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

Measurement configuration within the laboratory

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:
  • instrument
  • operator
  • temperature
  • humidity
The effect of uncontrollable environmental conditions in the laboratory can often be estimated from check standard data taken over a period of time, and methods for calculating components of uncertainty are discussed on other pages. Uncertainties resulting from controllable factors, such as operators or instruments chosen for a specific measurement, are discussed on this page.
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,

\( \large{ Y_{11}, Y_{12}, \cdots, Y_{1I}, \cdots, Y_{Q1}, Y_{Q2}, \cdots, Y_{QI} }\)

for each of Q artifacts and I instruments, the pooled standard deviation that describes the differences among instruments is:

\( \large{ s_{inst} = \sqrt{\frac{1}{Q} \left( \frac{1}{I-1} \right) \displaystyle \sum_{q=1}^Q \sum_{i=1}^I (Y_{qi} - \overline{Y}_{q \scriptsize{\, \bullet}})^2} }\)


\( \displaystyle \large{ \overline{Y}_{q \scriptsize{\, \bullet}} = \frac{1}{I} \sum_{i=1}^I Y_{qi} \,\, . } \)
Example of resistivity measurements on silicon wafers A two-way table of resistivity measurements ( 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.


 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

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