
• A process engineer watching a quality characteristic for a slow drift that a standard control chart is too slow to flag
• A maintenance engineer using CUSUM on equipment sensor readings (vibration, temperature, current draw) to detect early-stage degradation before failure
• A quality manager comparing CUSUM sensitivity against an existing Shewhart chart to decide which monitoring approach better fits a specific process

Common Mistakes to Avoid

• Resetting the CUSUM after every signal without investigating the cause — a signal indicates a real, sustained shift; resetting without root-cause analysis just delays discovering the underlying problem
• Using an outdated or incorrect target value — CUSUM is very sensitive to the target (μ₀); if the true process target has changed (e.g., after a legitimate process improvement), the old target will cause continuous false signals
• Estimating standard deviation from a dataset that already contains the shift you are trying to detect — this inflates sigma and can mask the very shift the chart is meant to catch; ideally sigma should come from a stable, in-control reference period
• Treating CUSUM values as if they were the raw measurements — the cumulative sums are diagnostic statistics, not physical quantities, so plot and interpret them on their own scale, not against the original data's specification limits
• Ignoring a data point after a chart is reset — some CUSUM implementations reset both sums to zero after a signal; forgetting to note this on the chart can make later points misleadingly appear to start a shift from a false zero baseline

How to Interpret Results

• No signal (shift: none) means the process is running consistently near the target value — no sustained shift has been detected within the data range provided
• A positive shift means the process has drifted sustainedly above the target — investigate root causes like equipment degradation, setpoint drift, or a process change
• A negative shift means the process has drifted sustainedly below the target — while sometimes desirable (e.g., lower energy consumption), it should still be investigated to confirm it is intentional and sustainable
• The point where a signal first appears (from the signals list) marks approximately when the sustained shift began — investigate conditions and events around that point in time, not just the most recent data
• A "both" shift (both C⁺ and C⁻ signaling) suggests either two separate shift events in the monitored period or a mismatch between the target/sigma used and the actual process behavior — review the full time series before acting

Related Standards & References

• Page, E.S. (1954), "Continuous Inspection Schemes", Biometrika — the original publication introducing the CUSUM control chart method
• ISO 50001:2018 — Energy management systems, which references CUSUM as a technique for energy performance indicator (EnPI) monitoring
• Montgomery, D.C., "Introduction to Statistical Quality Control" — standard SPC textbook reference for CUSUM design parameters (K, H) and comparison with Shewhart charts

Frequently Asked Questions

How should I choose the allowance (K) and decision interval (H)?

A common industry default is K = σ/2 (half a standard deviation) and H = 5σ, which this calculator uses automatically when left blank — this combination is tuned to detect a 1-sigma shift within about 8-10 points on average. Smaller K values make the chart more sensitive to small shifts but increase false alarms; larger H values reduce false alarms but delay detection of real shifts.

When should I use CUSUM instead of a standard Shewhart control chart?

Use CUSUM when you need to detect small, sustained shifts (roughly 0.5 to 2 standard deviations) as quickly as possible — Shewhart charts (X-bar, I-MR) are better suited to detecting large, sudden shifts but are slow to react to gradual drift. Many practitioners run both chart types side by side: Shewhart for sudden spikes, CUSUM for slow trends.

What does it mean if both C⁺ and C⁻ signal in the same dataset?

A signal on both sides usually indicates the process shifted in one direction and later shifted back (or overshot) in the other, or that the target/standard deviation used does not match the process well. It is worth reviewing the raw data and confirming the target value is still appropriate before treating this as two independent shifts.