CUSUM, EWMA or Shewhart: which should I use, and when?

Three charts, three ways of looking

A Shewhart chart compares each point or subgroup statistic with control limits, usually at ±3σ. It reacts quickly to a large jump, but a small shift inside the limits can go unnoticed for a long time.

CUSUM accumulates deviations: each value's difference from target, minus a slack k, feeds a running sum that cannot fall below zero. When the sum exceeds a decision limit h, the chart signals. EWMA averages the data with exponentially decreasing weights set by λ, so recent points count more. Both detect small sustained shifts earlier than Shewhart.

Common starting points for CUSUM are k around 0.5σ and h around 4 to 5σ. For EWMA they are λ between 0.05 and 0.25 and L around 2.7 to 3. Check them against your data; they are not standards.

A hypothetical CUSUM table

Hypothetical: μ0 = 10, σ = 1, k = 0.5, so each step adds x − 10.5.

• t = 1, x = 10.2: S⁺ = max(0, 0 − 0.3) = 0
• t = 2, x = 10.8: S⁺ = max(0, 0 + 0.3) = 0.3
• t = 3, x = 11.0: S⁺ = 0.3 + 0.5 = 0.8
• t = 4, x = 10.1: S⁺ = max(0, 0.8 − 0.4) = 0.4
• t = 5, x = 11.4: S⁺ = 0.4 + 0.9 = 1.3
• t = 6, x = 11.6: S⁺ = 1.3 + 1.1 = 2.4

None of the six values is beyond ±3σ, so Shewhart gives no signal. With h = 4σ = 4, CUSUM also gives no signal yet, but the sum has risen from 0 to 2.4 and every value sits above target. Four more points near 11 push it past 4.

Tuning and reading mistakes

The first mistake is ignoring autocorrelation. Measurements that depend on the previous one, such as a slowly moving temperature, can make CUSUM and EWMA signal too often, so check the residuals before setting limits. The second is estimating sigma during a period that was already drifting; use a stable in-control period instead. The third is choosing k, h, λ and L by feel. They should follow from the shift you care about and the average run length (ARL) you accept.

Follow-up questions