EWMA Chart Calculator — Exponentially Weighted Moving Average
Paste a column of measurements and get the exponentially weighted moving average with exact time-varying control limits, NIST’s asymptotic limits one click away, and the ordinary individuals chart of the same data drawn underneath so you can see what the smoothing bought.
When to reach for an EWMA chart instead of a Shewhart chart
Use it when the failure mode you are worried about is a small sustained shift — a tool wearing, a reagent degrading, a supplier quietly changing something. A Shewhart individuals chart looks at one reading at a time, so a process that has moved by one standard deviation still puts almost every reading inside its 3σ limits and the chart can run for dozens of points without a signal. An EWMA looks at a weighted average of everything so far, whose limits are much tighter, and it walks out of them.
Do not use it as a replacement. Smoothing is exactly as good at hiding a single large excursion as it is at revealing a slow drift: one wild reading contributes only λ of its size to the statistic. The two charts answer different questions, and the reason this page draws both is that neither answer is complete on its own. The sentence between the panels reports which of them signalled, and when.
The rough division most texts settle on: below about 1.5σ the time-weighted charts (EWMA and CUSUM) detect a sustained shift several times faster; above about 2σ the Shewhart chart is at least as fast and is the one whose false-alarm behaviour everybody has a table for.
Data
Live preview — unlock to use your own data
The sample dataset below is real and the chart beside it is live: change any option and watch the limits move, then export the figure. Entering your own measurements into EWMA Chart needs a licence — a one-time payment, no account, no subscription.
This is the sample dataset, shown in full. The chart beside it is live — change any option and watch it redraw.
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0 rows of data
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Enter data on the left to chart it
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Options
This plot has no adjustable options yet.
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How the statistic and the limits are computed
EWMAₜ = λ·Yₜ + (1 − λ)·EWMAₜ₋₁, starting from EWMA₀ = the centre line. Unrolled, that makes the statistic a weighted sum of every observation so far with weights decaying geometrically — which is why λ is the setting that matters most, and why it is printed on the figure.
The limits are the centre line ± k·σ̂·√(λ/(2−λ)·(1 − (1−λ)^2ᵗ)). The λ/(2−λ) term is the variance of the weighted average relative to a single observation; at λ = 0.2 its square root is 0.333, so the limits sit at a third of the width of the individuals limits on the same σ̂. The (1 − (1−λ)^2ᵗ) term corrects for the fact that at t = 1 the statistic has not accumulated its full variance yet, so the early limits are narrower still.
Most published examples drop that second term and draw a flat pair at the asymptotic width — including the NIST/SEMATECH worked example this tool reproduces. Both forms are offered, the exact one is the default because it is the correct standard error at every point, and the figure always says which is drawn. If an existing chart disagrees with this one over the first few points, this is why.
σ̂ scales the limits and never touches the plotted points. Changing the sigma source moves the limits and leaves the curve exactly where it was — worth knowing, because it means a disagreement about σ̂ and a disagreement about λ look completely different on the chart.
Choosing λ, and the one thing it is not
0.2 is the default here and in Minitab, and 0.05 to 0.25 is the usual working range. Smaller λ gives the statistic a longer memory: better at the smallest shifts, slower to react to a large one, more sensitive to the assumption that the process was on target when charting began. Larger λ moves the chart back towards the raw data, and at λ = 1 it *is* the raw data — the chart collapses onto the individuals chart exactly, which is worth trying once for the demonstration.
λ is not a smoothing preference. It is a decision about which size of shift you are buying detection for, and it is chosen together with the sigma multiple: every published average-run-length table for this chart is indexed by the pair, never by either alone. Changing λ after seeing the data is how a chart becomes an argument.
Why the run rules are switched off, and not quietly
Nelson, Western Electric and ISO 7870-2 all count patterns across consecutive points — nine in a row on one side, six in a row rising, four of five beyond 1σ. Every one of those tests assumes the points are independent. Consecutive EWMA values share all but one of their inputs; at λ = 0.2 two neighbouring points are about 96% the same number, and a long run on one side of the centre line is what a weighted moving average *does*, not evidence about the process.
So only the beyond-the-limits test is applied. The rest are listed on the figure as suppressed, with the reason, rather than silently reported as passed — the same treatment the run chart gives the tests that need a sigma it does not have. A tool that returns "no violations" for a test it never ran is worse than a tool that offers no tests.
What the tool will not do quietly
It will not centre the chart on data that contains the shift. The centre line is also EWMA₀, so it moves every plotted point and not just the horizontal rule. If the mean is taken over a series that has already drifted, the centre line lands between the two levels and the chart flags the settled period as low and the drifted period as high — two signals, neither of which locates the change. Set a target, or set a baseline over the period you believe was stable, and the figure will say which you did.
It will not carry the statistic across a declared process change. Mark the first row of a new campaign and the recursion restarts at the centre line, because an EWMA carried over a deliberate change is mostly a memory of the process that ended.
Common questions
- What lambda should I use for an EWMA chart?
- 0.2 with limits at 3σ is the conventional starting pair and this tool’s default. Use 0.05–0.1 if the shift you fear is under half a sigma and you can tolerate a slower response to anything larger; use 0.3–0.4 if you mostly want a lightly smoothed individuals chart. Pick it before you look at the data, and record it — the figure prints it for exactly that reason.
- Why don’t my EWMA control limits match the ones in my textbook?
- Almost always the limit form. Textbooks and the NIST handbook usually draw the asymptotic limits, a flat pair at ±k·σ̂·√(λ/(2−λ)). This tool defaults to the exact time-varying limits, which are narrower over the first several points and converge to that pair. Switch "Limit form" to asymptotic and they agree. The second most common cause is the sigma: an EWMA chart’s limits are the individuals limits scaled, so any disagreement about σ̂ shows up here too.
- Is an EWMA chart better than an I-MR chart?
- It is better at one thing: detecting a sustained shift of roughly 0.5σ to 1.5σ, where it signals several times sooner. It is worse at catching a single large excursion, because the smoothing that helps in the first case hurts in the second. Run both — this page draws both — and quote the Shewhart chart when both signal, because that is the one an auditor has a reference for.
- Can I use an EWMA chart on subgroup averages?
- Yes, and it is common. Paste the averages one per row and use a σ that is the standard error of the average rather than the process standard deviation — paste the raw σ and the limits come out about √n too wide. The CUSUM page’s worked example does exactly this with subgroups of four.
- Is my data uploaded anywhere?
- No. The whole calculation runs as JavaScript in this browser tab, and nothing on this site accepts a measurement — the only server code here takes a payment. That is why the tool is usable on data you are contractually barred from putting in a cloud service.