Z-Score
Overview
A Z-Score is a statistical measure that describes how far an observation is from a target mean in units of standard deviation. Z-Scores are widely used in quality control, laboratory monitoring, manufacturing, and statistical analysis to identify observations that are unusually high or low relative to expected process performance.
Unlike CUSUM and EWMA, which accumulate information over time, a Z-Score evaluates each observation independently.
In Cloud QC, μ is the CRM’s certified value. σ is its certified standard deviation when “Use CRM provided statistics” is ticked; if the CRM has no certified SD, or the box is unticked, σ is the standard deviation of that CRM’s results. The legend shows which was used (“CRM SD” or “computed SD”). Because every result is expressed in standard deviations, CRMs of different grades can be compared on one chart.
When to use
Z-Scores are commonly used for laboratory quality control, assay performance monitoring, outlier detection, proficiency testing and statistical data analysis.
Formula
Z = (x − μ) / σ
| Symbol | Description |
|---|---|
| Z | Z-Score |
| x | Observed value |
| μ | Target mean |
| σ | Standard deviation |
Interpretation
| Z-Score | Interpretation |
|---|---|
| 0 | Exactly at the mean |
| ±1 | One standard deviation from the mean |
| ±2 | Two standard deviations from the mean |
| ±3 | Three standard deviations from the mean |
| > +3 | Unusually high result |
| < −3 | Unusually low result |
Many quality monitoring systems use the following ranges:
| Z-Score range | Assessment |
|---|---|
| −2 ≤ Z ≤ +2 | In control |
| −3 ≤ Z < −2 or +2 < Z ≤ +3 | Warning |
| Z < −3 or Z > +3 | Action required |
Cloud QC uses these limits: the chart is shaded green within ±2, amber from 2 to 3 and red beyond ±3.
Worked example
Target mean μ = 100 and standard deviation σ = 2, so Z = (x − 100) / 2.
| Obs | Value (x) | Calculation | Z-Score | Status |
|---|---|---|---|---|
| 1 | 99.8 | (99.8 − 100) / 2 | −0.10 | In control |
| 2 | 100.5 | (100.5 − 100) / 2 | 0.25 | In control |
| 3 | 101.2 | (101.2 − 100) / 2 | 0.60 | In control |
| 4 | 98.9 | (98.9 − 100) / 2 | −0.55 | In control |
| 5 | 102.1 | (102.1 − 100) / 2 | 1.05 | In control |
| 6 | 103.0 | (103.0 − 100) / 2 | 1.50 | In control |
| 7 | 99.5 | (99.5 − 100) / 2 | −0.25 | In control |
| 8 | 96.8 | (96.8 − 100) / 2 | −1.60 | In control |
| 9 | 100.8 | (100.8 − 100) / 2 | 0.40 | In control |
| 10 | 106.5 | (106.5 − 100) / 2 | 3.25 | Action required |
For Observation 6, Z = (103.0 − 100) / 2 = 1.5: the observation is 1.5 standard deviations above the target mean.
For Observation 10, Z = (106.5 − 100) / 2 = 3.25. Since Z > 3, the observation exceeds the 3σ limit and should be investigated as a potential outlier or process shift.
- Observations 1–9 remain within ±2 standard deviations of the target mean and indicate normal process variation.
- A Z-Score above +3 is unlikely to occur from common process variation alone and may indicate a special cause affecting the process.
- Negative Z-Scores indicate results below the target mean, while positive Z-Scores indicate results above the target mean.
Summary
The Z-Score is a simple and effective statistical tool for assessing how far an observation deviates from a target mean. By expressing deviations in standard deviation units, Z-Scores provide a standardised method for detecting unusual results, comparing measurements across datasets, and identifying potential outliers. Results outside predefined control limits, typically ±3 standard deviations, should be investigated for possible process issues or special causes of variation.