TAR, TUR, Tolerance Consumption, and Capability

Test Accuracy Ratio (TAR) vs. Test Uncertainty Ratio (TUR)

Calibration laboratories must quantitatively evaluate whether their calibration process has sufficient capability to verify the tolerance of a Unit Under Test (UUT). Historically, this was evaluated using the Test Accuracy Ratio (TAR). In modern accredited calibration (ISO/IEC 17025 and ANSI/NCSL Z540.3), TAR has been largely superseded by the Test Uncertainty Ratio (TUR).

Test Accuracy Ratio (TAR)

Originating in military standards such as MIL-STD-45662A, TAR is defined as the ratio of the tolerance limit of the UUT to the tolerance limit of the calibration standard:

$$\text{TAR} = \frac{\text{Tolerance}_{\text{UUT}}}{\text{Tolerance}_{\text{STD}}} = \frac{\text{Span}_{\text{UUT}}}{\text{Span}_{\text{STD}}}$$

For two-sided symmetric tolerances ($\pm T_{\text{UUT}}$ and $\pm T_{\text{STD}}$):

$$\text{TAR} = \frac{T_{\text{UUT}}}{T_{\text{STD}}}$$

The Fatal Limitation of TAR:

TAR compares stated accuracy or error bounds using a declared convention. It does not assume the reference has zero error; its limitation is that a reference bound alone omits other process uncertainty contributions. Evaluate the setup, corrections, reference uncertainty, and UUT effects instead of substituting a catalog ratio for that analysis.

• It does not account for the standard's calibration drift since its last calibration.
• It completely ignores ambient environmental fluctuations (such as temperature gradients in the calibration room).
• It omits the resolution limit of the UUT and the comparator.
• It omits operator technique, fixturing deformation, and lead resistance.

Consequently, a TAR of 4:1 does not guarantee that the calibration process has an uncertainty 4 times smaller than the UUT tolerance.

Test Uncertainty Ratio (TUR)

To address the deficiencies of TAR, national and international standards—including ANSI/NCSL Z540.3-2006 and ILAC-G8:09/2019—describe uncertainty-aware capability and decision approaches; TUR is a useful ratio, not a universal ISO mandate. TUR is defined as the ratio of the tolerance span of the UUT attribute being measured to twice the expanded measurement uncertainty ($k = 2$) of the calibration process:

$$\text{TUR} = \frac{\text{Tolerance Span}_{\text{UUT}}}{2 \cdot U} = \frac{\text{USL} - \text{LSL}}{2 \cdot U}$$

where:

• $\text{USL}$ is the Upper Specification Limit of the UUT.
• $\text{LSL}$ is the Lower Specification Limit of the UUT.
• $\text{Tolerance Span}_{\text{UUT}} = \text{USL} - \text{LSL}$ (for a symmetric tolerance of $\pm T$, the span is $2T$).
• $U$ is the expanded measurement uncertainty of the calibration process evaluated at a $95\%$ confidence level (conventionally coverage factor $k = 2$).

For a symmetric two-sided tolerance ($\pm T$):

$$\text{TUR} = \frac{2 \cdot T}{2 \cdot U} = \frac{T}{U}$$

For the following one-sided illustration, explicitly define a nominal-to-limit distance and use that distance in the ratio. Other conventions can differ; state numerator width and uncertainty convention. A ratio alone is not a conformity decision or risk probability.

$$\text{TUR} = \frac{|\text{Specification Limit} - \text{Nominal}|}{U}$$

Crucially, the expanded uncertainty $U$ in the denominator is not just the uncertainty of the standard; it is the total combined expanded uncertainty of the calibration process, encompassing the reference standard, environmental influences, UUT resolution, and short-term repeatability.

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Capability targets and their limits

A 4:1 TUR corresponds to $U=T/4$ for a symmetric tolerance; a 10:1 TUR corresponds to $U=T/10$. These numbers can be program targets, but neither proves a universal false-accept probability. Tolerance is a permissible limit, not a probability distribution or variance. Squaring a ratio of uncertainty to tolerance does not give a universal fraction of “tolerance variance.”

