CUSUM chart
The CUSUM (Cumulative Sum) chart is a time-weighted control chart that detects small, sustained shifts in the process mean by accumulating deviations from a target value over time. Unlike Shewhart charts, CUSUM is sensitive to gradual drift and persistent shifts that would otherwise go unnoticed.
Map the data fields
- Subgroup/Lot: The sequence, time, or batch identifier for each observation.
- Process value: The measured values to be monitored.
- UCL: Optional field for a custom upper control limit (overrides the calculated UCL).
- LCL: Optional field for a custom lower control limit (overrides the calculated LCL).
- Central line: Optional field for a custom center line (overrides the default center of 0).
Calculation Methodology
Sigma Estimation
When no standard deviation is provided, sigma is estimated from the average moving range:
MR[i] = |y[i] - y[i-1]| for i = 1 … n
MR̄ = mean of all MR[i]
σ = MR̄ / 1.128 (d₂ constant for subgroup size n = 2)
Target
Target = user-specified target OR ȳ (mean of all observations)
Allowance (k) and Decision Interval (h)
k = K × σ (K is the allowance multiplier, typically 0.5)
h = H × σ (H is the decision interval multiplier, typically 4 or 5)
Cumulative Sums
Two one-sided statistics are tracked simultaneously:
Upper CUSUM (C⁺) — detects upward shifts:
C⁺[0] = max(0, y[0] - (Target + k))
C⁺[i] = max(0, y[i] - (Target + k) + C⁺[i-1])
Lower CUSUM (C⁻) — detects downward shifts:
C⁻[0] = max(0, (Target - k) - y[0])
C⁻[i] = max(0, (Target - k) - y[i] + C⁻[i-1])
Plotted Line
Primary line[i] = C⁺[i] - C⁻[i]
Control Limits
UCL = +h
LCL = -h
Center line = 0
Notes:
- If custom limits are provided, they override the calculated values
- A signal occurs when C⁺[i] > h (upward shift detected) or C⁻[i] > h (downward shift detected)
- Smaller K detects smaller shifts but produces more false alarms; K = 0.5 is the standard choice for detecting a 1σ shift
- Larger H reduces false alarms at the cost of a slower response; H = 4–5 is common