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

2.5.3.3.1.

Inconsistent bias

Strategy for inconsistent bias -- apply a zero correction

If there is significant bias but it changes direction over time, a zero correction is assumed and the standard deviation of the correction is reported as a type A uncertainty; namely,

\( \displaystyle \large{ s_{correction} = \frac{1}{\sqrt{3}} MaxBias } \)

Computations based on uniform or normal distribution

The equation for estimating the standard deviation of the correction assumes that biases are uniformly distributed between {-max |bias|, + max |bias|}. This assumption is quite conservative. It gives a larger uncertainty than the assumption that the biases are normally distributed. If normality is a more reasonable assumption, substitute the number '3' for the 'square root of 3' in the equation above.

Example of change in bias over time

The results of resistivity measurements with five probes on five silicon wafers are shown below for probe #283, which is the probe of interest at this level with the artifacts being 1 ohm.cm wafers. The bias for probe #283 is negative for run 1 and positive for run 2 with the runs separated by a two-month time period. The correction is taken to be zero.

          Table of biases (ohm.cm) for probe 283
            Wafer Probe    Run 1       Run 2

            -----------------------------------

              11   283   0.0000340  -0.0001841
              26   283  -0.0001000   0.0000861
              42   283   0.0000181   0.0000781
             131   283  -0.0000701   0.0001580
             208   283  -0.0000240   0.0001879

          Average  283  -0.0000284   0.0000652

A conservative assumption is that the bias could fall somewhere within the limits ± a, with a = maximum bias or 0.0000652 ohm.cm. The standard deviation of the correction is included as a type A systematic component of the uncertainty.

\( \displaystyle \large{ s_{correction} = \frac{1}{\sqrt{3}} MaxBias = 0.000038 \, \mbox{ohm.cm} } \)