1. Comparing Theory and Measurement
Agreement between theory and experiment does NOT
imply correctness.
Counter-examples include:
bad theory agreeing with bad data
bad theory agreeing with good data by coincidence
good theory agreeing with bad data because a variable
was not considered or controlled in the experiment
Scientific information can be misused selectively.
Comparisons must be made within the context of
uncertainty.

2. How Sure Are We ?
When a physical process is quantified, uncertainties associated with describing the process occur.
Uncertainties result from
Experiments Modeling

3. Systematic and Random Uncertainties
An error is the difference between the measured and
the true value.
An uncertainty is an estimate of the error.
Uncertainties are categorized as either systematic (bias)
or random (precision).
An uncertainty is assumed to be systematic if no
statistical information is provided.

4. Systematic and Random Uncertainties
Systematic, Bi:
arises from comparisons with standards (calibration)
involves no statistics; the number is given alone
related to the accuracy (‘to within ±Bi units’)
Random, Pi:
based upon repeated measurements
involves statistics ( )
related to the precision (scatter)
for one more measurement
for multiple measurements

5. Systematic and Random Uncertainties
Figure 9.2

6. Precision and Accuracy
Precision Accuracy
good poor
good good
poor poor

7. Measurement Uncertainty Analysis
Overall goal:
Obtain an estimate of Ux, where
x' = x ± Ux (%C)
x can be either a single value or an average value.
The magnitude of Ux depends upon the percent
confidence (%C), the contributing uncertainties, and
how the contributing uncertainties are combined.

8. Measurement Uncertainty Analysis
The overall uncertainty, Ux, is related to the combined
standard uncertainty, uc, through a coverage factor, where
For most experiments, N ≥ 10  tn,95 ≈ 2 (<10 % error).
This implies Ux,95 ≈ 2uc = This is called the large
scale approximation.

9. Student’s t Table
Gives the value of t for a given n and P % confidence .
What is t for N = 12 ?
Table 8.4

10. Quadrature Combination of Uncertainties
The combined standard uncertainty, uc, comes from the
combined estimated variance, uc2, which is expressed as:
assuming each xi is independent of the other J-1 variables.
r denotes a result (a variable that is a function of one or
more measurands)
qi is the absolute sensitivity coefficient, which weights the
uncertainty contribution of xi to the result.
qi = 1 when the result is simply a measurand.