CUSUM Chart
The CUSUM (cumulative sum) chart is a SPC Chart PRO chart type for catching small, sustained shifts in a process far sooner than a Shewhart chart can.
A Shewhart chart (XmR, Xbar-R) reacts well to large, sudden shifts but is slow to catch a small one that persists. Because each point is judged on its own, a drift of half a sigma can run for dozens of points before any single value lands outside the limits. CUSUM accumulates the deviation of every point from a target, so a small, steady bias adds up and crosses a decision boundary far sooner.
Select CUSUM chart (time-weighted) in the Chart Type dropdown.
How It Works
CUSUM runs two cumulative sums in parallel, each drawn as its own line:
- Upper sum (C⁺) accumulates deviations above the target and detects an upward shift.
- Lower sum (C⁻) accumulates deviations below the target and detects a downward shift.
Both start at zero and are held there while the process is on target — only a sustained deviation makes one of them climb away from the center line. The two sums share a center line of 0 and a symmetric pair of control limits, the decision interval ±H. When either sum crosses its limit, the point is flagged: the process has drifted.
Two parameters — the reference value and the decision interval — set how large a shift the chart is tuned to detect and how quickly it responds. They are usually chosen together from an ARL (average run length) table; the defaults, k = 0.5 and h = 4, are the standard choice for detecting a shift of one sigma.
Panel Options
CUSUM options appear in the SPC section when the CUSUM chart type is selected.
| Option | Description | Default |
|---|---|---|
| Reference value (k) | The slack, in sigma units — half the size of the shift you want to detect. Smaller values detect smaller shifts. | 0.5 |
| Decision interval (h) | The distance from the center line to the control limits, in sigma units. Larger values reduce false alarms at the cost of slower detection. | 4 |
| Target | The process target the deviations are measured from. When empty, the mean of the data is used. | Mean of data |
| Historical σ | The process standard deviation. When empty, it is estimated from within-subgroup variation. | Estimated |
| Fast initial response | Start the sums at ±H⁄2 so a shift already present at start-up is caught sooner. | Off |
Individuals or Subgroups
CUSUM works on individual measurements or on subgroups. Set the Subgroup size in the SPC section: a size of 1 treats each row as one point, while a larger size accumulates the deviation of each subgroup mean and tightens the decision interval by √n.
Reading the Chart
While the process is centered, both sums rest on the zero line. A sustained shift pulls one of them away — the upper sum for a shift up, the lower sum for a shift down — until it crosses the decision interval and the point is ringed. The direction of the crossing tells you which way the process moved, and the point at which the sum first left zero points back to when the shift began.
Methods and Formulas
For each point i, with subgroup mean x̄ᵢ, target T, and the standard error of the mean σx̄ = σ ⁄ √n:
- Upper sum: C⁺ᵢ = max( 0, C⁺ᵢ₋₁ + (x̄ᵢ − T) − K )
- Lower sum: C⁻ᵢ = min( 0, C⁻ᵢ₋₁ + (x̄ᵢ − T) + K )
where the reference value K = k·σx̄ and the decision interval H = h·σx̄. A point is out of control when C⁺ᵢ rises above H or C⁻ᵢ falls below −H. With Fast Initial Response the sums begin at C⁺₀ = +H⁄2 and C⁻₀ = −H⁄2 instead of zero.
The within-subgroup sigma is estimated the same way as for the matching Shewhart chart — the average moving range for individuals, the pooled subgroup deviation for subgroups — unless a Historical σ is supplied.
Run Rules on CUSUM
Because each CUSUM point already carries the history of every point before it, the pattern-based Nelson rules — runs, trends, and zone tests — do not apply; they would fire on the accumulation itself. For a CUSUM chart the panel therefore shows only Rule 1, which flags any point beyond the decision interval — exactly the CUSUM out-of-control signal. The remaining rules are hidden automatically.
Point Exclusion on a CUSUM Chart
Point exclusion behaves a little differently on a CUSUM chart than on a Shewhart chart, because the plotted statistic is cumulative.
Following Minitab's omit from estimation, an excluded point is dropped from the target and sigma estimates, so the whole chart — both sums and the decision interval — re-baselines to the estimate without it. The point itself stays plotted and remains in the cumulative sum: an excluded spike is still visible on the accumulated line. In other words, excluding a point removes its influence on the limits and target, not from the running total.
This is what you want when a known assignable cause — a bad measurement, a one-off upset — has inflated the target or the spread and you need the chart to reflect the process without it. The EWMA chart handles exclusion the same way.
When to Use CUSUM
| Scenario | Chart |
|---|---|
| Detecting a small, sustained shift (roughly 0.5σ–2σ) as early as possible | CUSUM |
| Detecting large, sudden shifts | XmR / Xbar-R / Xbar-S |
| Holding a target precisely over time (chemical, pharmaceutical, semiconductor) | CUSUM |
The CUSUM parameters, the target, and the subgroup size are saved with the dashboard. Like every chart type, CUSUM works with both a time and a numeric X-axis.
CUSUM is part of SPC Chart PRO. For requirements, licensing, and support, see the SPC Chart PRO documentation.