Measurement System Analysis

Repeatability and Bias Index Cgk (MSA Type 1)

Evaluates repeatability and systematic deviation (bias) of a measurement instrument in a single index.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Quality Tools > Gage Study > Type 1 Gage Study

Description

The index Cgk extends Cg by accounting for the systematic deviation (bias) of the measurement instrument from the reference value. It is calculated by reducing the half allowed tolerance fraction by the absolute bias and dividing by the weighted spread. Cgk is always ≤ Cg: an instrument with low spread (high Cg) can still have a low Cgk if a large bias is present. A Cgk ≥ 1.33 is considered capable, indicating that both repeatability and accuracy (bias) are adequate.

Formulas

Cgk

C_{g k} = \frac{\frac{k _{1} \cdot T}{2} - ∣ x ˉ - x _{re f} ∣}{k _{2} \cdot s _{g}}

Repeatability and bias index: half allowed tolerance fraction minus absolute bias, divided by weighted spread

Bias

Bias = \overset{x}{ˉ} - x_{re f}

Systematic deviation: difference between measurement mean and reference value

Bias%

Bias% = \frac{x ˉ - x _{re f}}{T} \cdot 100

Bias as percentage of tolerance width

T = U S L - L S L

Tolerance width

s_{g} = \frac{1}{n - 1} i = 1 \sum n (x_{i} - \overset{x}{ˉ})^{2}

Sample standard deviation of repeated measurements

Cg vs. Cgk

C_{g k} \leq C_{g} always

Cgk is always ≤ Cg — the difference reveals the bias contribution

Assumptions

Measurements under repeatability conditions (one operator, one instrument, one reference part)
At least 25 measurements (n ≥ 25)
Reference part is calibrated with known true value (x_ref)
Measurements are approximately normally distributed

Limitations

Cgk can become negative when the bias exceeds half the allowed tolerance fraction
Requires a calibrated reference part — bias determination is not possible without a known true value
With k₁ = 0.20 and k₂ = 4, Cgk ≥ 1.33 corresponds to a maximum bias of about 1.7% of tolerance

References

VDA Band 5 — Prüfprozesseignung: Eignung von Messsystemen, Mess- und Prüfprozessen, Erweiterte Messunsicherheit, 3. Auflage
AIAG — Measurement Systems Analysis Reference Manual, 4th Edition (MSA-4)
Dietrich, E. & Schulze, A. — Statistische Verfahren zur Maschinen- und Prozessqualifikation, 7. Auflage, Hanser