Topic 1.2 Uncertainties and
Error
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Essential idea:
Scientists aim towards designing experiments that
can give a “true value” from their measurements, but
due to the limited precision in measuring devices,
they often quote their results with some form of
uncertainty.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Nature of science:
Uncertainties: “All scientific knowledge is uncertain.
When the scientist tells you he does not know the
answer, he is an ignorant man. When he tells you he
has a hunch about how it is going to work, he is
uncertain about it. When he is pretty sure of how it is
going to work, he still is in some doubt. And it is of
paramount importance, in order to make progress,
that we recognize this ignorance and this doubt.
Because we have the doubt, we then propose
looking in new directions for new ideas.”
– Feynman, Richard P. 1998. The Meaning of
It All: Thoughts of a Citizen-Scientist.
Reading, Massachusetts, USA. Perseus. P 13.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Understandings:
• Random and systematic errors
• Absolute, fractional and percentage uncertainties
• Error bars
• Uncertainty of gradients and intercepts
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Applications and skills:
• Explaining how random and systematic errors can be
identified and reduced
• Collecting data that include absolute and/or fractional
uncertainties and stating these as an uncertainty
range (expressed as: best estimate ± uncertainty
range)
• Propagating uncertainties through calculations
involving addition, subtraction, multiplication, division
and raising to a power
• Determining the uncertainty in gradients and intercepts
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Guidance:
• Analysis of uncertainties will not be expected for
trigonometric or logarithmic functions in
examinations
Data booklet reference:
• If y = a  b then y = a + b
• If y = a · b / c then y / y = a / a + b / b + c / c
• If y = a n then y / y = | n · a / a |
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Theory of knowledge:
• “One aim of the physical sciences has been to give an
exact picture of the material world. One achievement
of physics in the twentieth century has been to prove
that this aim is unattainable.” – Jacob Bronowski.
• Can scientists ever be truly certain of their
discoveries?
Jacob Bronowski was a
mathematician, biologist, historian
of science, theatre author, poet and
inventor. He is probably best
remembered as the writer of The
Ascent of Man.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Utilization:
• Students studying more than one group 4 subject will
be able to use these skills across all subjects
Aims:
• Aim 4: it is important that students see scientific errors
and uncertainties not only as the range of possible
answers but as an integral part of the scientific
process
• Aim 9: the process of using uncertainties in classical
physics can be compared to the view of
uncertainties in modern (and particularly quantum)
physics
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Random and systematic errors
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).
EXAMPLE: Consider the following line whose length we
wish to measure. How long is it?
0
1
SOLUTION: It is closer
1 cm
1 mm
to 1.2 cm than to 1.1 cm,
so we say it measures 1.2 cm (or 12 mm or 0.012 m).
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Random and systematic errors
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.
EXAMPLE: Consider the following line whose length we
wish to measure. How long is it?
0
1
SOLUTION: It is closer
1 cm
1 mm
to 1.2 cm than to 1.1 cm,
so we say it measures 1.18 cm (11.8 mm or 0.0118m)
m).
FYI We record L = 11.8  .5 mm.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Random and systematic errors
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.
EXAMPLE: Suppose Bob measures L = 11.0 mm
 .5 mm and Ann measures L = 12.1 mm  .5 mm.
0
1
Then 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.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Random and systematic errors
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.
FYI
The only way to minimize random error
is to take many readings of the same
measurement and to average them all together.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Random and systematic errors
Systematic error is error due to the instrument being
“out of adjustment.”
A voltmeter might have a zero offset error.
A meter stick might be rounded on one end.
Now Bob measures the same line at 13 mm  1 mm.
Worn
1
off end 0
Furthermore, every measurement Bob makes will be off
by that same amount.
FYI
Systematic errors are usually difficult to detect.
Types of Experimental Errors:
• So what can be done about random errors?
– Don’t rush through your measurements! Be careful!
– Take as many trials as possible—the more trials you do, the
less likely one odd result will impact your overall lab results
Types of Experimental Errors:
• Systematic Errors:
– Errors that are inherent to the system or the
measuring instrument
– Results in a set of data to be centered around
a value that is different than the accepted
value
• Some Examples:
– Non-calibrated (or poorly calibrated)
measuring tools
– A “zero offset” on a measuring tool, requiring
a “zero correction”
– Instrument parallax error
Types of Experimental Errors:
• What can be done to reduce these?
– Unfortunately, nothing…HOWEVER:
– We can account for the systematic errors sometimes:
• i.e. if there’s a zero offset, make sure all your data has been adjusted
to account for that.
– Recognizing systematic errors will impact the size of your
absolute uncertainty (more details soon )
Are these “errors”?
• Misreading the scale on a triple-beam balance
• Incorrectly transferring data from your rough data table to
the final, typed, version in your report
• Miscalculating results because you did not convert to the
correct fundamental units
• Miscalculations because you use the wrong equation
Are these “errors”?
• NONE of these are experimental errors
• They are MISTAKES
• What’s the difference?
– You need to check your work to make sure these mistakes
don’t occur…ask questions if you need to (of your lab partner,
me, etc.)
– Do NOT put mistakes in your error discussion in the conclusion
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Random and systematic errors
The following game where a catapult launches darts
with the goal of hitting the bull’s eye illustrates the
difference between precision and accuracy.
