EWMA
Overview
An EWMA (Exponentially Weighted Moving Average) control chart is a statistical process control tool used to detect small and gradual shifts in a process mean.
Unlike a Shewhart control chart, which evaluates each observation independently, an EWMA chart gives greater weight to recent observations while still retaining information from historical data. This makes EWMA particularly sensitive to subtle process changes that may develop over time.
In Cloud QC, μ₀ and σ are the CRM’s certified value and standard deviation when “Use CRM provided statistics” is ticked; otherwise they are the mean and standard deviation of the results on the chart.
When to use
EWMA charts are commonly used for:
- Laboratory quality control
- Analytical chemistry and assay monitoring
- Detection of small process shifts (typically less than 2σ)
Data requirements
| Parameter | Description |
|---|---|
| xi | Current observation |
| μ0 | Target or reference mean |
| σ | Process standard deviation |
| λ | Weighting factor (0 < λ ≤ 1); Cloud QC “Weighting factor (Lambda)”, 0.05 to 1, default 0.20 |
Typical λ values:
| λ | Characteristics |
|---|---|
| 0.05 | Very sensitive to small shifts |
| 0.10 | Commonly used |
| 0.20 | Faster response |
| 0.30 | Greater emphasis on recent data |
EWMA formula
The EWMA statistic is calculated recursively, starting from the target:
Zi = λ xi + (1 − λ) Zi−1, with Z0 = μ0
| Symbol | Description |
|---|---|
| Zi | EWMA value at observation i |
| xi | Current observation |
| Zi−1 | Previous EWMA value |
| λ | Weighting factor |
Control limits
The EWMA control limits change slightly during the start-up period and eventually stabilise.
UCLi = μ0 + L σ √( λ / (2 − λ) × [1 − (1 − λ)2i] )
LCLi = μ0 − L σ √( λ / (2 − λ) × [1 − (1 − λ)2i] )
| Symbol | Description |
|---|---|
| L | Width factor (typically 3; Cloud QC “Number of Sigmas (L)”, default 2.5) |
| σ | Process standard deviation |
For large values of i, the limits approach their steady-state values, μ0 ± L σ √(λ / (2 − λ)). Cloud QC draws these limits point by point, so they start narrow and widen towards the steady-state value. EWMA points beyond them are shown in red.
Signal rule
An out-of-control condition occurs when Zi > UCLi or Zi < LCLi. This indicates a potential shift in the process mean.
Worked example
Target mean μ0 = 100, standard deviation σ = 2, weighting factor λ = 0.20 and control limit width L = 3. Starting condition Z0 = 100, so Zi = 0.2 xi + 0.8 Zi−1.
| Obs | Value (x) | EWMA (Z) |
|---|---|---|
| 0 | 100.0 | 100.000 |
| 1 | 100.2 | 100.040 |
| 2 | 99.9 | 100.012 |
| 3 | 100.5 | 100.110 |
| 4 | 101.0 | 100.288 |
| 5 | 101.2 | 100.470 |
| 6 | 101.3 | 100.636 |
| 7 | 101.5 | 100.809 |
| 8 | 101.6 | 100.967 |
| 9 | 101.8 | 101.134 |
| 10 | 102.0 | 101.307 |
For Observation 4, with Z3 = 100.110 and x4 = 101.0:
Z4 = (0.2 × 101.0) + (0.8 × 100.110) = 20.2 + 80.088 = 100.288
Interpretation
- Observations 1–3 remain close to the target mean.
- Beginning at Observation 4, values are consistently above the target.
- The EWMA steadily increases because recent observations are higher than the process mean.
- Although no individual observation appears extreme, the EWMA statistic accumulates evidence of the shift.
- If the EWMA eventually exceeds the upper control limit, the process is considered statistically out of control.
- Here the upper limit at Observation 10 is 101.99 with L = 3, or 101.66 with Cloud QC’s default L = 2.5, so Z₁₀ = 101.31 has not yet signalled. If results continued at 102, the EWMA would cross the default limit at Observation 14.
This illustrates why EWMA charts are effective for detecting small, persistent changes that may not trigger traditional Shewhart control chart rules.
Summary
The EWMA control chart monitors process stability by calculating a weighted average in which recent observations receive greater emphasis than older observations. The method is particularly useful for detecting small or gradual shifts in process performance. By comparing the EWMA statistic to dynamically calculated control limits, process changes can be identified earlier than with conventional control charts, allowing timely investigation and corrective action. The EWMA control chart, in terms of sensitivity to identifying drift, sits between the Shewhart Control Chart and the CUSUM chart.