CUSUM Control Chart

Cumulative Sum Control Chart

Parameters

Measurement Data

Comma or newline separated

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Target Value (leave blank to use mean)

k (reference value)0.5

Detection sensitivity, typically 0.5σ

h (decision threshold)5

Signal when cumulative sum exceeds this, typically 5σ

CUSUM Formulas

C⁺ᵢ = max(0, C⁺ᵢ₋₁ + xᵢ − μ₀ − K)

C⁻ᵢ = max(0, C⁻ᵢ₋₁ + μ₀ − K − xᵢ)

K = k·σ，H = h·σ

Process is out of control when C⁺ > H or C⁻ > H.

Far superior to Shewhart charts for detecting mean shifts of 0.5σ–1.5σ.

Data Points

Target μ₀

11.570

Param k

0.5

Signals

CUSUM Trend Chart

Data Table

No.	Observed	C⁺	C⁻	Status
1	10.200	0.0000	0.7811	OK
2	9.800	0.0000	1.9621	OK
3	10.100	0.0000	2.8431	OK
4	10.300	0.0000	3.5242	OK
5	9.900	0.0000	4.6052	OK
6	10.400	0.0000	5.1863	OK
7	10.700	0.0000	5.4673	OK
8	11.100	0.0000	5.3484	OK
9	11.400	0.0000	4.9294	OK
10	11.800	0.0000	4.1105	OK
11	12.000	0.0000	3.0915	OK
12	12.300	0.1411	1.7726	OK
13	12.100	0.0821	0.6536	OK
14	11.900	0.0000	0.0000	OK
15	12.400	0.2411	0.0000	OK
16	12.600	0.6821	0.0000	OK
17	12.900	1.4231	0.0000	OK
18	13.100	2.3642	0.0000	OK
19	13.400	3.6052	0.0000	OK
20	13.000	4.4463	0.0000	OK

Use Case

CUSUM charts are especially sensitive to small sustained shifts in the process mean, making them ideal for detecting 0.5σ~1.5σ drift in precision manufacturing (semiconductor, aerospace parts). They alarm earlier than traditional Shewhart charts, reducing out-of-spec output. In continuous chemical processes, parameters such as concentration or pH often drift slowly over time. CUSUM triggers investigation early in the trend, preventing large batch scrap.

Example

Set target μ₀ = 10.0, K = 0.5 (to detect a 1σ shift), H = 5 First 10 points normal; from point 11 the mean rises to 10.3 (+0.3 = 0.95σ shift): S+₁₀ = 0 (initial) S+₁₁ = max(0, 0 + (10.3-10.0) - 0.5) = max(0, -0.2) = 0 → Accumulation continues: point 15 S+ = 1.2, point 18 S+ = 3.5, point 21 S+ = 5.2 > H=5 → CUSUM alarms at point 21 → Detects the shift 4-6 points earlier than a Shewhart chart

Further reading: CUSUM & EWMA charts for detecting small shifts

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FAQ

Related Tools

EWMA Control Chart

Both CUSUM and EWMA are sensitive to small shifts but compute differently. EWMA uses exponentially weighted smoothing and handles autocorrelated data better.

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Control Chart (X̄-R)

Shewhart charts detect large shifts; CUSUM detects small ones. Used together they monitor both sudden shifts and slow drift.

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I-MR Control Chart

CUSUM is typically applied to individual measurements (same scenario as I-MR) and complements I-MR when early detection of slow drift is needed.

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