What does λ (lambda) control in an EWMA chart?

λ is the weight given to the newest point; 1−λ is how much of the running average survives each step. Small λ (0.05–0.2) gives long memory and the best sensitivity to small persistent shifts; large λ approaches a raw-point chart. A common default is λ = 0.2 with L = 3 limits, which is what ValidAnytime's warning tier runs.

Why do EWMA control limits widen over time?

Because the statistic's own variance grows from zero (E_0 = 0) toward its steady state: σ_E(t) = √(λ/(2−λ)·(1−(1−λ)^{2t})). Exact charts use these time-varying limits so early points are judged fairly; charts that use only the asymptotic limit are slightly too lenient at the start.

Is an EWMA chart's false-alarm rate guaranteed?

No — its calibration assumes independent, correctly standardized observations, and it is not anytime-valid. On autocorrelated production metrics the real false-warning rate can be far above nominal. That is why ValidAnytime runs the EWMA chart as an unbudgeted warning and delivers pages from the e-process tier, whose false-alarm control is a theorem rather than a model assumption.

EWMA vs a rolling average with bands?

A rolling band re-fits its mean and sigma on a trailing window, so it adapts to — and eventually absorbs — a slow regression. An EWMA chart standardizes against a frozen baseline, so a slow drift keeps accumulating distance and eventually fires. If your failure mode is the quiet slide rather than the spike, the frozen baseline is the feature.

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EWMA chart

An EWMA control chart tracks an exponentially weighted moving average of a metric and fires when that smoothed value leaves its control limits — which makes it the classical chart of choice for small, persistent shifts that individual-point rules never see. Each new point gets weight λ and history decays geometrically, so the chart has one interpretable knob: how long its memory is.

WARN

Also known as: exponentially weighted moving average control chart

In ValidAnytime, the EWMA chart runs in the warning tier: early and smooth, with false-warning calibration that is model-based (independent Gaussian data) rather than guaranteed — the budgeted page belongs to the e-process tier.

The rule, exactly

z_t = (x_t − μ̂) / σ̂        μ̂, σ̂ frozen after the first 30 points
E_t = λ·z_t + (1 − λ)·E_{t−1}        E_0 = 0
σ_E(t) = √( λ/(2−λ) · (1 − (1−λ)^{2t}) )
warn when |E_t| > L·σ_E(t), on the crossing

λ is the memory knob (small λ = long memory = sensitive to small shifts); L sets the limit width. The cloud suite runs λ = 0.2 with L = 3 and exact time-varying limits, and warns on the crossing only — no re-warn until the statistic re-enters its limits.

Live playground — synthetic data, labeled as such; runs entirely in your browser

break size3σ

memory λ0.20limit width L3.0

0 warnings on the 80 calm points · catches the break 1 point after onset.

Synthetic standard-normal noise (seeded, 160 uncadenced points — points, not minutes), with a labeled break at point 81 when one is enabled. The detector code running here is the same TypeScript implementation the /try bake-off runs, parity-tested against the cloud suite.

WARN

What is — and is not — guaranteed

Its calibration is a model-based promise: with independent Gaussian z-scores, L = 3 limits put the in-control false-warning rate near the familiar 3σ textbook figure.

That promise is not anytime-valid and not distribution-free: autocorrelation is the EWMA chart's known blind spot — the smoothed statistic inherits the data's serial dependence, and the false-warning rate on realistic textures can exceed nominal by orders of magnitude, measured openly in our benchmark.

Warning tier: classical control charts — sensitive and fast, but calibrated against a model rather than a guarantee, so warnings arrive unbudgeted. See how it scores in the benchmark.

When it wins

The shift is small and persistent: exponential weighting pools evidence across points, catching drifts of a fraction of a sigma that no single-point rule can see.
Noise is spiky: one outlier only moves the average by λ·z, so the chart shrugs off isolated glitches that fire a static threshold.
You want one knob: λ maps directly to "how many recent points matter", which makes tuning explainable to a reviewer.

When it lies

The data is autocorrelated: the EWMA statistic smooths *and* correlates, so its nominal limits are quietly wrong on exactly the wandering, seasonal metrics production systems emit.
The shift is a large, abrupt step: smoothing that protects against noise also dilutes a jump, so a Shewhart-style rule or CUSUM often catches big steps a few points sooner.
The chart is already outside its limits when the real incident begins: it warns on the crossing only, so a statistic parked out-of-bounds pages once and then goes silent.

Where it comes from

The EWMA control chart was introduced by S. W. Roberts in 1959 in Technometrics (“Control chart tests based on geometric moving averages”) and remains, with CUSUM, one of the two workhorse memory charts of statistical process control. The version here — exact time-varying limits, frozen training window — is the warning-tier chart the ValidAnytime cloud suite ships; the playground runs a line-for-line TypeScript port of it, golden-tested against the cloud implementation's own outputs.

Questions engineers ask

Related detectors

In the glossary: Changepoint detection · The peeking problem

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