2. Measurement Process Characterization
2.5. Uncertainty analysis
2.5.3. Type A evaluations

## Material inhomogeneity

Purpose of this page The purpose of this page is to outline methods for assessing uncertainties related to material inhomogeneities. Artifacts, electrical devices, and chemical substances, etc. can be inhomogeneous relative to the quantity that is being characterized by the measurement process.
Effect of inhomogeneity on the uncertainty Inhomogeneity can be a factor in the uncertainty analysis where
1. an artifact is characterized by a single value and the artifact is inhomogeneous over its surface, etc.
2. a lot of items is assigned a single value from a few samples from the lot and the lot is inhomogeneous from sample to sample.
An unfortunate aspect of this situation is that the uncertainty from inhomogeneity may dominate the uncertainty. If the measurement process itself is very precise and in statistical control, the total uncertainty may still be unacceptable for practical purposes because of material inhomogeneities.
Targeted measurements can eliminate the effect of inhomogeneity It may be possible to measure an artifact very carefully at a specific site and direct the user to also measure at this site. In this case there is no contribution to measurement uncertainty from inhomogeneity.
Example Silicon wafers are doped with boron to produce desired levels of resistivity (ohm.cm). Manufacturing processes for semiconductors are not yet capable (at least at the time this was originally written) of producing 2" diameter wafers with constant resistivity over the surfaces. However, because measurements made at the center of a wafer by a certification laboratory can be reproduced in the industrial setting, the inhomogeneity is not a factor in the uncertainty analysis -- as long as only the center-point of the wafer is used for future measurements.
Random inhomogeneities Random inhomogeneities are assessed using statistical methods for quantifying random errors. An example of inhomogeneity is a chemical compound which cannot be sufficiently homogenized with respect to isotopes of interest. Isotopic ratio determinations, which are destructive, must be determined from measurements on a few bottles drawn at random from the lot.
Best strategy The best strategy is to draw a sample of bottles from the lot for the purpose of identifying and quantifying between-bottle variability. These measurements can be made with a method that lacks the accuracy required to certify isotopic ratios, but is precise enough to allow between-bottle comparisons. A second sample is drawn from the lot and measured with an accurate method for determining isotopic ratios, and the reported value for the lot is taken to be the average of these determinations. There are therefore two components of uncertainty assessed:
1. component that quantifies the imprecision of the average
2. component that quantifies how much an individual bottle can deviate from the average.
Systematic inhomogeneities Systematic inhomogeneities require a somewhat different approach. Roughness can vary systematically over the surface of a 2" square metal piece lathed to have a specific roughness profile. The certification laboratory can measure the piece at several sites, but unless it is possible to characterize roughness as a mathematical function of position on the piece, inhomogeneity must be assessed as a source of uncertainty.
Best strategy In this situation, the best strategy is to compute the reported value as the average of measurements made over the surface of the piece and assess an uncertainty for departures from the average. The component of uncertainty can be assessed by one of several methods for evaluating bias -- depending on the type of inhomogeneity.
Standard method The simplest approach to the computation of uncertainty for systematic inhomogeneity is to compute the maximum deviation from the reported value and, assuming a uniform, normal or triangular distribution for the distribution of inhomogeneity, compute the appropriate standard deviation. Sometimes the approximate shape of the distribution can be inferred from the inhomogeneity measurements. The standard deviation for inhomogeneity assuming a uniform distribution is:

$$\displaystyle \large{ s_{inh} = \frac{1}{\sqrt{3}} MaxDeviation }$$