EWMA Control Chart

Exponentially Weighted Moving Average Chart

Parameters

Measurement Data

Comma or newline separated

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λ (smoothing constant)0.20

Smaller λ = longer memory, more sensitive to slow drift

L (control limit multiplier)3.00

UCL/LCL = μ ± L·σ·√(…), typically 3

EWMA Formula

Zᵢ = λ·xᵢ + (1−λ)·Zᵢ₋₁

UCL/LCL = μ₀ ± L·σ·√(λ/(2−λ))·√(1−(1−λ)²ⁱ)

Z₀ = μ₀ (sample mean)

Process is out of control when Zᵢ exceeds UCL or LCL.

λ = 0.05–0.25 is most sensitive to small drifts (0.5σ–1.5σ).

Mean μ

11.580

Std Dev σ

1.304

Smooth λ

0.20

Signals

EWMA Trend Chart

No out-of-control signals detected. Process is in control.

Data Table

No.	Observed	EWMA	UCL	LCL	Status
1	10.100	11.2840	12.3627	10.7973	OK
2	9.900	11.0072	12.5823	10.5777	OK
3	10.200	10.8458	12.7005	10.4595	OK
4	10.000	10.6766	12.7700	10.3900	OK
5	9.800	10.5013	12.8125	10.3475	OK
6	10.300	10.4610	12.8389	10.3211	OK
7	10.500	10.4688	12.8555	10.3045	OK
8	10.800	10.5351	12.8660	10.2940	OK
9	11.200	10.6680	12.8727	10.2873	OK
10	11.500	10.8344	12.8770	10.2830	OK
11	11.900	11.0476	12.8797	10.2803	OK
12	12.100	11.2580	12.8814	10.2786	OK
13	12.400	11.4864	12.8825	10.2775	OK
14	12.200	11.6291	12.8832	10.2768	OK
15	12.600	11.8233	12.8837	10.2763	OK
16	12.900	12.0387	12.8840	10.2760	OK
17	13.100	12.2509	12.8842	10.2758	OK
18	13.300	12.4607	12.8843	10.2757	OK
19	13.600	12.6886	12.8844	10.2756	OK
20	13.200	12.7909	12.8844	10.2756	OK

Use Case

Temperature data from continuous chemical processes often exhibits strong autocorrelation, causing traditional Shewhart charts to produce excessive false alarms. The EWMA chart applies exponentially weighted smoothing to suppress random noise and is more sensitive to small process drifts (0.5σ~1.5σ), making it well suited for precision machining and semiconductor process monitoring.

Example

Set λ = 0.2, L = 3.054 (ARL₀ ≈ 500) Target μ₀ = 10.0, process std dev σ = 0.5 EWMA₀ = μ₀ = 10.0 Measurement X₁ = 10.6: EWMA₁ = 0.2 × 10.6 + 0.8 × 10.0 = 10.12 Measurement X₂ = 10.8: EWMA₂ = 0.2 × 10.8 + 0.8 × 10.12 = 10.256 UCL (steady-state) = μ₀ + L×σ×√(λ/(2-λ)) = 10.0 + 3.054×0.5×√(0.1111) = 10.51 → Point 8 EWMA exceeds UCL, triggering an alarm

Further reading: CUSUM and EWMA charts — detecting small shifts

View articles →

FAQ

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CUSUM Control Chart

Both CUSUM and EWMA are sensitive to small shifts with similar performance in most scenarios; choose based on familiarity and software support.

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Control Chart (X̄-R)

Shewhart charts detect large shifts while EWMA detects small shifts; used together they provide comprehensive process monitoring coverage.

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I-MR Control Chart

EWMA is typically applied to individual measurements, sharing the same use cases as I-MR charts and serving as the preferred complement for detecting slow drift.

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