MSA Type 4 (linearity)

Purpose of the Study

MSA Type 4 (linearity) checks whether a measurement system reads consistently across its entire operating range — or whether the systematic deviation drifts as the reference value grows. While MSA Type 1 only evaluates a single point on the scale, Type 4 covers the complete range between the lower and upper specification limits.

Linearity: Change of bias across the measurement range. A linear measurement system shows the same (ideally zero) bias at every reference point. A non-linear system exhibits drift: small values may read too low, large values too high.

Bias: Systematic deviation at a single reference point: mean of the repeat measurements minus the reference value. Computed separately for each reference point.

Process Variation (PV): Reference figure used to normalize the linearity deviation. Two modes: Tolerance (PV = USL − LSL) for capability studies, or 6·${\sigma }_P$ derived from ongoing process spread. AIAG prefers PV, VDA 5 always uses the tolerance.

A Type 4 failure is insidious: the system may be flawless in the middle of the range (e.g. at the calibration point) while systematically off near the specification edges. Classic causes are non-linear sensors, worn scales or single-point software calibrations.

Data Requirements

• Two columns from the worksheet, long/tidy layout: one reference column, one measured** column. Each row is one individual measurement.
• At least five distinct reference values that cover the whole measurement range — ideally evenly spaced between LSL and USL.
• At least three repeat measurements per reference; 10–12 are common. More repeats stabilize the standard deviation per point.
• One operator, one instrument, short time frame — repeatability conditions as in Type 1.
• Traceable reference parts (certified master set); otherwise bias assessment is blind.

Long/tidy layout instead of wide: Do not create one column per reference point. Instead create two long columns: one for the reference, one for the measured value. This is the only layout the module accepts and matches how Minitab and JMP handle Type 4 data.

Tolerance vs. σ_P: For capability studies (release of a new gage) enter LSL/USL. For process-uncertainty budgets (VDA 5, running production) use the process spread ${\sigma }_P$ instead of the tolerance; the formula switches internally to PV = 6·${\sigma }_P$.

Formulas (AIAG + VDA 5)

Type 4 boils down to a linear regression of individual bias values $b_{ij}=y_{ij}-x_{ref,i}$ against the reference value $x_{ref,i}$. Both the AIAG KPIs and the VDA 5 uncertainty contribution are derived from this regression.

Regression model: $b_{ij}$ = a · $x_{ref,i}$ + b̂ + ε. Slope a ≠ 0 means the bias grows systematically with the reference (linearity error). Intercept b̂ ≠ 0 means a constant offset across the whole range.

Per-reference bias: For each reference $x_{ref,i}$ with $n_i$ repeats: $b_i={\bar{y}}_i-x_{ref,i}$. The one-sample t-test $t=b_i\sqrt{n_i}/s_i$ tests $H_0:b_i=0$. If the p-value < α, bias at that point is statistically significant.

AIAG %Linearity: %Lin = 100 · |a| · Range($x_{ref}$) / PV. Range is the difference between the largest and smallest reference. PV is either the tolerance T = USL − LSL or 6·${\sigma }_P$.

AIAG peak %Bias: $\%{\text{Bias}}_{\max }=100\cdot {\max }_i|b_i|/PV$. Shows the worst individual bias point normalized to the reference figure.

VDA 5 uncertainty contribution u_BI: $u_{BI}=\sqrt{{\max }_i|b_i{|}^2+(SE(a)\cdot \text{Range}{)}^2}$. Combines the largest observed bias magnitude with the slope uncertainty extrapolated across the measurement range.

VDA 5 expanded uncertainty U: U = 2 · $u_{BI}$ (coverage factor k = 2 for approximately 95 % coverage). U is used directly against the tolerance in the traffic-light verdict.

VDA 5 Q_MS,BI: $Q_{MS,BI}$ = U / T · 100 % — share of the tolerance T consumed by the bias/linearity contribution. Together with the other Q-shares (repeatability, reproducibility, resolution) it makes up the total $Q_{MS}$.

Verdict & Traffic Lights

The module displays one traffic light for the active standard. AIAG evaluates %Linearity and peak %Bias, VDA 5 evaluates $Q_{MS,BI}$. The two standards typically agree, but may diverge in borderline cases — the customer or release specification then decides.

AIAG light — green: Both %Linearity and peak %Bias below 5 %, and neither slope nor intercept is statistically significant (p ≥ α). The system reads linearly and without bias across the entire range.

AIAG light — yellow: The worse of the two values (%Linearity or peak %Bias) lies between 5 % and 10 %, with no significant regression coefficient. Conditionally usable; investigate the cause before releasing the system.

AIAG light — red: At least one of the two values exceeds 10 %, or slope / intercept are significant (p < α). The system is non-linear or systematically shifted — do not release; check calibration or replace the sensor.

VDA 5 light — green: $Q_{MS,BI}$ ≤ 15 %. The bias contribution to measurement uncertainty is small enough not to burden process capability.

VDA 5 light — yellow: $Q_{MS,BI}$ between 15 % and 30 %. Conditionally usable; the system may be used if the remaining Q-shares and the process margin allow it.

VDA 5 light — red: $Q_{MS,BI}$ > 30 %. The linearity/bias contribution eats more than 30 % of the tolerance — not usable, rework or replace the gage.

The per-reference table additionally shades each row: green when the t-test is not significant, yellow when bias is significant but below 10 % PV, red when bias is significant and above 10 % PV. This makes it obvious at a glance which reference point tips the light.

Examples

This module ships with the following example datasets — load any of them in the app with a single click.

Available in the following cycles

• DMAIC: Measure
• DMADV: Measure
• 8D: D2 — Describe the Problem