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Measurement Decision Rule False Accept Calculator

A reading of 24.998 on an upper limit of 25.000 looks like a pass, but if the standard uncertainty is 0.002 there is a real chance the part is over. This calculator turns margin and uncertainty into that probability, and into a simple cost comparison.

Your numbers

Margin inside the limit

Distance between the measured value and the nearest specification limit.

Standard uncertainty

Standard uncertainty u, not the expanded value. Divide expanded U by k first.

Cost of rejecting a good item

currency

Cost of shipping a bad item

currency

Your results

Probability true value is out of tolerance

6.7%

Assumes a normal distribution about the measured value and no prior information about the process.

Confidence of conformity

93.3%

Margin in standard uncertainties

1.5

Expected escape cost per accepted item

$167

Expected cost if rejected instead

$37

Chance the item was good multiplied by the cost of rejecting it.

Single-sided limit and a flat prior. Bayesian methods with process knowledge, as in JCGM 106, give different and usually lower risk values.

Save results:

False accept probability vs margin

Sweeps Margin inside the limit from 0 to 0.008, holding your other inputs fixed.

Record decision rules in your quality system

Optional: discuss storing uncertainty and decision rules with inspection results.

Optional. The tool and every export work without this. No spam.

How to use it

Enter how far inside the specification limit the measured value sits and the standard uncertainty of the measurement. If you only have expanded uncertainty at k=2, halve it before entering. The result is the probability that the true value lies beyond the limit. Add costs of rejecting a good item and shipping a bad one to compare expected costs, which helps when choosing between simple acceptance and a guard band.

Use standard uncertainty here, not expanded uncertainty.
Apply the calculation to the limit nearest the reading.
Treat the cost comparison as indicative, not a policy.

The method

With a normal distribution centered on the measured value, the chance that the true value exceeds the limit equals one minus the normal cumulative probability at the margin divided by standard uncertainty. At zero margin the probability is 50 percent, at one standard uncertainty it is about 16 percent, at two about 2.3 percent and at three about 0.13 percent. This is the quantity that ISO 14253-1 guard banding aims to control through a documented decision rule.

Worked example

Margin 0.003 mm and u of 0.002 mm give a ratio of 1.5. The probability of being beyond the limit is about 6.7 percent. With a shipping cost of 2,500 and a good-item rejection cost of 40, the expected escape cost of accepting is about 167, versus about 37 for rejecting. In this setting rejection or retest is cheaper, which supports a wider guard band or a better gage.

Common mistakes

Do not confuse the probability computed here with the share of parts that are bad. It is the risk for a single reading at a given margin. Do not feed expanded uncertainty as if it were u, which would understate risk by a factor of two. Also remember that if your process is well centered and capable, prior knowledge reduces the true risk, which is why a Bayesian treatment gives lower numbers. Use this as a conservative screen.

Frequently Asked Questions

What is probability of false accept?

It is the chance that an item you accepted on the basis of a measurement is actually outside its specification. It depends on the margin to the limit and the measurement uncertainty.

Why is the risk 50 percent at the limit?

If the measured value exactly equals the limit, the true value is equally likely to be on either side under a symmetric uncertainty distribution.

How does this relate to guard banding?

A guard band sets a minimum margin before accepting. Choosing the multiplier is choosing the maximum false accept probability you will tolerate for readings at the acceptance limit.

Is this the same as consumer risk in sampling?

No. Sampling consumer risk concerns lots and defect rates. This is the risk for one item measured with an uncertain instrument.

If you want pass or fail decisions in your ERP quality module to record uncertainty and decision rule used, we can help design that.

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