CUSUM and EWMA Control Charts

Shewhart control charts (X-bar, I-MR) are excellent at detecting large shifts (≥ 2σ) but slow to detect small, sustained shifts (0.5–1.5σ). CUSUM and EWMA charts are designed specifically to catch these small shifts quickly.

CUSUM (Cumulative Sum)

CUSUM accumulates deviations from the target. A sustained shift causes the cumulative sum to drift steadily in one direction, triggering an out-of-control signal when it exceeds a threshold h.

Two-sided CUSUM: C⁺ᵢ = max(0, C⁺ᵢ₋₁ + (xᵢ − μ₀ − k)) C⁻ᵢ = max(0, C⁻ᵢ₋₁ − (xᵢ − μ₀ + k)) Signal: C⁺ or C⁻ > h Typical: k = 0.5σ, h = 5σ → ARL₀ ≈ 465 (false alarm rate similar to 3σ Shewhart) → ARL₁ ≈ 10 for a 1σ shift (vs ARL₁ ≈ 44 for Shewhart)

EWMA (Exponentially Weighted Moving Average)

EWMAᵢ = λxᵢ + (1 − λ)EWMAᵢ₋₁ λ = smoothing constant (0 < λ ≤ 1) Small λ (e.g., 0.1): more weight on history, detects small shifts Large λ (e.g., 0.4): more weight on recent data, faster response Control limits: UCL = μ₀ + Lσ√(λ/(2−λ)) LCL = μ₀ − Lσ√(λ/(2−λ))

CQE exam tips

Both CUSUM and EWMA have Average Run Length (ARL) properties; smaller ARL₁ = faster detection.
For a large (≥ 3σ) shift, a standard Shewhart chart detects it faster — use CUSUM/EWMA when the shift is expected to be small.
EWMA with λ = 1 is identical to a Shewhart I-chart — the connection between charts.

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