1. Uncertainty and Its Propagation Through Calculations
Engineering Experimental Design
Valerie L. Young

2. Uncertainty No measurement is perfect
Our estimate of a range likely to include the true value is called the uncertainty (or error)
Uncertainty in data leads to uncertainty in calculated results
Uncertainty never decreases with calculations, only with better measurements
Reporting uncertainty is essential
The uncertainty is critical to decision-making
Estimating uncertainty is your responsibility

3. Today’s Topics . . . How to report uncertainty
The numbers
The text
Identifying sources of uncertainty
Estimating uncertainty when collecting data
Uncertainty and simple comparisons
Propagation of error in calculations

4. Reporting Uncertainty – The Numbers
Experimental data and results always shown as
xbest ± x
Uncertainty gets 1 significant figure
Or 2 if it’s a 1, if you like
Best estimate gets rounded consistent with uncertainty
Keep extra digits temporarily when calculating

5. Examples Right Wrong (6050 ± 30) m/s (10.6 ± 1.3) gal/min (-16 ± 2) °C
(1.61 ± 0.05)  1019 coulombs
Wrong
( ± 32.21) m/s
( ± 2) °C

6. Fractional Uncertainty
x / |xbest |
Also called “relative uncertainty”
x is “absolute uncertainty”
x / |xbest | is dimensionless (no units)
Example
(-20 ± 2) °C  2 / |-20| = 0.10
-20 °C ± 10 %

7. Reporting Uncertainty – The Text
You must explain how you estimated each uncertainty. For example:
The reactor temperature was (35 ± 2) °C. The uncertainty. . .
. . .is estimated based on the thermometer scale.
. . .is given by the manufacturer’s specifications for the thermometer.
. . .is the standard deviation of 10 measurements made over the 30 minutes of the experiment.
. . .represents the 95% confidence limits for 10 measurements made over the 30 minutes of the experiment.

8. Reporting Uncertainty – The Text
You must explain how you estimated each uncertainty. For example:
The reactor temperature was (35 ± 2) °C. The uncertainty. . .
. . .is the standard deviation of 10 measurements made over the 30 minutes of the experiment.
. . .represents the 95% confidence limits for 10 measurements made over the 30 minutes of the experiment.
These estimates of uncertainty include both the precision of temperature control on the reactor and the precision of the measurement technique. They do not account for the accuracy of the measurement technique.

9. Precision vs. Accuracy
Precision
Accuracy

10. Estimating Uncertainty from Scales

11. Estimating Uncertainty from Scales

12. Graphical Display of Data and Results
Figure 1. Cell reproduction declines exponentially as the mass of growth inhibitor present increases. Vertical error bars represent standard deviation of 5 replicate measurements for one growth plate.

13. Experimental Results and Conclusions
A single measured number is uninteresting
An interesting conclusion compares numbers
Measurement vs. expected value
Measurement vs. theoretical prediction
Measurement vs. measurement
Do we expect exact agreement?
No, just “within experimental uncertainty”

14. Comparison and Uncertainty

15. Comparison and Uncertainty

16. Comparison and Uncertainty
xbest ± x means . . .
xtrue is probably between xbest - x and xbest + x
(later we’ll make “probably” quantitative)
Two values whose uncertainty ranges overlap are not significantly different
They are “consistent with one another”
A value just outside the uncertainty range may not be significantly different
More on this later (hypothesis testing)

17. Propagation of Uncertainties
We often do math with measurements
Density = (m ± m) / (V ± V)
What is the uncertainty on the density?
“Propagation of Error” estimates the uncertainty when we combine uncertain values mathematically
NOTE: don’t use error propagation if you can measure the uncertainty directly (as variation among replicate experiments)

18. Simple Rules Addition / Subtraction, q = x1 + x2 – x3 – x4
q = sqrt((x1)2+(x2)2+(x3)2+(x4)2)
Multiplication / Division, q = (x1x2)/(x3x4)
q/|q| = sqrt((x1/x1)2+(x2/x2)2+(x3/x3)2+(x4/x4)2)
1-Variable Functions, q = ln(x)
q = |dq/dx| x  |1/x| x

19. General Formula for Error Propagation
q = f(x1,x2,x3,x4)
q = sqrt(((q/ x1) x1)2 + ((q/ x2) x2)2 + ((q/ x3) x3)2 + ((q/ x4) x4)2 )

20. User Beware!
Error propagation assumes that the relative uncertainty in each quantity is small
Weird things can happen if it isn’t, particularly for functions like ln
e.g., ln(0.5 ± 0.4) = -0.7 ± 0.8
In this case, I suggest assuming that the relative error in x is equal to the relative error in f(x)
Don’t use error propagation if you can measure the uncertainty directly (as variation among replicate experiments)

21. Sample Calculation You pour the following into a batch reactor:
(100 ± 1) ml of 1.00 M NaOH in water
(1000 ± 1) ml of water
What is the concentration of NaOH in the batch reactor?

22. Sample Calculation ([NaOH] VNsOH) / (VNaOH+Vwater+Vwater))
The concentration of NaOH in the reactor is ( ± )M. The uncertainty was estimated by propagation of error, using the measurement uncertainties in the volumes added, and assuming an uncertainty of ± 0.01 M in the concentration of the 1.00 M NaOH solution. Note that writing 1.00 M implies an uncertainty of 0.01 M.