The decision rule and risk model must account for result location and uncertainty. A result exactly at a specification boundary can have substantial specific risk even at high TUR. Low TUR can be managed through an agreed guard band, better measurement capability, adjusted scope, or reporting results without a conformity statement. Determine what the contract permits; ISO/IEC 17025 does not automatically ban every process below 4:1.

Percent of Tolerance Consumed

During routine calibration, technicians calculate the percent of tolerance consumed to assess how close an indicated reading is to the allowable specification limits:

$$\% \text{ of Tolerance} = \frac{|\text{Measured Error}|}{\text{Allowable Tolerance}} \times 100\% = \frac{|y - x_{\text{ref}}|}{T_{\text{allowable}}} \times 100\%$$

The ratio indicates the distance from nominal relative to the chosen tolerance, not future stability or legal permission to use the instrument. For $E=+0.007\text{ mm}$ and a symmetric tolerance of $\pm0.010\text{ mm}$, 70% is consumed. If $U=0.004\text{ mm}$ and the agreed acceptance rule requires $|E|+U\le T$, the result is outside the acceptance zone despite its error being inside tolerance.

A laboratory can choose alert levels to prompt interval review or adjustment, but 70% is not a universal action threshold. Trend, duty cycle, environment, and historical data inform the next interval. An observed error beyond the stated tolerance triggers the applicable nonconforming-work evaluation and decision rule.

Gage Capability Indices ($C_g$ and $C_{gk}$)

While process capability indices ($C_p, C_{pk}$) evaluate manufacturing processes against part tolerances, gage capability indices ($C_g, C_{gk}$) evaluate the suitability of the measurement equipment itself. A Type 1 Gage Study isolates the measurement system from part-to-part variation by repeatedly measuring a single calibrated reference standard.

Repeatability Index ($C_g$)

The $C_g$ index evaluates the inherent precision (repeatability dispersion) of the measuring instrument relative to the total product tolerance band:

$$C_g = \frac{0.20 \cdot T}{6 \cdot s_g}$$

where:

• $T = \text{USL} - \text{LSL}$ is the total tolerance band of the attribute to be measured.
• The 20% tolerance allocation in this illustrative capability convention is not a universal industry or ISO requirement. Declare the adopted formula, units, tolerance width, study conditions, and acceptance criterion before interpreting an index.
• $s_g$ is the sample standard deviation of the number of repeated measurements performed on the reference standard under repeatable conditions.
• $6 \cdot s_g$ represents the full $6\sigma$ spread of the gage's repeatability distribution ($99.73\%$ coverage).

Repeatability and Bias Index ($C_{gk}$)

The $C_{gk}$ index evaluates both the precision (repeatability) and the trueness (systematic bias) of the measuring instrument:

$$C_{gk} = \frac{0.10 \cdot T - |\bar{x}_g - x_{\text{ref}}|}{3 \cdot s_g}$$

where:

• $0.10 \cdot T$ represents half of the allowable gage variation ($10\%$ of product tolerance allocated to each side).
• $\bar{x}_g$ is the arithmetic mean of the repeated gage measurements: $\bar{x}_g = \frac{1}{n}\sum_{i=1}^n x_{g,i}$.
• $x_{\text{ref}}$ is the accepted calibrated value of the reference standard.
• $|\bar{x}_g - x_{\text{ref}}|$ represents the absolute systematic bias of the gage.
• $3 \cdot s_g$ represents the half-width ($3\sigma$) of the gage repeatability spread.

Acceptance Criteria

If the applicable study plan adopts a 20% tolerance allocation and 1.33 thresholds, interpret the results as follows; these are study-policy choices, not universal requirements:

• $C_g \ge 1.33$ and $C_{gk} \ge 1.33$: The gage is capable and approved for production inspection.
• $C_g < 1.33$: The gage has excessive random variation (inadequate repeatability or resolution).
• $C_g \ge 1.33$ but $C_{gk} < 1.33$: The gage has excellent repeatability but possesses a significant uncorrected systematic bias (offset), requiring zero adjustment, recalibration, or mechanical re-alignment.

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