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

2.5.3.3.2.

Consistent 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:

Corrected result = Measurement - Estimate of bias

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 two-month 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,

\( \large{ Y_{qi}(q = 1, \cdots, Q, \,\, I = 1, \cdots, I) } \)

on Q artifacts with I instruments, the average bias for instrument, I' say, is

\( \displaystyle \large{ \overline{B}_{I'} = \frac{1}{Q} \sum_{q=1}^Q \left( Y_{qI'} - \overline{Y}_{q \, \scriptsize{\bullet}} \right) } \)

where

\( \displaystyle \large{ \overline{Y}_{q \, \scriptsize{\bullet}} = \frac{1}{I} \sum_{i=1}^I Y_{qi} \,\, . } \)

Computation of correction

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

\( \large{ Y_{corrected} = Y_{measured} - \overline{B}_{I'} } \)

Type A uncertainty of the correction

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

\( \displaystyle \large{ s_{correction} = \frac{1}{\sqrt{Q}} s_{bias} = \frac{1}{\sqrt{Q}} \sqrt{ \frac{1}{Q-1} \sum_{q=1}^Q \left( Y_{qI'} - \overline{Y}_{q \, \scriptsize{\bullet}} - \overline{B}_{I'} \right)^2 } } \)

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.