What Are EWMA and CUSUM Charts, and When Should You Use Advanced SPC?

Standard Shewhart control charts (like the X-bar and R chart) are excellent at detecting large, sudden process shifts. However, they are notoriously slow at detecting small, persistent shifts (typically less than 1.5 sigma). If your process drifts gradually—due to tool wear, raw material lot changes, or operator fatigue—you need more sensitive tools. This is where Advanced SPC comes in, specifically through EWMA and CUSUM control charts.

What It Is

Advanced SPC refers to a set of control charting techniques designed for high-sensitivity process monitoring. Unlike Shewhart charts that only use the most recent data point, EWMA (Exponentially Weighted Moving Average) and CUSUM (Cumulative Sum) charts incorporate historical data with a weighting scheme. This makes them far more effective at detecting small shifts while remaining robust to false alarms.

These methods are formally defined in the AIAG SPC Manual (2nd Edition, Section on Advanced Control Charts) and the international standard ISO 7870 series. They are the go-to tools when your process requires tight control and early warning of subtle degradation.

How It Works: The Formulas

### EWMA Chart

The EWMA statistic is defined as:

\[
z_t = \lambda x_t + (1 - \lambda) z_{t-1}
\]

Where:

• \( x_t \) = the current sample mean or observation
• \( z_{t-1} \) = the previous EWMA value (starting with \( z_0 = \mu_0 \), the process target)
• \( \lambda \) = a weighting constant, typically 0.2 (per AIAG recommendation for detecting small shifts)

The control limits for EWMA are time-varying (they widen as \( t \) increases) and are calculated as:

\[
UCL/LCL = \mu_0 \pm L \sigma \sqrt{\frac{\lambda}{(2-\lambda)} [1 - (1-\lambda)^{2t}]}
\]

Where \( L \) is the width of the control limits (commonly 2.7 for a false alarm rate similar to Shewhart's 3-sigma, or 3.0 for stricter control).

### CUSUM Chart

The CUSUM statistic accumulates deviations from a target value. The tabular (V-mask) form uses two one-sided statistics:

\[
C^+_t = \max(0, x_t - (\mu_0 + K) + C^+_{t-1})
\]
\[
C^-_t = \max(0, (\mu_0 - K) - x_t + C^-_{t-1})
\]

Where:

• \( K = \frac{\delta}{2} \sigma \) is the "allowance" or reference value (often set to detect a shift of \( \delta = 1\sigma \))
• \( H = 5\sigma \) is the decision interval (typical AIAG value)

An out-of-control signal occurs when either \( C^+_t \) or \( C^-_t \) exceeds \( H \).

A Worked Illustrative Example

Example data (illustrative only): Suppose a process has a target mean \( \mu_0 = 10.0 \) and known standard deviation \( \sigma = 1.0 \). You collect 5 consecutive sample means: 10.2, 10.5, 10.3, 10.8, 11.0. The process is actually drifting upward by about 0.2σ per sample.

Using EWMA with \( \lambda = 0.2 \):

• \( z_0 = 10.0 \)
• \( z_1 = 0.2(10.2) + 0.8(10.0) = 10.04 \)
• \( z_2 = 0.2(10.5) + 0.8(10.04) = 10.13 \)
• \( z_3 = 0.2(10.3) + 0.8(10.13) = 10.17 \)
• \( z_4 = 0.2(10.8) + 0.8(10.17) = 10.30 \)
• \( z_5 = 0.2(11.0) + 0.8(10.30) = 10.44 \)

The UCL at \( t=5 \) with \( L=2.7 \) is approximately \( 10 + 2.7 \times 1 \times \sqrt{0.2/1.8} \times \sqrt{1 - (0.8)^{10}} \approx 10.55 \). The EWMA value of 10.44 is still within limits, but the trend is clear—the chart would signal within 2 more samples.

Using CUSUM with \( K = 0.5 \) and \( H = 5 \):

• \( C^+_1 = \max(0, 10.2 - 10.5 + 0) = 0 \)
• \( C^+_2 = \max(0, 10.5 - 10.5 + 0) = 0 \)
• \( C^+_3 = \max(0, 10.3 - 10.5 + 0) = 0 \)
• \( C^+_4 = \max(0, 10.8 - 10.5 + 0) = 0.3 \)
• \( C^+_5 = \max(0, 11.0 - 10.5 + 0.3) = 0.8 \)

CUSUM accumulates the evidence slowly, but once the shift persists, \( C^+ \) will grow linearly and cross \( H = 5 \) after about 10 more samples. A Shewhart chart would not signal until the mean exceeds the 3-sigma limit (around 13.0), which could take 15+ samples.

Common Pitfalls

1. Using EWMA/CUSUM for large shifts: These charts are slower than Shewhart for detecting big, sudden jumps. Use them only when you care about small shifts.
2. Ignoring the memory effect: EWMA and CUSUM charts have "inertia"—once a process shifts, the chart may take time to respond, and after a sustained shift, it may be slow to return to center. Reset or restart the chart after an assignable cause is found.
3. Choosing wrong parameters: Using \( \lambda = 0.8 \) (too high) makes EWMA behave like a Shewhart chart, defeating its purpose. Stick to \( \lambda = 0.2 \) for small shifts.
4. Applying to autocorrelated data: Both charts assume independent observations. If your data is autocorrelated (common in continuous processes), you must model the autocorrelation first or you will get excessive false alarms.

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Ready to detect small shifts before they become defects? Use the free Advanced SPC tool at https://www.6sq.com/tools/adv_spc/ to automatically generate EWMA and CUSUM charts from your data, with parameters aligned to the AIAG SPC Manual and ISO 7870.