# Measurement Decision Rule False Accept Calculator > Source: https://netray.co/tools/measurement-decision-rule-false-accept-calculator > Publisher: Netray (netray.co), AI and integration engineering for ERP and manufacturing systems. > Section: Free Tools ## Short answer Compute the chance that an item measured inside a limit is truly outside it, as a function of margin and standard uncertainty. ## Category quality ## Industry Precision Manufacturing ## Audience - Metrologist - Quality engineer - Calibration lab technician ## Tool **Kind:** calculator **Inputs:** #### Margin inside the limit **Key:** margin **Type:** number **Default:** 0.003 **Min:** 0 **Step:** 0.0005 **Unit:** mm **Help:** Distance between the measured value and the nearest specification limit. #### Standard uncertainty **Key:** u **Type:** number **Default:** 0.002 **Min:** 0 **Step:** 0.0005 **Unit:** mm **Help:** Standard uncertainty u, not the expanded value. Divide expanded U by k first. #### Cost of rejecting a good item **Key:** cost_scrap **Type:** number **Default:** 40 **Min:** 0 **Step:** 5 **Unit:** currency #### Cost of shipping a bad item **Key:** cost_escape **Type:** number **Default:** 2500 **Min:** 0 **Step:** 50 **Unit:** currency **Outputs:** #### Margin in standard uncertainties **Key:** z **Expr:** margin / u **Format:** decimal #### Probability true value is out of tolerance **Key:** pfa **Expr:** (1 - normsdist(z)) * 100 **Format:** percent **Primary:** true **Description:** Assumes a normal distribution about the measured value and no prior information about the process. #### Confidence of conformity **Key:** conf **Expr:** 100 - pfa **Format:** percent **Primary:** true #### Expected escape cost per accepted item **Key:** exp_escape **Expr:** pfa / 100 * cost_escape **Format:** currency #### Expected cost if rejected instead **Key:** exp_reject **Expr:** conf / 100 * cost_scrap **Format:** currency **Description:** Chance the item was good multiplied by the cost of rejecting it. **Chart:** #### False accept probability vs margin **Type:** line **X:** margin **From:** 0 **To:** 0.008 **Steps:** 24 **Series:** - pfa **Disclaimer:** Single-sided limit and a flat prior. Bayesian methods with process knowledge, as in JCGM 106, give different and usually lower risk values. ## Content **Intro:** 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. **Sections:** #### How to use it **Body:** 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. **Bullets:** - 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 **Body:** 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 **Body:** 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 **Body:** 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. **Cta:** If you want pass or fail decisions in your ERP quality module to record uncertainty and decision rule used, we can help design that. **Faq:** #### What is probability of false accept? **Answer:** 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? **Answer:** 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? **Answer:** 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? **Answer:** No. Sampling consumer risk concerns lots and defect rates. This is the risk for one item measured with an uncertain instrument. ## Search terms probability of false accept calculator, pfa measurement uncertainty, conformity assessment decision rule, iso 14253-1 decision rule, risk of false acceptance calibration, measured value close to limit, uncertainty and conformance, false accept risk --- Talk to an engineer: https://netray.co/contact?intent=erp_ai | Machine catalog: https://netray.co/agent/catalog.json