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2. Measurement Process Characterization
2.5. Uncertainty analysis
2.5.3. Type A evaluations Material inhomogeneity Data collection and analysis

Purpose of this page The purpose of this page is to outline methods for:
  • collecting data
  • testing for inhomogeneity
  • quantifying the component of uncertainty
Balanced measurements at 2-levels The simplest scheme for identifying and quantifying the effect of inhomogeneity of a measurement result is a balanced (equal number of measurements per cell) 2-level nested design. For example, K bottles of a chemical compound are drawn at random from a lot and J (J > 1) measurements are made per bottle. The measurements are denoted by

\( \large{ Y_{11}, Y_{12}, \cdots, Y_{1j}; \, \cdots; \, Y_{k1}, Y_{k2}, \cdots, Y_{kj} }\)

where the k index runs over bottles and the j index runs over repetitions within a bottle.

Analysis of measurements The between (bottle) variance is calculated using an analysis of variance technique that is repeated here for convenience.

\( \large{ \displaystyle s_{inh}^2 = \frac{1}{K-1} \sum_{k=1}^K ( \overline{Y}_{k \, \scriptsize{\bullet}} - \overline{Y}_{\scriptsize{\bullet \bullet}} )^2 - \frac{1}{KJ(J-1)} \sum_{k=1}^K \sum_{j=1}^J ( Y_{kj} - \overline{Y}_{k \, \scriptsize{\bullet}} )^2 } \)


    \( \large{ \displaystyle \overline{Y}_{k \, \scriptsize{\bullet}} = \frac{1}{J} \sum_{j=1}^J Y_{kj} } \)
    \( \large{ \displaystyle \overline{Y}_{\scriptsize{\bullet \bullet}} = \frac{1}{K} \sum_{k=1}^K \overline{Y}_{k \, \scriptsize{\bullet}} \,\, . } \)
Between bottle variance may be negative If this variance is negative, there is no contribution to uncertainty, and the bottles are equivalent with regard to their chemical compositions. Even if the variance is positive, inhomogeneity still may not be statistically significant, in which case it is not required to be included as a component of the uncertainty.

If the between-bottle variance is statistically significantly (i.e., judged to be greater than zero), then inhomogeneity contributes to the uncertainty of the reported value.

Certification, reported value and associated uncertainty The purpose of assessing inhomogeneity is to be able to assign a value to the entire batch based on the average of a few bottles, and the determination of inhomogeneity is usually made by a less accurate method than the certification method. The reported value for the batch would be the average of N repetitions on Q bottles using the certification method.

The uncertainty calculation is summarized below for the case where the only contribution to uncertainty from the measurement method itself is the repeatability standard deviation, s1 associated with the certification method. For more complicated scenarios, see the pages on uncertainty budgets.

If \( \large{ s_{inh}^2 \le 0 \Rightarrow s_{reported \, value} = \frac{1}{\sqrt{QN}} s_1 } \)

If \( \large{ s_{inh}^2 > 0 } \) , we need to distinguish two cases and their interpretations:

  1. The standard deviation

    \( \displaystyle \large{ s_{reported \, value} = \sqrt{ \frac{Q+1}{Q} s_{inh}^2 + \frac{1}{QN} s_1^2} } \)

    leads to an interval that covers the difference between the reported value and the average for a bottle selected at random from the batch.

  2. The standard deviation

    \( \displaystyle \large{ s_{reported \, value} = \sqrt{ \frac{Q+1}{Q} s_{inh}^2 + \frac{QN+1}{QN} s_1^2} } \)

    allows one to test the instrument using a single measurement. The prediction interval for the difference between the reported value and a single measurement, made with the same precision as the certification measurements, on a bottle selected at random from the batch. This is appropriate when the instrument under test is similar to the certification instrument. If the difference is not within the interval, the user's instrument is in need of calibration.

Relationship to prediction intervals When the standard deviation for inhomogeneity is included in the calculation, as in the last two cases above, the uncertainty interval becomes a prediction interval ( Hahn & Meeker) and is interpreted as characterizing a future measurement on a bottle drawn at random from the lot.
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