Fifty readings with the process moving 0.9 units — 0.9 sigma — at reading 26. Not one reading crosses a three-sigma limit, and an individuals chart on this data flags nothing at all, by any of the eight Nelson rules. CUSUM signals at reading 26, one after the shift. EWMA at λ = 0.2 signals at 28, three after it. This is the regime these charts exist for: a shift too small for a Shewhart chart and too sustained to ignore.
50 readings · λ = 0.2 · CUSUM k = 0.5, h = 4
EWMA never signals, CUSUM at 33
The EWMA statistic carries about 9.0 readings of history and its limits settle at ±0.8873 around 100.5974. An individuals chart on the same readings flags 0 of 50 by any of the eight rules, with 0 outside its limits. Sigma is estimated from the average moving range at 0.8873.
EWMA first signal
none
0 readings beyond
CUSUM first signal
33
3 readings signalling
Shewhart comparison
0 beyond
0 flagged by any rule
Effective window
9.0
steady limits ±0.8873
How λ changes the chart
The EWMA chart at four smoothing constants
λ
History carried
Steady half-width
First signal
0.05
39.0 readings
±0.4263
never
0.1
19.0 readings
±0.6107
never
0.2
9.0 readings
±0.8873
never
0.4
4.0 readings
±1.3310
never
A smaller λ carries more history and draws narrower limits, which detects a smaller shift and reacts more slowly to a sudden one. The two effects partly cancel, so a bigger λ does not reliably signal sooner — on this data all four settle within a couple of readings of each other.
Both statistics, reading by reading
The EWMA statistic and both CUSUM sums at each reading
#
Value
EWMA
Its limits
CUSUM high
CUSUM low
1
100.350
100.5479
100.065 to 101.130
0.000
0.000
2
101.650
100.7683
99.916 to 101.279
0.686
0.000
3
100.030
100.6207
99.835 to 101.360
0.000
0.139
4
99.890
100.4745
99.788 to 101.407
0.000
0.437
5
99.780
100.3356
99.759 to 101.436
0.000
0.858
6
101.300
100.5285
99.741 to 101.454
0.292
0.000
7
98.460
100.1148
99.730 to 101.465
0.000
1.909
8
100.610
100.2138
99.723 to 101.472
0.000
1.395
9
99.420
100.0551
99.718 to 101.477
0.000
2.222
10
101.010
100.2461
99.715 to 101.480
0.000
1.257
11
99.460
100.0888
99.713 to 101.481
0.000
2.038
12
100.630
100.1971
99.712 to 101.483
0.000
1.502
13
100.990
100.3557
99.711 to 101.483
0.000
0.559
14
100.360
100.3565
99.711 to 101.484
0.000
0.327
15
100.190
100.3232
99.711 to 101.484
0.000
0.286
16
98.970
100.0526
99.710 to 101.484
0.000
1.620
17
99.610
99.9641
99.710 to 101.484
0.000
2.233
18
99.790
99.9293
99.710 to 101.485
0.000
2.643
19
100.600
100.0634
99.710 to 101.485
0.000
2.140
20
99.520
99.9547
99.710 to 101.485
0.000
2.854
21
101.660
100.2958
99.710 to 101.485
0.698
1.156
22
100.460
100.3286
99.710 to 101.485
0.043
0.811
23
100.960
100.4549
99.710 to 101.485
0.000
0.000
24
102.080
100.7799
99.710 to 101.485
1.171
0.000
25
100.950
100.8139
99.710 to 101.485
1.068
0.000
26
101.400
100.9311
99.710 to 101.485
1.473
0.000
27
100.140
100.7729
99.710 to 101.485
0.457
0.015
28
102.660
101.1503
99.710 to 101.485
2.282
0.000
29
101.610
101.2423
99.710 to 101.485
2.923
0.000
30
101.270
101.2478
99.710 to 101.485
3.181
0.000
31
100.620
101.1223
99.710 to 101.485
2.707
0.000
32
101.670
101.2318
99.710 to 101.485
3.415
0.000
33
102.080
101.4014
99.710 to 101.485
4.586 *
0.000
34
101.060
101.3332
99.710 to 101.485
4.608 *
0.000
35
101.100
101.2865
99.710 to 101.485
4.674 *
0.000
36
99.860
101.0012
99.710 to 101.485
3.343
0.331
37
100.410
100.8830
99.710 to 101.485
2.632
0.042
38
99.470
100.6004
99.710 to 101.485
0.861
0.813
39
100.970
100.6743
99.710 to 101.485
0.781
0.000
40
102.140
100.9674
99.710 to 101.485
2.020
0.000
41
101.150
101.0040
99.710 to 101.485
2.142
0.000
42
101.790
101.1612
99.710 to 101.485
2.986
0.000
43
100.110
100.9509
99.710 to 101.485
1.937
0.049
44
99.400
100.6407
99.710 to 101.485
0.088
0.899
45
100.180
100.5486
99.710 to 101.485
0.000
0.869
46
100.680
100.5749
99.710 to 101.485
0.000
0.276
47
100.280
100.5159
99.710 to 101.485
0.000
0.134
48
101.100
100.6327
99.710 to 101.485
0.066
0.000
49
100.880
100.6822
99.710 to 101.485
0.000
0.000
50
99.110
100.3677
99.710 to 101.485
0.000
1.176
The EWMA limits widen over the first few readings because the statistic has not yet accumulated its full history. The CUSUM sums reset to zero whenever they would go negative, which is what keeps the chart from accumulating credit during a good spell and then spending it during a bad one.
