EWMA vs CUSUM Charts: Choosing a Small-Shift Detection Method When Shewhart Isn't Enough

Your X-bar chart has run in control for three weeks. No points outside the limits, no obvious runs. But the last dozen subgroup averages have all landed a hair above the center line, and your scrap rate crept up half a percent. The chart says nothing is wrong. A 0.75σ drift is exactly the kind of shift a Shewhart chart is statistically built to ignore — and the two charts designed to catch it are the CUSUM and the EWMA.

Both detect small, sustained shifts far faster than a Shewhart chart, and for a 1σ shift they perform almost identically: roughly ten subgroups to signal, versus about forty-four for a Shewhart chart watching for a single point beyond 3σ. The real question is not whether to use one — it is which one fits your process, your shift size, and the people reading the chart. This post lays out how each works, where they diverge, and a verdict by the kind of work you do.

Why a Shewhart chart misses the drift

A Shewhart chart — the X-bar R, I-MR, or X-bar S chart you run daily — evaluates each subgroup against the control limits in isolation. Each point is judged on its own; the chart has no memory. That is a deliberate design choice: it keeps the chart simple and easy to read, but it also means a shift smaller than about 1.5σ rarely pushes a single point past the 3σ limit. The process can drift and stay drifted for a long time before one point happens to land outside.

Pattern rules close part of the gap. The Western Electric and Nelson run rules add memory by flagging patterns — eight points on one side of the center line, six points trending — so a sustained small shift eventually trips a rule. But the run rules inflate the false-alarm rate when you stack several of them, and they still lag a chart engineered specifically for small shifts. When small-shift detection is the whole point, the CUSUM and EWMA are the purpose-built tools. (For why the Shewhart chart loses this race in the first place, see the companion comparison on CUSUM versus Shewhart charts.)

How the CUSUM chart works

The cumulative sum (CUSUM) chart accumulates the deviation of each measurement from the target. Instead of plotting the measurement, it plots a running tally that resets to zero whenever the process is on target and grows whenever the process drifts. The tabular (one-sided) form tracks an upper and lower sum:

Key Formula $$C_i^{+} = \max\left(0,\; x_i - (\mu_0 + K) + C_{i-1}^{+}\right)$$ $$C_i^{-} = \max\left(0,\; (\mu_0 - K) - x_i + C_{i-1}^{-}\right)$$

Here $\mu_0$ is the process target, $K$ is the reference value (the slack), and the chart signals when either sum exceeds the decision interval $H$. The two tuning constants are usually set in standard-deviation units: $K = k\sigma$ with $k = 0.5$, and $H = h\sigma$ with $h = 4$ or $5$. Setting $K$ to half the shift you most want to catch is what makes the CUSUM optimal for that specific shift size — the NIST/SEMATECH e-Handbook section on CUSUM charts works through the parameter selection in full.

Worked Example A process targets μ₀ = 100 with σ = 2. You want fast detection of a 1σ (2-unit) shift, so set K = 0.5σ = 1 and H = 5σ = 10. The upper sum uses a reference of μ₀ + K = 101. For subgroup averages 100, 101, 102, 101, 103, 102 the upper CUSUM runs 0, 0, 1, 1, 3, 4 — climbing steadily toward the H = 10 signal as the small upward drift accumulates. A Shewhart chart, with its 3σ limit at 106, would not have flinched at any of these points.

How the EWMA chart works

The exponentially weighted moving average (EWMA) chart plots a weighted average of all measurements to date, with the most recent reading weighted most heavily and older readings fading geometrically. The statistic is a single recursive line:

Key Formula $$z_i = \lambda x_i + (1 - \lambda) z_{i-1}, \qquad z_0 = \mu_0$$

The smoothing constant $\lambda$ (between 0 and 1) sets how much memory the chart carries: a small $\lambda$ weights history heavily and reacts to tiny shifts, a large $\lambda$ behaves more like a Shewhart chart. Practitioners typically use $\lambda$ between 0.05 and 0.25, with 0.2 a common default (the NIST/SEMATECH e-Handbook section on EWMA charts details the limit derivation). The control limits tighten as data accumulates and settle at an asymptotic width:

Key Formula $$\text{UCL},\,\text{LCL} = \mu_0 \pm L\,\sigma\sqrt{\frac{\lambda}{2-\lambda}}$$

Worked Example Same process: μ₀ = 100, σ = 2. With λ = 0.2 and L = 3, the asymptotic limit half-width is $3 \times 2 \times \sqrt{0.2 / 1.8} = 6 \times 0.333 = 2.0$, giving an EWMA UCL of 102.0 and LCL of 98.0. The EWMA line only has to clear ±2 units to signal — a third of the ±6-unit gap an individuals Shewhart chart imposes. That tighter band is exactly why the EWMA catches the half-percent drift the X-bar chart slept through.

EWMA vs CUSUM, side by side: where they actually differ

For the common designs — CUSUM with $k=0.5, h=5$ and EWMA with $\lambda=0.2, L=3$, both tuned to an in-control average run length (ARL) near 465–480 — detection speed is nearly a tie across the whole range of shift sizes. The differences that matter in practice are about tuning, interpretation, and edge behavior.

The ARL values above are the approximate, widely-published figures for these standard designs (Montgomery, Introduction to Statistical Quality Control; NIST/SEMATECH e-Handbook). Read them as "average subgroups until the chart signals" — both small-shift charts cut the 1σ detection time to roughly a quarter of the Shewhart chart's.

Criterion	CUSUM	EWMA
Best at	A shift size you can name in advance (set K to half of it)	Very small shifts (±0.5–1σ) and unknown shift sizes
Tuning parameters	Two (k and h) — less intuitive	One main one (λ) — easier to explain
What you plot	A cumulative sum that resets at zero	A smoothed average near the data scale
Operator readability	Lower — the sum is abstract	Higher — the line tracks the measurement
Behavior with non-normal data	Sensitive to the normality assumption	More tolerant, especially at small λ
Reaction to a large spike	Signals quickly	Smooths it — can lag a sudden large shift

Common Mistake Running a CUSUM or EWMA without first proving the process is stable and the measurement system is adequate. Both charts assume a known, reliable target μ₀ and σ. If your Gage R&R is marginal, a small-shift chart will faithfully detect measurement drift and route your team to chase a phantom process problem.

The verdict, by what you do

Neither chart is universally better — their detection curves nearly overlap. Pick on context:

If you know the shift you fear (a tool-wear signature that always drifts about 1σ before a part goes out of spec): use a CUSUM and set K to half that shift. Naming the target shift is the CUSUM's home advantage.
If you want one chart for a range of small shifts and don't want to retune per failure mode: use an EWMA with λ = 0.2. It is the more forgiving general-purpose small-shift chart.
If operators read the chart on the floor: lean EWMA. Its line tracks near the measurement scale, so a technician can see the trend without translating an abstract cumulative sum.
If your data are noticeably non-normal (counts, skewed dimensions): lean EWMA at a small λ, which tolerates departures from normality better than the CUSUM.
If you mostly face sudden, large shifts: you may not need either — a Shewhart chart already catches those in one or two points. Reserve CUSUM and EWMA for the slow drift it misses, and consider running a Shewhart chart alongside for large excursions.

A practical pattern in many shops is to run a Shewhart chart for gross excursions and overlay an EWMA or CUSUM for the slow drift — the two answer different questions. If you are still deciding which base chart fits your data type and subgroup structure, the control chart selection guide covers that decision first. Once your data are loaded, the control chart tool computes the EWMA and CUSUM statistics and their limits from the standard design parameters, so you can compare the two detection curves on your own process rather than on a textbook example.