1. Assigning and Propagating Uncertainties

2. Assigning Uncertainties
In the laboratory experiments you will be performing you will have to make measurements; the results of these measurements will always be uncertain to some extent
For example, suppose you use a balance to determine that the mass (m) of some chemical constituent used in your experiment is 15.87g; but your “weighing” apparatus only had two decimal places and the mass could be anywhere from to 15.88g
Your best estimate was 15.87g and there was uncertainty of 0.01g
You would then report your result as:
m = 15.87g ± 0.01g
NOTE: THE UNITS (grams) APPLY TO BOTH THE MEASUREMENT AND THE UNCERTAINTY

3. Assigning Uncertainties
YOU SHOULD ASSIGN AN UNCERTAINTY TO EVERY MEASUREMENT THAT YOU MAKE!!!
In the example on the previous slide, the uncertainty resulted from a limitation in the resolution of the instrument used; you might use a balance with a higher resolution (three decimal places) and find m = g ± 0.001g

4. Assigning Uncertainties
For example, you may measure the time (t) that it takes for a reaction to occur
On successive tries you find t = 1.3, 1.4, 1.2, 1.5, 1.4, 1.4, 1.3, and 1.5 seconds
It might then be reasonable to report your results as: t = 1.4 ± 0.1 seconds, since the range from 1.3 to 1.5 seconds encompasses almost all of the experimental readings

5. Assigning Uncertainties
No hard and fast rules are possible for assigning uncertainties, instead you must be guided by common sense.
If the space between the scale divisions is large, you may be comfortable in estimating to 1/5 or 1/10 of the least count.
If the scale divisions are closer together, you may only be able to estimate to the nearest 1/2 of the least count
If the scale divisions are very close you may only be able to estimate to the least count.

6. Assigning Uncertainties
A solid rule of thumb when trying to determine an uncertainty for a given measuring instrument is:
“Half the smallest unit measured”
For example, when using a graduated cylinder that has 1 mL gradations the uncertainty would be ±0.5mL
It is also helpful to look at the measurement instrument and see if an uncertainty is printed on the instrument

7. Absolute and Relative Uncertainty
Uncertainty can be expressed in two different ways
Absolute uncertainty refers to the actual uncertainty in a quantity
For example, the have three mass measurements:
6.3302g
6.3301g
6.3303g
The average is ±0.0001g
The absolute uncertainty is g

8. Absolute and Relative Uncertainty
Relative uncertainty expresses the uncertainty as a fraction of the quantity of interest
Other ways of expressing relative uncertainty are in per cent, parts per thousand, and part per million
For the following example of an object with mass ±0.0001g, the relative uncertainty is g / g which is equal to 2 × 10-3 percent, or 2 parts in 100, 000 or 20 parts per million
Relative uncertainty is a good way to obtain a qualitative idea of the precision of your data and results

9. Absolute and Relative Uncertainty
In general, results of observations should be reported in such a way that the last digit given is the only one whose value is uncertain due to random errors
Note that systematic and random errors refer to problems associated with making measurements.
Mistakes made in the calculations or in reading the instrument are not considered in error analysis.
It is assumed that the experimenters are careful and competent!

10. Definition of Propagation of Uncertainties
Once you have assigned uncertainties to your experimentally measured quantities, you will probably have to combine these quantities in some way in order to determine some derived quantity
For example, you measure the distance that an object traveled, say, 1.00 ± 0.01 meters and the time during which it moved, say 2.0 ± 0.1 seconds; now you want to know the average speed of that object

11. Definition of Propagation of Uncertainties
How will you combine the uncertainties in the two measured quantities in order to find the uncertainty in speed?
The manner in which uncertainties combine to give the resultant uncertainty is called propagation of uncertainties

12. Propagation of Uncertainties
There are three different ways of calculating or estimating the uncertainty in calculated results