Detects small shifts EWMA and CUSUM together Shewhart shown for contrast Slow on a sudden jump
What this tool shows
The shipped preset shifts by 0.9 sigma at reading 26, and a Shewhart chart never finds it — not by a limit and not by any of the eight Nelson rules, across all fifty readings. CUSUM signals at reading 26, one after the shift. EWMA at λ = 0.2 signals at 28. Both charts run on the same data here, with the Shewhart result printed beside them, so the comparison is measured rather than asserted.
EWMA with a settable smoothing constant and limits that widen over the first few readings
A tabular CUSUM on the same data, with both one-sided sums shown
An individuals chart computed alongside, as the baseline both are meant to beat
A λ sensitivity table: history carried, steady-state limit width and first signal
Every reading with its EWMA statistic and both CUSUM sums
A stable preset where all three charts correctly signal nothing
Small shifts EWMA and CUSUM Shewhart for contrast λ sensitivity table
Set the target and sigma from a stable baseline. Estimating them from the data hides the shift.
Updated 13 September 2026 · Works in any browser, no installation
An EWMA chart plots a weighted average of every reading so far, with the weight decaying geometrically into the past. A Shewhart chart looks at one reading at a time and throws the rest away, which makes it fast on a large jump and nearly blind to a small sustained one. Carrying the past forward reverses that trade. CUSUM does the same thing differently — accumulating how far readings have strayed from target — and the two usually agree to within a reading or two.
At a glance
Formula shown
EWMA: zᵢ = λxᵢ + (1 − λ)zᵢ₋₁ starting from the target, with limits μ ± L·σ·√[(λ/(2−λ))(1 − (1−λ)^{2i})]. Those limits widen over the first readings and settle at L·σ·√(λ/(2−λ)). The statistic carries roughly (2 − λ)/λ readings of history. CUSUM: Cᵢ⁺ = max(0, Cᵢ₋₁⁺ + zᵢ − k) and Cᵢ⁻ = max(0, Cᵢ₋₁⁻ − zᵢ − k), signalling when either exceeds h. The reset at zero is what stops a good spell from banking credit against a later bad one.
Scenario support
Detecting slow process drift, monitoring a chemical or pharmaceutical process where a small sustained shift matters, watching a service metric that degrades gradually, any automated monitor where a Shewhart chart alarms too late, and validating that a process change actually took effect.
Educational estimate
Planning support from the values you enter — not professional advice.
A shift no Shewhart rule finds
The first preset is built to fail every Shewhart detector at once, and the failure is complete rather than marginal.
Fifty readings, a 0.9-sigma shift at reading 26. With a known target of 100 and sigma of 1, not one reading crosses a three-sigma limit.
An individuals chart with limits estimated from the data flags nothing at all, by any of the eight Nelson rules — not a run, not a two-of-three, not a four-of-five.
CUSUM signals at reading 26. One reading after the shift began.
EWMA at λ = 0.2 signals at 28. Three readings after. Both are on the page, and so is the Shewhart result, so none of this has to be taken on trust.
λ is a choice about what you are looking for
The smoothing constant is the only real decision an EWMA chart asks for, and it trades two things against each other in a way that partly cancels.
A small λ carries more history. At 0.05 the statistic holds about 39 readings; at 0.4, about 4.
More history means narrower limits. The steady-state half-width goes from 0.4804 at λ = 0.05 to 1.5000 at λ = 0.4, on the same sigma.
So a small λ detects a smaller shift and reacts more slowly to a sudden one, and the two effects work against each other — which is why all four settings on the shipped data signal within a couple of readings of each other.
The usual range is 0.05 to 0.25, with smaller values for smaller shifts. λ = 1 reduces the chart to a Shewhart chart exactly, which is the useful way to see what the memory is buying.
EWMA or CUSUM
They perform almost identically, and they are not interchangeable in practice for reasons that have little to do with statistics.
CUSUM is tuned for one shift size. k is half the shift you care about in sigmas, so k = 0.5 targets a one-sigma shift, and it is close to optimal at that size and worse elsewhere.
EWMA is tuned by λ and is more forgiving across a range of shift sizes. It is the better default when the shift size is not known in advance.
EWMA plots on the original scale. Its statistic is in the units of the measurement and sits near the process mean, which operators read easily; a CUSUM sum is a dimensionless accumulation that needs explaining.
CUSUM makes the change point visible. The reading where the sum left zero is an estimate of when the shift began, which an EWMA does not give directly — and which is often the first question after a signal.
