2.
Measurement Process Characterization
2.5. Uncertainty analysis 2.5.3. Type A evaluations 2.5.3.1. Type A evaluations of random components


Purpose of this page 
The purpose of this page is to outline options for estimating
uncertainties related to the specific measurement configuration
under which the test item is measured, given other possible
measurement configurations. Some of these may be controllable and
some of them may not, such as:


First, decide on context for uncertainty  The approach depends primarily on the context for the uncertainty statement. For example, if instrument effect is the question, one approach is to regard, say, the instruments in the laboratory as a random sample of instruments of the same type and to compute an uncertainty that applies to all results regardless of the particular instrument on which the measurements are made. The other approach is to compute an uncertainty that applies to results using a specific instrument.  
Next, evaluate whether or not there are differences  To treat instruments as a random source of uncertainty requires that we first determine if differences due to instruments are significant. The same can be said for operators, etc.  
Plan for collecting data  To evaluate the measurement process for instruments, select a random sample of I (I > 4) instruments from those available. Make measurements on Q (Q >2) artifacts with each instrument.  
Graph showing differences among instruments  For a graphical analysis, differences from the average for each artifact can be plotted versus artifact, with instruments individually identified by a special plotting symbol. The plot is examined to determine if some instruments always read high or low relative to the other instruments and if this behavior is consistent across artifacts. If there are systematic and significant differences among instruments, a type A uncertainty for instruments is computed. Notice that in the graph for resistivity probes, there are differences among the probes with probes #4 and #5, for example, consistently reading low relative to the other probes. A standard deviation that describes the differences among the probes is included as a component of the uncertainty.  
Standard deviation for instruments 
Given the measurements,
for each of Q artifacts and I instruments, the pooled standard deviation that describes the differences among instruments is:
where


Example of resistivity measurements on silicon wafers 
A twoway table of resistivity
measurements (ohm.cm) with 5 probes on 5 wafers (identified as: 138,
139, 140, 141, 142) is shown below. Standard deviations for probes with
4 degrees of freedom each are shown for each wafer. The pooled standard
deviation over all wafers, with 20 degrees of freedom, is the type A
standard deviation for instruments.
Wafers Probe 138 139 140 141 142  1 95.1548 99.3118 96.1018 101.1248 94.2593 281 95.1408 99.3548 96.0805 101.0747 94.2907 283 95.1493 99.3211 96.0417 101.1100 94.2487 2062 95.1125 99.2831 96.0492 101.0574 94.2520 2362 95.0928 99.3060 96.0357 101.0602 94.2148 Std dev 0.02643 0.02612 0.02826 0.03038 0.02711 DF 4 4 4 4 4 Pooled standard deviation = 0.02770 DF = 20 