What Is Bias and Linearity in Measurement System Analysis (MSA)?

Measurement System Analysis (MSA) is a cornerstone of quality engineering, ensuring that the data you collect truly reflects the process you are measuring. Two fundamental properties of any measurement system are bias and linearity. This article explains what they are, how to calculate them, and why they matter, following the authoritative frameworks of the AIAG MSA Manual (4th Ed.) and VDA 5.

What It Is

Bias (also called accuracy) is the difference between the observed average of measurements and a known reference value. It tells you if your measurement system is consistently over- or under-stating the true value.

Linearity describes how consistent that bias is across the entire expected operating range of the measurement system. A system may have low bias at one end of its range but high bias at the other. Linearity quantifies this variation in bias.

In short:

• Bias = systematic error at a single point.
• Linearity = how that systematic error changes across the measurement range.

Both concepts are defined in the AIAG MSA Manual (4th Edition) and VDA 5, which serve as the global references for measurement system capability.

How It Works: Formula and Steps

### Calculating Bias

Per AIAG MSA methodology, bias is typically assessed at a single reference value. The formula is:

\[
\text{Bias} = \bar{x} - \text{Reference Value}
\]

Where:

• \(\bar{x}\) = average of repeated measurements (typically \(n \geq 10\))
• Reference Value = the known, traceable standard (e.g., a calibrated master part)

Often, bias is expressed as a percentage of the reference value or of the process variation (tolerance). The AIAG manual recommends comparing bias to the process variation (usually % of tolerance or % of study variation) to decide if it is acceptable.

### Assessing Linearity

Linearity is evaluated by selecting multiple parts (typically 5 or more) that span the working range of the gauge. For each part, you:

1. Establish the reference value (via a high-level measurement method).
2. Take repeated measurements (e.g., 10–12 per part) with the gauge under study.
3. Calculate the bias for each part.
4. Plot bias against the reference values.

The linearity is then determined by fitting a regression line:

\[
\text{Bias}_i = a + b \times \text{Reference}_i
\]

Where:

• \(b\) (slope) indicates how bias changes per unit of reference value.
• \(a\) (intercept) is the bias at a reference value of zero.

Linearity is often reported as the absolute value of the slope times the process variation, expressed as a percentage. A slope close to zero means the bias is constant across the range (good linearity). A large slope means bias grows or shrinks as you move across the range.

### Acceptance Criteria (per AIAG/VDA practice)

• Bias: Ideally, the bias should be statistically insignificant (i.e., the confidence interval of the bias includes zero). If significant, it should be small relative to tolerance (commonly <10% of tolerance).
• Linearity: The % linearity (slope × process variation) should be < 5% to be considered acceptable, per common industry practice. AIAG recommends that the linearity be evaluated with a hypothesis test on the slope.

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A Worked Illustrative Example

Example data (illustrative only) – not from any real factory.

Suppose you are validating a digital caliper with a working range of 0–100 mm. You select five master blocks with reference values of 10, 30, 50, 70, and 90 mm. You take 10 repeated measurements per block.

Reference (mm) | Observed Mean (mm) | Bias (mm)

• 10 | 10.05 | +0.05
• 30 | 30.02 | +0.02
• 50 | 50.00 | 0.00
• 70 | 69.95 | −0.05
• 90 | 89.90 | −0.10

Bias at 50 mm = 50.00 − 50.00 = 0.00 mm (good).

Linearity: Plotting bias vs. reference gives a slope \(b \approx −0.0017\) mm per mm. If the process variation (e.g., 6σ) is 20 mm, then:

\[
\text{Linearity} = |b| \times \text{Process Variation} = 0.0017 \times 20 = 0.034 \text{ mm}
\]

As a percentage of process variation: \(0.034 / 20 \times 100\% = 0.17\%\), which is well below the 5% threshold — excellent linearity.

However, the bias at 90 mm (−0.10 mm) may be significant relative to a tight tolerance. This example shows why you must check both bias and linearity together.

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Common Pitfalls

1. Using only one reference value – This tells you about bias but hides linearity problems.
2. Ignoring the confidence interval – A bias may look small but still be statistically significant if measurement variation is tiny.
3. Confusing linearity with repeatability – Linearity is about systematic error across range; repeatability is about random error.
4. Not using traceable reference values – Your reference must be calibrated to a known standard, or your bias calculation is meaningless.
5. Forgetting to test the slope – Even if % linearity looks small, always perform the hypothesis test on the slope as recommended by AIAG.

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Closing

Bias and linearity are essential checks for any measurement system. They reveal whether your gauge is accurate and whether that accuracy holds across the full range of use. To simplify your analysis, use our free, interactive Bias and Linearity (Type 1) analysis tool at https://www.6sq.com/tools/type1/ — it follows AIAG/VDA methodology and gives you instant results with clear pass/fail criteria.