A CUSUM (Cumulative Sum) chart is an SPC tool that detects small, sustained shifts in a process mean by tracking the cumulative sum of deviations from a target value. Unlike Shewhart charts that evaluate each point independently, CUSUM accumulates evidence over time, making it highly sensitive to persistent drifts.

How CUSUM Works

• Define a target value $\mu_0$ (the desired process mean).
• For each observation $x_i$, calculate the deviation: $x_i - \mu_0$.
• Accumulate these deviations:**
• Upper CUSUM: $C_i^+ = \max(0, C_{i-1}^+ + (x_i - \mu_0 - K))$ — detects upward shifts.
• Lower CUSUM: $C_i^- = \max(0, C_{i-1}^- - (x_i - \mu_0 + K))$ — detects downward shifts.
• $K$ is the reference value (allowance), typically set at half the shift size you want to detect: $K = \delta\sigma / 2$.
• Signal when $C^+$ or $C^-$ exceeds the decision interval $H$ (typically 4–5 times $\sigma$).

Why CUSUM Is Powerful

• Cumulative Memory: Small deviations that individually look normal accumulate over time. A consistent 0.5σ drift will eventually push the CUSUM past the threshold.
• Optimal for Small Shifts: CUSUM is theoretically the most efficient fixed-sample-size test for detecting a sustained shift of known magnitude.
• V-Mask Alternative: An equivalent graphical approach uses a V-shaped mask placed on the cumulative sum plot — the process is out of control if the plotted path crosses the mask boundaries.

CUSUM vs. EWMA vs. Shewhart

Semiconductor Applications

• Etch Rate Drift: Detecting gradual etch rate changes of 0.5–1% that accumulate over many lots.
• Film Thickness Trends: Identifying CVD deposition rate drift before it impacts yield.
• Overlay Monitoring: Detecting systematic overlay drift between lithography maintenance cycles.
• Tool Degradation: Monitoring gradual performance degradation that signals upcoming maintenance needs.

Practical Considerations

• Resetting: After an alarm and corrective action, the CUSUM is reset to zero.
• Two-Sided: Separate upper and lower CUSUMs detect shifts in both directions.
• ARL (Average Run Length): The key performance metric — how quickly (in number of samples) the CUSUM detects a shift. Smaller ARL = faster detection.

CUSUM is the mathematically optimal method for detecting small persistent process shifts — it is the gold standard when sensitivity to drift matters more than simplicity.