Measurement System Analysis

Bias per Reference Point (one-sample t-test)

For each reference point, tests whether the bias differs significantly from zero.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Quality Tools > Gage Study > Gage Linearity and Bias Study (per-reference bias)

Description

Groups the long/tidy measurements by reference value x_ref,i and computes per group: mean ȳ_i, standard deviation s_i (Bessel-corrected), bias b_i = ȳ_i − x_ref,i, t-statistic t_i = b_i · √n_i / s_i, and the two-sided p-value at df = n_i − 1. Lets you pinpoint which reference tips the AIAG/VDA 5 traffic light.

Formulas

Bias

b_{i} = \overset{y}{ˉ}_{i} - x_{ref, i}

Mean of the repeats minus the reference value

Standardabweichung

s_{i} = \frac{1}{n _{i} - 1} j = 1 \sum n_{i} (y_{ij} - \overset{y}{ˉ}_{i})^{2}

Bessel-corrected spread of the repeats at reference i

t-Statistik

t_{i} = \frac{b _{i}}{s _{i} / n _{i}}

One-sample t-test H₀: b_i = 0

p-Wert

p_{i} = 2 \cdot P (∣ T_{n_{i} - 1} ∣ > ∣ t_{i} ∣)

Two-sided p-value at df = n_i − 1

Assumptions

Repeat measurements are approximately normally distributed
≥ 3 repetitions per reference (more repeats increase test power)
Repeatability conditions — no change of operator, time or instrument during the series

Limitations

For very small n_i (3–5) the t-test has low power — small biases go undetected
Multiple simultaneous tests: interpret p-values with a Bonferroni or FDR correction

References

AIAG — Measurement Systems Analysis Reference Manual, 4th Edition (MSA-4)
VDA Band 5 — Prüfprozesseignung, 3. Auflage