**Question**

I have a concentration measurement for an ingredient in a bulk final product. This measurement is close to one of the spec limits. Given each measurement has an inherent error (or range), how do I calculate the probability this measurement is actually within the specs? Or, in other words, how do I calculate the probability that I’m right or wrong in accepting or discarding the batch based on that measurement?

**Response**

The total variability is the combination of process variation and measurement variability. Measurement variability is further defined as the sum of reproducibility and repeatability. Unfortunately, in many situations the measurement and process variation cannot be parsed from the total, especially when only a single value is collected for a measurement. Assuming a measurement study (MSA) was conducted, the repeatability from this study can be used to estimate the confidence interval (or tolerance interval) around a measurement. This would require knowing the degrees of freedom from the estimate of the repeatability from the measurement study to effective estimate the probability.

The reality is that you never know the probability of being right or wrong, as the assumption is that the measurement variability is acceptable, and that any reading close to the specification is not different than any other reading. If the measurement system is capable, we must use the value obtained from the measurement as the true value, without any probability of making a wrong decision.

I will use the following example to clarify the point.

Assume we have a repeatability from a measurement study of 0.25 (standard deviation of sigma error is 0.25) and one with a S=0.05. Assuming an upper specification of 79.25, the probability using the standard normal of a mean exceeding the specification when the observed value is below the specification is seen below.

Observed Value |
S=0.25 |
S=.05 |

79.00 | 15.9% | 0.0% |

79.10 | 27.4% | 0.1% |

79.20 | 42.1% | 15.9% |

79.25 | 50.0% | 50.0% |

Assume we have a repeatability from a measurement study of 0.25 (standard deviation of sigma error is 0.25) and one with a S=0.05. Assuming an upper specification of 79.25, the probability using the standard normal of a mean being less than the specification when the observed value is above the specification is seen below.

Observed Value |
S=0.25 |
S=.05 |

79.30 | 42.1% | 15.9% |

79.40 | 27.4% | 0.1% |

79.50 | 15.9% | 0.0% |

Based on the criticality of the measurement, the measurement system should be improved to minimize the probability of making either a Type 1 or Type II error.

Steven Walfish

Secretary, U.S. TAG to ISO/TC 69

ASQ CQE

Principal Statistician, BD

http://statisticaloutsourcingservices.com

The reply describes the calculated result, not how to calculate it, which is what the questioner asks.

The calculation is easily done in Excel using the formula = NORMDIST(A1,A2,A3,TRUE) with the inputs for true value, observed value and standard deviation in cells A1, A2 and A3, respectively.