Complete your MSA suite. Check whether a measurement system is biased across its range, and whether inspectors agree on pass/fail calls — bias regression, linearity plot, agreement percentages and kappa, all client-side.
Pick a study type, enter your data and press Calculate to run the gage linearity & bias or attribute agreement analysis.
Your gauge reads slightly off — but is the error the same across the range, or worse at the extremes? Enter the lab to check its linearity and bias.
Pick Linearity & Bias for variable (measured) data, where you have reference parts of known value, or Attribute Agreement for pass/fail and categorical inspection where you have a known standard for each part.
For linearity, paste the reference value and how many times each gauge reads it. For attribute agreement, paste the known answer and each person's rating — one part per row.
For linearity, you'll see how much the gauge is off (biased) at each reference point, and whether that error gets bigger or smaller as the true value rises. For attribute agreement, you'll get how often each person agrees with themselves, how often different people agree with each other, and how often they match the known right answer — plus a bonus score showing whether that agreement is better than random guessing.
Gage R&R tells you how much variation comes from your measurement system, but it does not tell you whether the gage is accurate, or whether your inspectors agree. Linearity and bias check accuracy across the measurement range; attribute agreement checks whether pass/fail judgements are consistent and correct. Together they complete the measurement systems analysis (MSA) toolkit.
Bias is the difference between the average measured value and the true reference value — a systematic offset that makes the gage read consistently high or low. Linearity is how that bias changes across the measurement range. A good gage has near-zero bias everywhere; a linearity problem means it is accurate for small parts but biased for large ones (or vice versa). Attribute agreement is the equivalent check for pass/fail inspection.
For linearity and bias, the tool measures known reference parts and works out the bias — how far each reading is from the true value. It then fits a line to those biases across the measuring range: a flat line near zero means the gauge is accurate everywhere, while a sloping line means the error changes across the range (a linearity problem). A p-value flags whether that slope is real. For pass/fail gauges it instead measures agreement — the share of times appraisers match — and kappa adjusts that figure for the matches you would expect by pure luck.
A calibration engineer measures reference standards from 10mm to 100mm. At 10mm the calliper reads 0.1mm too low. At 50mm it reads correctly. At 100mm it reads 0.2mm too high.
This is linearity bias — the gauge drifts across its range. Calibrating it at one point in the middle gives a false sense of accuracy at the extremes.
What to do with this: Either apply a correction factor at each end of the range or replace the calliper. Any measurement near the extremes is currently unreliable.
A biased gage silently scraps good parts and ships bad ones. If a caliper reads high on large parts you may reject in-spec components at the top of the range and pass defects elsewhere. Disagreeing inspectors create the same chaos for go/no-go decisions. Quantifying bias, linearity and agreement protects every decision that depends on the measurement.
Run linearity & bias in the Measure phase of DMAIC after Gage R&R, whenever a gage covers a wide range or a calibration is in question. Run an attribute agreement study whenever decisions rely on human pass/fail or category judgements — visual inspection, defect classification, or grading. Both confirm your measurement system is fit before you trust the data.
Linearity, bias and attribute agreement sit alongside Gage R&R in the MSA toolkit. A capable measurement system is a prerequisite for capability studies, hypothesis tests and control charts — if the gage is biased or inspectors disagree, every downstream statistic is built on sand. MSA is a core Measure-phase gate in any Six Sigma project.
If bias or linearity is significant, recalibrate or service the gage and re-run the study; a linearity problem usually needs a multi-point calibration rather than a single zero adjustment. If attribute agreement is weak, retrain appraisers, clarify the operational definitions and boundary samples, then repeat the study. Document the accepted measurement system in your control plan, and only then proceed to capability analysis and control charting with confidence in your data.
Pair this calculator with a full Gage R&R study and capability analysis to build a complete, defensible measurement systems analysis.
Gage linearity describes how the bias of a measurement system changes across its operating range. A perfectly linear gage shows the same bias whether you are measuring a small part or a large one. You assess it by measuring several reference parts of known value many times, calculating the bias (measured minus reference) at each point, then regressing bias on the reference value. A non-zero slope means the gage drifts as the measured value changes — for example, reading increasingly high for larger parts.
Bias is the difference between the average measured value and the true (reference) value at a single point — a constant offset. Linearity is the change in that bias across the full measurement range. A gage can have bias but no linearity problem (a fixed offset everywhere), or it can be accurate in the middle of its range but biased at the extremes (a linearity problem). Both are tested by checking whether the regression intercept (overall bias) and slope (linearity) differ significantly from zero.
An attribute agreement analysis (attribute MSA) checks whether inspectors making pass/fail or categorical judgements agree — with themselves, with each other, and with a known standard. Each appraiser rates the same set of parts several times. The study reports within-appraiser agreement (consistency on repeat looks), between-appraiser agreement (do inspectors agree with one another), and agreement versus the known standard (are they correct). It is the attribute equivalent of Gage R&R for go/no-go inspection.
Kappa is a statistic that measures agreement while correcting for the agreement you would expect by chance. It ranges from below 0 to 1: a value of 1 is perfect agreement and 0 is no better than guessing. A common rule of thumb is that kappa above 0.75 indicates excellent agreement, 0.40 to 0.75 is fair to good, and below 0.40 is poor. Because raw percentage agreement can look high purely by luck — especially when one category dominates — kappa gives a more honest picture of how reliable an inspection process really is.
A capable measurement system underpins every reliable improvement. Structured training shows you how to run, interpret and act on a full MSA.
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Want to assess gauge bias and linearity inside a full DMAIC project? The Black Belt covers linearity and bias studies, measurement systems analysis, and the advanced Measure phase toolkit.
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