CUSUM
Overview
A two-sided CUSUM (Cumulative Sum) chart is used to detect small shifts in a process mean that may not be visible on a traditional control chart.
The method maintains two cumulative sums:
- Positive CUSUM (C⁺) detects upward shifts in the process mean.
- Negative CUSUM (C⁻) detects downward shifts in the process mean.
As each new observation is received, both cumulative sums are updated. When either cumulative sum exceeds a decision limit, the process may be considered out of control.
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
Two-sided CUSUM charts are commonly used for:
- Laboratory quality control
- Analytical chemistry and assay performance tracking
- Detecting slow drift in a laboratory’s CRM results, such as calibration drift, before any single result fails
Formula
For each observation xi:
Positive: Ci+ = max(0, Ci−1+ + (xi − μ0 − k))
Negative: Ci− = max(0, Ci−1− + (μ0 − xi − k))
| Symbol | Description |
|---|---|
| xi | Current observation |
| μ0 | Target or reference mean |
| k | Allowance = Shift Detection setting × σ (Cloud QC default 0.5σ) |
| Ci+ | Positive cumulative sum |
| Ci− | Negative cumulative sum |
If the calculated value becomes negative, it is reset to zero.
Decision limit
A decision limit H is set by the Decision Interval setting × σ (Cloud QC default 3σ). The process is flagged when:
Ci+ > H or Ci− > H
On the Cloud QC chart, C⁺ is drawn above zero against a line at +H, and C⁻ is drawn below zero (as a negative value) against a line at −H. Points beyond either line are shown in red.
Typical practice is k = δ / 2, where δ is the shift size to be detected, in units of σ. Cloud QC’s default Shift Detection of 0.5 therefore targets a shift of about 1σ.
Worked example
Target mean μ0 = 100 and standard deviation σ = 1, so k = 0.5σ = 0.5 and H = 5σ = 5 (in Cloud QC: Shift Detection 0.5, Decision Interval 5). C0+ = C0− = 0.
| Obs | Value (x) | x − μ₀ − k | μ₀ − x − k | C⁺ | C⁻ |
|---|---|---|---|---|---|
| 1 | 100.1 | −0.4 | −0.6 | 0.00 | 0.00 |
| 2 | 99.8 | −0.7 | −0.3 | 0.00 | 0.00 |
| 3 | 100.4 | −0.1 | −0.9 | 0.00 | 0.00 |
| 4 | 101.0 | 0.5 | −1.5 | 0.50 | 0.00 |
| 5 | 101.3 | 0.8 | −1.8 | 1.30 | 0.00 |
| 6 | 101.5 | 1.0 | −2.0 | 2.30 | 0.00 |
| 7 | 100.9 | 0.4 | −1.4 | 2.70 | 0.00 |
| 8 | 101.4 | 0.9 | −1.9 | 3.60 | 0.00 |
| 9 | 101.7 | 1.2 | −2.2 | 4.80 | 0.00 |
| 10 | 102.0 | 1.5 | −2.5 | 6.30 | 0.00 |
Interpretation
- Observations 1–3 remain close to the target and do not accumulate a CUSUM.
- Beginning at Observation 4, values are consistently above the target mean, and the positive CUSUM steadily increases.
- At Observation 10, C⁺ = 6.30, which exceeds the decision limit H = 5.0. The CUSUM chart therefore signals a potential upward shift in the process mean.
- The negative CUSUM remains zero throughout because no sustained downward trend is present.
Summary
The two-sided CUSUM method accumulates evidence of persistent changes from a target value. Positive and negative cumulative sums are tracked separately, allowing early detection of both upward and downward process shifts. When either cumulative sum exceeds the specified decision limit, the process should be investigated for a potential change in performance.
Whilst the Shewhart control chart can identify more broad shifts in process, CUSUM is able to identify more subtle shifts and flag before the process goes out of control, early in the process. The two charts complement each other: the Shewhart chart reacts faster to a single large jump.