Control Chart (Time-Weighted)

Time-Weighted Control Charts

Classic Shewhart charts (I-MR, X̄-R) are insensitive to small persistent shifts — a 1σ drift is often only detected after 40–50 points. Time-weighted charts integrate information across multiple observations and pick up exactly those patterns much faster.

EWMA — Exponentially Weighted Moving Average: Each value is a weighted average of the current observation and the previous EWMA: zᵢ = λ·xᵢ + (1−λ)·zᵢ₋₁. The smoothing parameter λ controls memory: small λ (0.05–0.1) detects very small shifts, larger λ (0.3–0.4) behaves more like a Shewhart chart.

CUSUM — Cumulative Sum: Accumulates deviations from the target but only starts "counting" once the deviation exceeds a slack k. Two one-sided sums C⁺ and C⁻ detect upward / downward shifts. A signal fires as soon as either sum crosses the limit h.

Both charts are used in the Control phase when a process is already running (target μ₀ and σ known) and small slow drifts must be detected early — typical use: chemical processes, tool wear, gauge drift.

Procedure and Parameter Choice

• Determine target μ₀ and standard deviation σ from a stable baseline.
• Prefer σ̂ from mean moving range MR̄/d₂ (more robust to special causes than the sample SD).
• Decide which shift size you want to detect early — e.g. 1σ.
• EWMA: λ ≈ 0.1 for 1σ shifts; use L = 2.7 (instead of 3.0) when λ is small.
• CUSUM: k = 0.5σ and h = 4σ or 5σ — this combination flags 1σ shifts within ~10 points.
• Run the chart and react to the first signal (point outside the limits).

Rule of thumb for shift size: to detect a Δ·σ shift, choose k = Δ/2. The CUSUM procedure is then tuned optimally to that shift.

Reading and Reacting

• EWMA: the zᵢ line crosses UCL/LCL → special cause. The limits widen at the start (transient) and approach the steady-state asymptotically.
• CUSUM: C⁺ rising → upward shift; C⁻ rising → downward shift. Signal as soon as either sum crosses the horizontal limit h.
• Both charts show the beginning of a shift later than its actual onset — the integration costs reaction time but gains sensitivity.
• After a signal: investigate the cause, correct the process, and reset the sums / EWMA.

Pitfalls

Wrong σ estimate: If σ is computed from a sample with special causes, the limits are too wide. Always estimate from a confirmed-stable baseline — preferably from MR̄/d₂.

EWMA on Shewhart problems: EWMA is sluggish for large, sudden jumps. For genuine outliers an I-MR chart reacts faster. Running EWMA and Shewhart in parallel is common (two-tier monitoring).

Not resetting CUSUM: After a correction, both C⁺ and C⁻ must be reset to 0 — otherwise the old deviation keeps contributing to the next signal.

Ignoring autocorrelation: Both methods assume independent observations. With autocorrelated data (batch processes, continuous measurements) they fire too many false alarms. Decorrelate first or use a model-residual chart.

Arbitrary k and h: Values other than the standard pairs (k=0.5/h=4 or 5) drastically change the ARL behavior. If you deviate, consult ARL tables or run a simulation.

Examples

This module ships with the following example datasets — load any of them in the app with a single click.

Available in the following cycles

• DMAIC: Control
• DMADV: Verify
• 8D: D6 — Implementation