The target must come from somewhere else
These charts are more sensitive than a Shewhart chart, and that sensitivity is entirely lost if the baseline is estimated from contaminated data.
A shift inside the data pulls the estimated centre toward itself, so the post-shift readings sit closer to the centre than they should and the chart has less to detect.
The same applies to sigma. Estimating it from the whole series inflates it and widens the limits, twice over.
So phase one matters more here than on a Shewhart chart: establish the target and sigma on a period known to be stable, then hold both fixed.
This calculator estimates both from the data in the box, which is the right default for exploring a dataset and the wrong one for running a process. The moving-range estimate is used rather than the series standard deviation precisely because it resists a shift.
Why the limits are curved at the start
The EWMA limits flare outward over the first readings rather than sitting flat, and it is not a cosmetic choice.
At reading one the statistic is a weighted average of one reading and the target, so its variance is much smaller than it will eventually be.
The exact variance carries a (1 − (1−λ)²ⁱ) factor that rises from λ/(2−λ) toward 1 as readings accumulate.
Using the steady-state limits from the start would make the chart far too insensitive early on, which is exactly when a newly started or newly adjusted process most needs watching.
The flare disappears within about 3/λ readings, so at λ = 0.2 the limits are effectively flat after fifteen.
What memory costs
Carrying the past forward is not free, and the cost falls in exactly the cases a Shewhart chart handles well.
A single wild reading barely moves an EWMA. At λ = 0.2 it contributes 20% of one point, so a spike that a Shewhart chart flags instantly can pass unnoticed.
Which is why these charts are usually run alongside a Shewhart chart rather than instead of one. Together they cover both failure modes.
They also inherit the inertia problem: after a large shift the statistic takes several readings to return, so a process corrected immediately can keep signalling.
And they are harder to explain. A point outside a limit on a Shewhart chart is a reading someone took; a point outside a limit on an EWMA is a statistic, and that difference decides whether a chart survives contact with a shop floor.
Reporting these charts
Four items, and the first two are what make the chart reproducible at all.
Give λ and L for an EWMA, or k and h for a CUSUM. Without them the chart cannot be redrawn and its sensitivity cannot be judged.
Say where the target and sigma came from. Estimated from this data or carried from a baseline — the two give materially different charts.
Say whether a Shewhart chart was run alongside. These charts are poor at spikes, and a monitoring scheme that uses only one of the two has a known blind spot.
And give the reading at which the signal occurred, not just that one occurred. On a CUSUM the point where the sum left zero estimates when the shift began, which is the actionable part.
Method. Both charts are computed on the same readings in the same pass, and an individuals chart is computed alongside them, because the claim on this page is a comparison and a comparison run on different data is not one. The EWMA limits use the exact time-varying variance rather than the steady-state approximation, so they flare over the first readings and settle afterwards; the suite asserts on 120 generated series that they widen monotonically and never exceed the steady-state width. Sigma comes from the average moving range rather than the series standard deviation, for the same reason it does on an individuals chart: a sustained shift inflates the second and leaves the first alone. The suite also asserts that a constant series produces a constant EWMA, that the effective window is exactly (2 − λ)/λ, and that a series sitting exactly on target accumulates nothing at all in either CUSUM sum. That engine is verified on every change against 139 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Individuals Control ChartIndividuals and moving range charts with sigma from the average moving range, both sigma estimates shown, and all eight Nelson rules counted.
X-bar and R ChartX-bar with R or S charts, every Shewhart constant computed from its definition rather than looked up, and all eight Nelson rules.
Moving AverageSimple, exponential, weighted and centred moving averages with the lag each one carries — exactly (n−1)/2 periods for a simple average, which is also the reason α = 2/(n+1) is in every EMA formula.
p Chartp and np charts with limits computed per point when the sample sizes differ, and the cost of flat limits measured on your own chart.
CpkCp, Cpk, Pp and Ppk with the defect rates they predict and the rate actually observed — including the built-in case where Cp is 2.05, Cpk is 0.57 and a quarter of the sample is already out of spec.
AutocorrelationACF and PACF at every lag with Bartlett bands that widen rather than a flat line, plus the shapes that identify a model.
An educational tool. EWMA and CUSUM are built for small sustained shifts and are poor at single spikes — a wild reading contributes only λ of its size to an EWMA — so they are normally run alongside a Shewhart chart rather than instead of one. Both also assume the target and sigma come from a stable baseline; estimating them from data that already contains the shift removes most of the sensitivity they exist for.
Launched EWMA and CUSUM computed on the same readings, with an individuals chart alongside as the baseline both are meant to beat.
Shipped a 0.9-sigma shift that crosses no Shewhart limit and breaks none of the eight Nelson rules across fifty readings; CUSUM signals one reading after it and EWMA three.
Used the exact time-varying EWMA variance so the limits flare over the first readings rather than sitting flat.
Added a lambda sensitivity table showing history carried, steady-state limit width and first signal together.
Documented what memory costs — a single spike contributes only lambda of its size, so these charts are run alongside a Shewhart chart rather than instead of one.
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