Uncertainty analysis involves quantifying errors in measurement systems and experiments. Total experimental error is determined by finding the precision error due to environmental factors and bias error due to equipment. If possible, the error of each experimental component should be determined to identify areas for minimizing error. Errors propagate in data processing based on functional relationships between variables. Proper experimental design and system calibration can help reduce bias and precision errors.

1. Uncertainty Analysis
ME 310 – Engineering
Experimentation I
Only two things are infinite,
the universe and human stupidity,
and I’m not sure about the former.
- Albert Einstein

2. Measurement Errors
• Now that we know how to quantify errors, we have to
apply these to sensors and DAQ systems and the
experiments we perform with them
• Just as we did with sample measurements, we wish to
express measurement uncertainty as
• To do this with some assurance, we would like to
determine the error of each component or portion of the
exper€im ent or procedure and determine the total error

3. Determination of Total Error
• At minimum, determine total error by finding error due to
equipment (bias) and due to environment (precision)
• If possible, find error of each component or portion of
experiment to determine where error may be minimized
• Errors propagate in post-processing phase
– Quantities that are non-linear functions of a
measurement or functions of multiple measurements
will errors based upon the functional relationship

4. Bias and Single-Sample Uncertainty
• Provides no direct evidence of either bias
error’s
magnitude or its presence
• Can only be found definitively by repeating
measurement
on another piece of equipment, which is
seldom done
• Estimated by examining
– Applicability of handbook or manufacturers
values for
physical properties used
– Calibration; standards used during
calibration
procedure

5. Contribution of Bias & Precision Error
• Both bias and precision errors are present in an experiment
• The precision is measured; the bias error is usually
determined from equipment vendor specs
• The total error is the vector sum of these errors
• Note that errors in estimating each error affect your value
of the total error

6. Propagation of Uncertainty
• Assume that a calculated result y is a function of several different
measured variables x1, x2, x3,…,xn having uncertainties u1, u2, u3,…,un
.
If the individual uncertainties are statistically independent the uncertainty
of y is:

7. Example: Determine the uncertainty of the estimate of a
cylinder’s density from measurements of its mass, diameter,
and
length:

8. Measurement Procedure & Sources of Error

9. Sources of Error
• Calibration
• Data Acquisition
• Post-Processing and data reduction
• Even assuming no mistakes are made, there are
multiple
possibilities of errors in each step
• Any of these can be bias or precision; in general
– Precision: random and can be statistically estimated
– Bias: typically fixed and must be estimated by
comparison

10. General Procedure
• Determine possible sources of error
• Obtain bias errors from vendor specs for each
component
or subsystem of the measurement system
• Calibrate the instruments before the experiments
• Determine maximum reasonable number of
measurements
possible
• Statistically find the precision errors; determine
confidence
• Find sums of precision and bias errors
• Find total error

11. Approaches for Minimizing Experimental Error
• Best time to minimize error is in experiment’s design stage
• Perform an uncertainty analysis before an experiment is built
• General precautions to observe during the design of an experiment
– Avoid approaches that require two large numbers to be measured in
order
to determine the small difference between them
– Design experiments or sensors that amplify the signal strength in
order to
increase sensitivity
– Build null designs where the output is measured as a change from
zero
(reduces bias and precision error – e.g. Wheatstone Bridge)
– Avoid experiments in which large correction factors are required
– Attempt to minimize the influence of measuring system on the
measurand
– Calibrate entire systems, rather than individual components, to
minimize
calibration-related bias error

12. Summary
Use z- or t-tables to determine confidence from means and
standard deviations
Total error = vector sum of precision and bias errors
Propagation of error is vector sum of all errors