CUSUM Control Chart: Cumulative Sums Detect Small Persistent Shifts
What is the CUSUM Control Chart?
CUSUM accumulates each sample's deviation from the target: C⁺ accumulates positive deviations and C⁻ negative ones, forming a combined C⁺/C⁻ chart. Tabular CUSUM updates C_i = max(0, C_{i−1} + (x_i − μ0 − k)) and signals when the cumulative sum exceeds the decision interval h. A small shift that is invisible in single points keeps growing the cumulative sum until it crosses the limit.
When to Use It
Use CUSUM when a persistent small mean shift (0.5–2σ) must be detected quickly: pharmaceutical and chemical continuous processes, and sensitive monitoring of critical characteristics. k is typically 0.5σ (half the shift to detect) and h is 4–5σ; when σ is unknown, estimate it from the moving range. CUSUM also applies to count data, so its scope is wider than often assumed.
How to Use It (Step by Step)
Enter the observations and the target μ0, set k and h, and the tool plots the C⁺ and C⁻ curves and flags any crossing of h. After a signal, confirm the assignable cause on the floor, then reset both cumulative sums to zero and continue monitoring — the standard Tabular CUSUM operating procedure.
Key Formulas / Example
C_i⁺ = max(0, C_{i−1}⁺ + x_i − μ0 − k) and C_i⁻ = max(0, C_{i−1}⁻ − x_i + μ0 − k); signal when either exceeds h. k = 0.5σ is optimal for detecting a 1σ shift, and h = 4 or 5 corresponds to a false-alarm rate around 0.2–0.5%: h = 4 is more sensitive, h = 5 alarms less.