1. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainty and error in measurement
1.2.1 Describe and give examples of random and systematic errors.
1.2.2 Distinguish between precision and accuracy.
1.2.3 Explain how the effects of random errors may be reduced.
1.2.4 Calculate quantities and results of calculations to the appropriate number of significant figures.
© 2006 By Timothy K. Lund

2. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainty and error in measurement
Error in measurement is expected because of the imperfect nature of us and our measuring devices.
For example a typical meter stick has marks at every millimeter (10-3 m or 1/1000 m).
Thus the best measurement you can get from a typical meter stick is to the nearest mm.
EXAMPLE: Consider the following line whose length we wish to measure. How long is it?
SOLUTION:
It is closer to 1.2 cm than 1.1 cm, so we say it measures 1.2 cm (or 12 mm or m).
© 2006 By Timothy K. Lund 1
1 mm
1 cm

3. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainty and error in measurement
Error in measurement is expected because of the imperfect nature of us and our measuring devices.
We say the precision or uncertainty in our measurement is  1 mm.
As a rule of thumb, use the smallest increment of your measuring device as your uncertainty.
EXAMPLE: Consider the following line whose length we wish to measure. How long is it?
SOLUTION:
It is closer to 1.2 cm than 1.1 cm, so we say it measures 1.2 cm (or 12 mm or m).
© 2006 By Timothy K. Lund
FYI
We record
L = 12 mm  1 mm. 1
1 mm
1 cm

4. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainty and error in measurement
Random error is error due to the recorder, rather than the instrument used for the measurement.
Different people may measure the same line slightly differently.
You may in fact measure the same line differently on two different occasions.
Suppose Bob measures the line at 11 mm  1 mm and Ann measures it at 12 mm  1 mm.
Thus Bob guarantees that the line falls between 10 mm and 12 mm.
Ann guarantees it is between 11 mm and 13 mm.
Both are absolutely correct.
© 2006 By Timothy K. Lund 1

5. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainty and error in measurement
Random error is error due to the recorder, rather than the instrument used for the measurement.
Different people may measure the same line slightly differently.
You may in fact measure the same line differently on two different occasions.
Perhaps the ruler wasn’t perfectly lined up.
Perhaps your eye was viewing at an angle rather than head-on. This is called a parallax error.
© 2006 By Timothy K. Lund
FYI
The only way to minimize random error is to take many readings of the same measurement and to average them all together.

6. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainty and error in measurement
Systematic error is error due to the instrument being used for the measurement being “out of adjustment.”
For example, a voltmeter might not be zeroed properly.
A meter stick might be rounded on one end.
Now Bob measures the same line at 13 mm  1 mm.
Furthermore, every measurement Bob makes will be off by that same amount.
© 2006 By Timothy K. Lund 1
Worn off end
FYI
Systematic errors are hardest to detect and remove.

7. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
FYI: RANDOM ERROR is where accuracy varies in a random manner.
Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
FYI: SYSTEMATIC ERROR is where accuracy varies in a predictable manner.
Uncertainty and error in measurement
The following game where a catapult launches darts with the goal of hitting the bull’s eye illustrates the difference between precision and accuracy.
First Trial
Second Trial
Third Trial
Fourth Trial
© 2006 By Timothy K. Lund
Low Precision
Low Precision
High Precision
High Precision
Hits not grouped
Hits not grouped
Hits grouped
Hits grouped
Low Accuracy
High Accuracy
Low Accuracy
High Accuracy
Average well below bulls eye
Average right at bulls eye
Average well below bulls eye
Average right at bulls eye

8. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainty and error in measurement
Significant figures are the reasonable number of digits that a measurement or a calculation should have.
For example, as illustrated before, a typical wooden meter stick has two significant figures.
The number of significant figures in a calculation reflects the precision of the least precise of the measured values.
© 2006 By Timothy K. Lund

9. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainty and error in measurement
Significant figure rules: Live with them!
(1) All non-zero digits are significant
438 g
26.42 m
1.7 cm
0.653 L 3 4 2
(2) All zeros between non-zero digits are significant
225 dm
12060 m cm 3 4 5
© 2006 By Timothy K. Lund
(3) Filler zeros to the left of an understood decimal place are not significant
220 L
60 g
30. cm 2 1
(4) Filler zeros to the right of a decimal place are not significant.
0.006 L
0.8 g 1
(5) All non-filler zeros to the right of a decimal place are significant.
8.0 L
60.40 g cm 2 4 5

10. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainty and error in measurement
A calculation must be rounded to the same number of significant figures as in the measurement with the fewest significant figures.
EXAMPLE CALCULATOR SIG. FIGS
(1.2 cm)(2 cm) 2.4 cm2 2 cm2
π(2.75 cm) cm cm2
5.350 m/2.752 s m/s m/s
© 2006 By Timothy K. Lund
( n)(6 m) nm nm
EXAMPLE CALCULATOR SIG. FIGS
1.2 cm + 2 cm 3.2 cm 3 cm
2×103 m m m m
5.30×10-3m – 2.10m m m

11. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in calculated results
1.2.5 State uncertainties as absolute, fractional and percentage uncertainties.
1.2.6 Determine the uncertainties in results.
© 2006 By Timothy K. Lund

12. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in calculated results
Absolute, fractional and percentage uncertainties.
Absolute error is the raw uncertainty or precision of your measurement.
EXAMPLE: A student measures the length of a line with a wooden meter stick to be 11 mm  1 mm. What is the absolute error or uncertainty in her measurement?
SOLUTION:
The  number is the absolute error. Thus 1 mm is the absolute error.
1 mm is also the precision.
1 mm is also the raw uncertainty.
© 2006 By Timothy K. Lund

13. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in calculated results
Absolute, fractional and percentage uncertainties.
Fractional error is given by
fractional error
Absolute Error
Measured Value
Fractional Error =
EXAMPLE: A student measures the length of a line with a wooden meter stick to be 11 mm  1 mm. What is the fractional error or uncertainty in her measurement?
SOLUTION:
Fractional error =
© 2006 By Timothy K. Lund 1 11
= .09

14. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in calculated results
Absolute, fractional and percentage uncertainties.
Percentage error is given by
percentage error
Absolute Error
Measured Value
Percentage Error =
· 100%
© 2006 By Timothy K. Lund
EXAMPLE: A student measures the length of a line with a wooden meter stick to be 11 mm  1 mm. What is the percentage error or uncertainty in her measurement?
SOLUTION:
Percentage error = 1 11
·100%
= 9%

15. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in calculated results
Absolute, fractional and percentage uncertainties.
PRACTICE: A student measures the voltage of a calculator battery to be 1.6 V  0.1 V. What are the absolute, fractional and percentage uncertainties of his measurement? Find the precision and the raw uncertainty.
Absolute uncertainty is 0.1 V.
Fractional uncertainty is (0.1)/1.6 = 0.06.
Percentage uncertainty is 0.06(100%) = 6%.
Precision is 0.1 V.
Raw uncertainty is 0.1 V.
© 2006 By Timothy K. Lund

16. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in calculated results
To find the uncertainty in a sum or difference you just add the uncertainties of all the ingredients.
In formula form we have
If y = a  b then ∆y = ∆a + ∆b
uncertainty in sums and differences
© 2006 By Timothy K. Lund
FYI
Note that whether or not the calculation has a + or a -, the uncertainties are ADDED.
Uncertainties DO NOT EVER REDUCE ONE ANOTHER.

17. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in calculated results
To find the uncertainty in a sum or difference you just add the uncertainties of all the ingredients.
EXAMPLE: A 9.51  0.15 meter rope ladder is hung from a roof that is  0.07 meters above the ground. How far is the bottom of the ladder from the ground?
SOLUTION:
y = a – b = = 3.05 m
∆y = ∆a + ∆b = = 0.2 m
Thus the bottom is 3.05  0.2 m from the ground.
© 2006 By Timothy K. Lund

18. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in calculated results
To find the uncertainty in a product or quotient you just add the percentage or fractional uncertainties of all the ingredients.
In formula form we have
If y = ab/c then ∆y/y = ∆a/a + ∆b/b + ∆c/c
uncertainty in products and quotients
© 2006 By Timothy K. Lund
FYI
Note that whether or not the calculation has a  or a /, the uncertainties are ADDED.
Since you can’t add numbers not having the same units, we use fractional uncertainties for products and quotients.

19. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in calculated results
To find the uncertainty in a product or quotient you just add the percentage or fractional uncertainties of all the ingredients.
EXAMPLE: A car travels 64.7  0.5 meters in 8.65  0.05 seconds. What is its speed during this time interval?
SOLUTION:
r = d/t = 64.7/8.65 = 7.48 m s-1
∆r/r = ∆d/d + ∆t/t = .5/ /8.65 =
∆r/7.48 = so that
∆r = 7.48(0.0135) = 0.1 m s-1.
Finally we can state that the car is traveling at 7.48  0.1 m s-1.
© 2006 By Timothy K. Lund

20. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in graphs
1.2.7 Identify uncertainties as error bars in graphs.
1.2.8 State random uncertainty as an uncertainty range (+/-) and represent it graphically as an ‘error bar.’
1.2.9 Determine the uncertainties in the gradient and intercepts of a straight-line graph.
© 2006 By Timothy K. Lund

21. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in graphs
Identify uncertainties as error bars in graphs.
IB has a requirement that when you conduct an experiment of your own design, you must have three trials for each variation in your independent variable.
This means that for each independent variable you will gather three values for the dependent variable.
The three values for each dependent variable will then be averaged.
The following slide shows a sample of a well designed table containing all of the information required by IB.
© 2006 By Timothy K. Lund

22. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in graphs
Identify uncertainties as error bars in graphs.
3 trials
Independent variable manipulated by you
Sheets
n / no units
n = 0
Rebound Height
hi / cm
hi = 0.2 cm
Average Rebound Height
h / cm
h = 2.0 cm 55
54.8
55.1
54.6
Trial 1
Trial 2
Trial 3 2 52
53.4
52.5
49.6 4 49
50.7
48.7
48.6 6 48
49.0
47.1
48.5 8 45
45.9
45.0
44.6 10 41
39.5
41.4
42.4 12 35
35.8
34.0
35.1 14 33
31.1
33.5
33.0 16 29
29.7
27.2
29.3
Dependent variable responding to your changes
© 2006 By Timothy K. Lund

23. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in graphs
Identify uncertainties as error bars in graphs.
In order to determine the uncertainty in the dependent variable we reproduce the first two rows of the previous table:
The uncertainty in the height was taken to be half the largest range in the trial data, corresponding to the row for 2 sheets of paper:
Sheets
n / no units
n = 0
Rebound Height
hi / cm
hi = 0.2 cm
Average Rebound Height
h / cm
h = 2.0 cm 55
54.8
55.1
54.6
Trial 1
Trial 2
Trial 3 2 52
53.4
52.5
49.6
© 2006 By Timothy K. Lund 2
= 2

24. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in graphs
Identify uncertainties as error bars in graphs.
The size of the error bar in the graph is then up two and down two at each point in the graph on the next slide…
© 2006 By Timothy K. Lund

25. Error bars go up 2 and down 2 at each point.
© 2006 By Timothy K. Lund

26. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors
Uncertainties in graphs
Determine the uncertainties in the gradient and intercepts of a straight-line graph.
Now to determine the uncertainty in the slope of a best fit line we look only at the first and last error bars, as illustrated here for a different set of data:
A sample of a well-done graph for the Bounce-Height lab is shown on the next slide:
mmax
mbest
mmin
© 2006 By Timothy K. Lund
uncertainty in slope
mmax - mmin 2
mbest  m = mbest 

27. mmax - mmin m = 2 m = m = 0.25 m = -1.6 0.3 -1.375 - -1.875 2
© 2006 By Timothy K. Lund
m = 0.25
m = -1.6 0.3