13. Propagation of Uncertainties
Significant Figures: the easy way out
Use the implicit uncertainty in each measurement’s significant figures to determine significant figures in the result by following the rules for adding, subtracting, multiplying and dividing
Useful when a more extensive uncertainty analysis is not needed. A reasonable estimate of the uncertainty should always be implied by the significant figures in any calculated result

14. Propagation of Uncertainties
2. Uncertainty Propagation: Not as bad as it looks
Uses uncertainty or precision of each measurement, arising from limitations of measuring devices. Contribution of each uncertainty to the final result is calculated
Useful for limited number or single measurements

15. Propagation of Uncertainties
3. Statistical Methods: When you have lots of numbers
Uses the spread in the values of many repeated results to estimate the uncertainty in their average
Useful for many repeated measurements and for fits to analytical equations, linear or otherwise

16. Propagation of Uncertainties

17. Propagation of Uncertainties
SIGNIFICANT FIGURE ANALYSIS:
Multiplication and division:
The results has the same number of significant figures as the smallest of the number of significant figures for any value used in the calculation
Addition and subtraction:
The result will have a last significant digit in the same place as the left-most of the last significant digits of all the numbers used in calculation

18. Propagation of Uncertainties
2. UNCERTAINTY PROPAGATION:
Addition and Subtraction:
When adding or subtracting numbers indicating uncertainties you also add the uncertainties
Example:
1384 ± 2g ± 2 g = 1495 ± 4g
45.34 ± 0.05ºC – ± 0.05ºC = 22.0 ± 0.1ºC

19. Propagation of Uncertainties
2. UNCERTAINTY PROPAGATION:
Multiplication and Division:
When multiplying or dividing numbers indicating uncertainties you add the relative uncertainties (percent uncertainties)
Example:
2810 ± 4g ÷ 7.43 ± 0.05 dm3
2810 g ± 0.14% ÷ 7.43dm3 ± 0.67% = 378 g dm-3 ± 0.81%
378 ± 3 g dm-3 (This is reasonable because the uncertainty matches the last place of significance)

20. Propagation of Uncertainties
2. UNCERTAINTY PROPAGATION:
Exponents and Square Roots:
Exponents are essentially multiplication so, for x2 the uncertainty is equal to 2 times the relative uncertainty (percent uncertainty)
2(Δx/x)
For a cubed unit the uncertainty would be 3 times the relative uncertainty
A square root would be ½ times the uncertainty

21. Propagation of Uncertainties
2. UNCERTAINTY PROPAGATION:
Exponents and Square Roots:
Exponents are essentially multiplication so, for x2 the uncertainty is equal to 2 times the relative uncertainty (percent uncertainty)
2(Δx/x)
For a cubed unit the uncertainty would be 3 times the relative uncertainty
A square root would be ½ times the uncertainty

22. Propagation of Uncertainties
2. UNCERTAINTY PROPAGATION:
Logarithms:
x ± a
log x
the new uncertainty is calculated by:
Xnew = (a/x)
ln x
xnew = a/x

23. Propagation of Uncertainties
2. UNCERTAINTY PROPAGATION:
Pure Numbers:
This refers to numbers such as 2, ½, π, etc.
In general, a division of a measurement by 2 will reduce the absolute uncertainty by 2
Example:
43.2 cm ± 2.6 cm x 3.0 =
= cm ± 7.8 cm

24. Propagation of Uncertainties
2. UNCERTAINTY PROPAGATION:
Uncertainty In A Mean:
An average value will have the same number of significant figures as the least precise measured value
A reasonable idea of the uncertainty can be obtained by dividing the range of values by the number of values
Uncertainty in mean ~ xmax – xmin / n

25. Propagation of Uncertainties
2. UNCERTAINTY PROPAGATION:
Uncertainty In A Mean:
Example:
Five temperature measurements were taken: 23.1ºC, 22.5ºC, 21.9ºC, 22.8ºC, 22.5ºC
Determine the mean and uncertainty for this data:
Mean = / 5 = 22.6ºC
Uncertainty in mean ~ 23.1 – 21.9/5 = 0.2
So the mean temperature is 22.6 ± 0.2ºC