CUSUM & EWMA Control Charts

CUSUM and EWMA detect small, sustained process shifts that standard Shewhart charts miss. By accumulating information across consecutive points, they react faster to gradual drift.

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Step 1 — Choose chart type

Chart	What it does
CUSUM	Accumulates deviations from target; very sensitive to small sustained shifts
EWMA	Weighted moving average; tunable sensitivity
Both	Side by side comparison

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Step 2 — Parameters

CUSUM:

• Target (in-control mean)
• k = 0.5 (reference value, in sigma units)
• h = 5 (decision interval, in sigma units)

EWMA:

• Target (in-control mean)
• lambda = 0.2 (smoothing constant)
• L = 3 (control limit width)

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Step 3 — Results

CUSUM: C+ and C- cumulative sums, signals when exceeding h, CUSUM plot.

EWMA: Smoothed statistic, time-varying control limits, signals when crossing limits, EWMA plot.

A UniversalChatBot is available for discussion.

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Statistical methods used

CUSUM:

z_i = (x_i - target) / sigma
C+_i = max(0, C+_{i-1} + z_i - k)
C-_i = max(0, C-_{i-1} - z_i - k)
Signal if C+ > h or C- > h

EWMA:

z_i = lambda * x_i + (1 - lambda) * z_{i-1}
UCL_i = target + L * sigma * sqrt((lambda/(2-lambda)) * (1-(1-lambda)^(2(i+1))))
Signal if z_i > UCL or z_i < LCL

EWMA limits widen then stabilise — correctly reflecting growing variance of the weighted average.

Comparison:

• Both excellent for small sustained shifts
• Shewhart better for large sudden shifts
• Best practice: run Shewhart alongside CUSUM/EWMA