2.
Measurement Process Characterization
2.5. Uncertainty analysis 2.5.3. Type A evaluations 2.5.3.3. Type A evaluations of bias


Consistent bias 
Bias that is significant and persists consistently over time for a
specific instrument, operator, or configuration should be corrected if
it can be reliably estimated from repeated measurements. Results with
the instrument of interest are then corrected to:
The example below shows how bias can be identified graphically from measurements on five artifacts with five instruments and estimated from the differences among the instruments. 

Graph showing consistent bias for probe #5  An analysis of bias for five instruments based on measurements on five artifacts shows differences from the average for each artifact 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. Notice that on the graph for resistivity probes, probe #2362, (#5 on the graph), which is the instrument of interest for this measurement process, consistently reads low relative to the other probes. This behavior is consistent over 2 runs that are separated by a twomonth time period.  
Strategy  correct for bias  Because there is significant and consistent bias for the instrument of interest, the measurements made with that instrument should be corrected for its average bias relative to the other instruments.  
Computation of bias 
Given the measurements,
on Q artifacts with I instruments, the average bias for instrument, I' say, is where 

Computation of correction 
The correction that should be made to measurements
made with instrument I' is


Type A uncertainty of the correction 
The type A uncertainty of the correction is the
standard deviation of the average bias or


Example of consistent bias for probe #2362 used to measure resistivity of silicon wafers 
The table below comes from the
table of resistivity measurements
from a type A analysis of random effects with the average for each wafer
subtracted from each measurement. The differences, as shown, represent
the biases for each probe with respect to the other probes. Probe #2362
has an average bias, over the five wafers, of 0.02724 ohm.cm. If
measurements made with this probe are corrected for this bias, the
standard deviation of the correction is a type A uncertainty.
Table of biases for probes and silicon wafers (ohm.cm) Wafers Probe 138 139 140 141 142  1 0.02476 0.00356 0.04002 0.03938 0.00620 181 0.01076 0.03944 0.01871 0.01072 0.03761 182 0.01926 0.00574 0.02008 0.02458 0.00439 2062 0.01754 0.03226 0.01258 0.02802 0.00110 2362 0.03725 0.00936 0.02608 0.02522 0.03830 Average bias for probe #2362 =  0.02724 Standard deviation of bias = 0.01171 with 4 degrees of freedom Standard deviation of correction = 0.01171/sqrt(5) = 0.00523 

Note on different approaches to instrument bias  The analysis on this page considers the case where only one instrument is used to make the certification measurements; namely probe #2362, and the certified values are corrected for bias due to this probe. The analysis in the section on type A analysis of random effects considers the case where any one of the probes could be used to make the certification measurements. 