CUSUM Chart Calculator — Tabular CUSUM with Reference Value k and Decision Interval h

How the tabular CUSUM is computed

Two accumulators, one for each direction, each floored at zero: S⁺(i) = max(0, S⁺(i−1) + xᵢ − μ̂₀ − k) and S⁻(i) = max(0, S⁻(i−1) + μ̂₀ − k − xᵢ). The process signals when either goes beyond h — strictly beyond, not equal to.

k is the slack. Subtracting it at every step is what stops an on-target process from random-walking away from zero: small deviations are absorbed and the sum sits at its floor. The design rule is k = δ/2 in sigma units, so k = 0.5σ tunes the chart for a one-sigma shift, and that is the default here.

h is the decision interval, and it is the whole answer. With k = 0.5, h = 4 and h = 5 are the two conventional choices; 4 signals sooner and is Minitab’s default, 5 buys a substantially longer in-control run length. The same data under the two values gives two different answers to "when did it shift", which is why both parameters are printed on the figure and survive into the export.

Both accumulators are non-negative numbers compared against the same positive h. The chart draws S⁻ downwards purely so the two curves do not overlap; the axis labels are the accumulator values, which read positive in both directions.

Why there is no V-mask, and what is here instead

The V-mask is the original graphical form of this chart: a wedge laid over the plain cumulative sum with its vertex ahead of the most recent point, signalling when an earlier point falls outside an arm. It is not drawn here, and that is a decision rather than an omission.

The NIST/SEMATECH handbook says it plainly — most users of CUSUM procedures prefer the tabular chart — and the tabular form is what Minitab produces by default, so it is what anybody comparing this output against an existing chart will have in front of them. Beyond that: a mask is anchored on the last observation and moves every time a point arrives, which makes a static exported figure unreadable without knowing where the vertex was, and its two arms encode exactly the decision the two accumulators already show.

What the tool does offer is the V-mask design arithmetic, because that is where a decision interval comes from when it is not taken from the rule of thumb: k = δσ/2, d = (2/δ²)·ln((1−β)/α), h = dk. Switch it on and the figure prints all three from your δ, α and β, and warns if the h they imply is not the h the chart is drawn with. Running that calculation on the handbook’s own stated α, β and δ does not reproduce its own printed h — the discrepancy and the arithmetic that does reproduce it are both recorded on the validation page rather than smoothed over.

Run rules do not apply here at all

Not one test in the Nelson, Western Electric or ISO sets can be applied to a cumulative sum, and the figure lists all of them as suppressed with the reason. The beyond-the-limits tests measure a distance from a centre line in multiples of σ, and this chart decides on h instead — two decision rules on one chart cannot both be the answer. Every other test counts a pattern across consecutive points, and each cumulative sum contains the whole of the one before it, so nine points in a row above zero is the mechanism rather than the evidence.

The decision interval is the test. That is also why the figure names the first crossing rather than every point beyond h: a CUSUM that has signalled stays signalled until the sum is reset, so eleven points beyond h are one event, not eleven.

Fast initial response, and why it is off by default

A headstart starts both accumulators at half of h instead of at zero, so a process that was already off target when charting began is caught within a few points rather than after the sum has climbed from nothing. It is a well-established option and it is offered here.

It is off by default for one reason: it changes every printed sum, and no published worked example this tool reproduces uses one. A default that quietly makes the tool disagree with every textbook costs more in support than it saves in detection. Switch it on deliberately, and the figure says the headstart is there and how large it is.

Common questions

What values of k and h should I use for a CUSUM chart?

k = 0.5σ with h = 4σ or 5σ. k = δ/2 tunes the chart for a shift of δ sigma, so 0.5 is the standard choice for detecting a one-sigma shift, and h = 4 with k = 0.5 is Minitab’s default. Choose the pair before you see the data: every published average-run-length table is indexed by (k, h) together, and adjusting h after a signal is how a chart stops being evidence.

Should I use a CUSUM or an EWMA chart?

For a sustained shift of about one sigma they perform almost identically at their conventional settings, and on the same data they usually signal within a point or two of each other. Prefer a CUSUM when you want a decision — "the process shifted at observation 14" — and an EWMA when you want a picture of where the process is now. Both tools are on this site and both take the same pasted column.

Why is my CUSUM signalling in both directions?

Almost always because the target is the mean of data that already contains a shift. That mean sits between the two levels, so the settled period reads low and the shifted period reads high, and neither signal locates the change. Enter a real target, or set a baseline over the period you believe was stable. The tool prints a warning when it detects this.

Can a CUSUM be used on subgroup averages?

Yes — the worked example this tool reproduces is exactly that: twenty averages of subgroups of four. The sigma that k and h are measured in must then be the standard error of the average, σ/√n, not the process standard deviation. Getting that wrong scales the decision interval by √n and the chart either never signals or never stops.

Is my data uploaded anywhere?

No. The whole calculation runs as JavaScript in this browser tab, and nothing on this site accepts a measurement — the only server code here takes a payment. That is why the tool is usable on data you are contractually barred from putting in a cloud service.