Test limit and guard band calculator

How it works

A test cannot see the true value of a parameter, only a reading that carries measurement uncertainty. Near a specification limit that means some bad parts pass (false accept) and some good parts fail (false reject). This tool sets test limits inside the specification, a guard band, and shows what each choice does to escapes, to yield and to the risk for a part that reads right on the limit.

1. Combine the measurement uncertainty. Enter the uncertainty as an expanded value with its coverage factor, as a standard uncertainty, or build it from the tester accuracy limit (turned into a standard uncertainty by the divisor for its distribution), the repeatability and the reference standard. The parts add as a root sum square. The expanded uncertainty is the standard uncertainty times the coverage factor.u = sqrt( (accuracy / divisor)^2 + repeatability^2 + (reference U / reference k)^2 ); U = k x u; TUR = (USL − LSL) / (2 U)
2. Model the true value and the reading. The true value X of a part follows the process distribution (normal, mean μ, standard deviation σ). The reading is Y = X + offset + e, where the error e is normal with standard deviation u. X and Y are then a bivariate normal pair with correlation ρ = σ / sqrt(σ^2 + u^2).X ~ N(μ, σ^2); Y = X + offset + e; e ~ N(0, u^2)
3. Set the test limits. The test limits sit inside the specification limits by a guard band g: the lower test limit is LSL + g and the upper is USL − g. The method decides g. None sets g to zero. The simple rule sets g to a multiple of U, and a multiple of 1 is the stringent acceptance rule of ISO 14253-1. The target rule searches for the smallest g whose false accept rate (global PFA) or shipped escape rate meets your target. The JCGM 106 specific risk rule places each test limit where a part reading exactly on it has the target probability of being out of specification.AL = LSL + g; AH = USL − g; target rule: smallest g with PFA(g) ≤ target; specific rule: P(X outside specification | Y = test limit) = α
4. Global risks by integration. The global false accept and false reject rates are probabilities over the whole population of parts. For a given true value x the chance of passing is Φ((AH − x − offset) / u) − Φ((AL − x − offset) / u). Multiplying by the process density and integrating over the values that are out of specification gives PFA; integrating the chance of failing over the values that are in specification gives PFR. The integral is done numerically, closed form where the chance of passing is 0 or 1 and Simpson's rule near the limits.PFA = ∫ over x outside [LSL, USL] of f(x) P(pass | x) dx; PFR = ∫ over x inside [LSL, USL] of f(x) P(fail | x) dx
5. Specific risks at the test limit. A specific risk belongs to one reading rather than the population. With the process distribution as the prior, the true value of a part that reads y is normal with a mean pulled towards the process mean and a smaller spread. The specific false accept risk is the chance that such a part is out of specification; the specific false reject risk of a part that fails by the same reading is the chance that it is in specification. A flat prior uses y alone with spread u.X | Y = y ~ N( μ + κ (y − μ − offset), κ u^2 ), κ = σ^2 / (σ^2 + u^2)
6. Yields and shipped escapes. True yield is the share of parts inside the specification. Test yield is the share that pass. The yield lost to the guard band is the drop in test yield against testing at the specification limits. The shipped escape rate is the share of passing parts that are out of specification.true yield = P(LSL ≤ X ≤ USL); test yield = P(AL ≤ Y ≤ AH); shipped DPPM = 1,000,000 x PFA / test yield

Sources

The methods and default figures rest on these public sources. The wording here is our own; read the source for its full text.

• JCGM 106:2012, Evaluation of measurement data: The role of measurement uncertainty in conformity assessment (opens in a new tab), Joint Committee for Guides in Metrology (BIPM), 2012. Specific and global consumer's and producer's risks, acceptance intervals and guard bands. Free from BIPM; adopted as ISO/IEC Guide 98-4.
• ISO 14253-1, Geometrical product specifications (GPS), Inspection by measurement of workpieces and measuring equipment, Part 1: Decision rules for verifying conformity or nonconformity with specifications, International Organization for Standardization, 2017 edition. Cited by name and part only. The simple guard band of one expanded uncertainty is described from the general practice of stringent acceptance; read the standard for its present default rule.
• A Guard-Band Strategy for False-Accept Management (opens in a new tab), Keysight Technologies (Dobbert), also published in NCSLI Measure, 2008. Guard bands chosen to hold the false accept probability at a target, with the false reject cost.
• NCSLI Recommended Practice RP-18 (section on the 4 to 1 test uncertainty ratio requirement), NCSL International, Current edition. Cited by name only. Discusses why a 4 to 1 test uncertainty ratio alone does not control the false accept risk.