Trial 1 RE
SE
Trial 2 RE
Trial 3 RE
SE
SE
Hits not grouped
Low Precision
High Precision
Hits not grouped Hits grouped
Low Accuracy
High Accuracy
Low Accuracy
Low Precision
Average well
below bulls eye
Average right
at bulls eye
Average well
below bulls eye
Trial 4 RE
SE
High Precision
Hits grouped
High Accuracy
Average right
at bulls eye
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Random and systematic errors
PRACTICE:
SOLUTION:
This is like the rounded-end ruler. It will produce a
systematic error.
Thus its error will be in accuracy, not precision.
Uncertainties in Measurement
• Limit of Reading:
– Equal to the smallest graduation of the scale on an instrument
• Degree of Uncertainty:
– Equal to half the limit of reading
– Gives an indication of the precision of the reading
Uncertainties in Measurement
• Absolute Uncertainty:
– The size of an error, including units
– The SMALLEST the uncertainty can be is equal to the degree of
uncertainty, however it is ALMOST ALWAYS BIGGER!
• The absolute uncertainty can NOT be more precise than
your measurement
• The absolute uncertainty is ALWAYS reported to only 1
sig. fig.
• Note: +/- is sometimes symbolized with  (Greek letter
Delta)
Examples:
• Acceptable:
1.62 +/- 0.01 m
• NOT acceptable:
1.62 +/- 0.005 m
More uncertainties in measurement
• Relative (fractional) uncertainty:
– Equal to the ratio of the absolute uncertainty to the
measurement:
absolute uncertaint y
measurement
• Percentage uncertainty:
– (fractional uncertainty) x 100 = %
In order to compare uncertainties, use is made of absolute,
fractional and percentage uncertainties.
1 mm is the absolute uncertainty
1/208 is the fractional uncertainty (0.0048)
0.48 % is the percentage uncertainity
Absolute/fractional errors and percentage
errors
We use ± to show an error in a measurement
(208 ± 1) mm is a fairly accurate measurement
(2 ± 1) mm is highly inaccurate
Uncertainty Propagation
• When we perform calculations with measurements that
have uncertainties, there is a certain amount of
uncertainty in our calculated answer as well.
• Carrying our errors through the calculations is called
“error propagation” or “uncertainty propagation”
Rule 1: Addition/Subtraction
• When you are adding or subtracting values in a
calculation, the uncertainty in the calculated answer is
equal to the sum of the absolute uncertainties for each
measurement added:
24.10 ± 0.05 g
+ 13.05 ± 0.02 g
= 37.15 ± 0.07 g
Rule #2: Multiplying/Dividing
• When multiplying or dividing, the
uncertainty in the calculated value is equal
to the sum of the percentage
uncertainties for each of the individual
measurements:
For example, let’s say we were to calculate the volume
from the following measurements:
(12.0 ± 0.2 cm)*(23.1 ± 0.2 cm)*(7.5 ± 0.1 cm)
• Step 1: determine the %
uncertainty for each
measurement
• Step 2: Add all of the %
uncertainties and round
to 1-2 sig figs (usually a
whole number, although if
under 1 keep as a one
s.f. decimal percentage:
0.2
 100  1.7%
12.0
0.2
 100  0.9%
23.1
0.1
 100  1.3%
7.5
1.7%  0.9%  1.3%
 3.9%  4%
• Step 3: convert the percentage
uncertainty in your answer back to an
absolute uncertainty:
(12.0 ± 0.2 cm)*(23.1 ± 0.2 cm)*(7.5 ± 0.1 cm)
= 2100 cm3 ± 4%
0.04 x 2100 cm3 = 84 cm3 ≈ 80cm3
(note: since the precision of the measurement and the
uncertainty MUST BE THE SAME, always go with the
least precise of the two when reporting your final
answer.)
V = 2100 ± 80 cm3
Express the answer with the
correct uncertainty
(45 +/- 3) + (12 +/- 2) = ??
57 +/- 5
Express the answer with the
correct uncertainty
(12.1 +/- .3) - (4.5 +/- .6) = ??
7.6 +/- .9
Express the answer with the
correct uncertainty
(11 +/- 3) - (7 +/- 2) = ??
4 +/- 5 Ummm?
Express the answer with the
correct % uncertainty
(6 +/- 3%) x (8 +/- 2%) = ??
6x8 = 48
uncty = 3% + 2% = 5%
48 + 5%
48 + 5%
W
Express the answer with the
correct % uncertainty
(72 +/- 5%)  (9 +/- 2%) = ??
72/9 = 8
uncty = 5% + 2% = 7%
8 + 7%
8 + 7%
W
Example:
A metal plate measures 21.1 + .5 cm by 15.3 + .1 cm.
What is its area?
Find the area:
21.1 x 15.3 = 322.83
Use the formula:
y
= .5 + .1
322.83
21.1 15.3
y = 9.76
323+ 10cm2
TOC
Express the answer with the
correct uncertainty
(45 +/- 3) x (12 +/- 2) = ??
12 x 45 = 540
uncty = 540(3/45+2/12) = 126
540 +/- 100
540 +/- 130
W
Express the answer with the
correct uncertainty
(30 +/- .7)  (1.25 +/- .1) = ??
30/1.25 = 24
uncty = 24(.7/30+.1/1.25) = 2.48
20+/- 2
24 +/- 2.5
W
Express the answer with the correct uncertainty
A circle has a radius of 4.5 + .1 cm. What is its
area?
Area = r2 =  x r x r
r2 = 4.52 = 63.6173
uncty = 63.6173(.1/4.5+.1/4.5) = 2.83
64+ 3cm2
63.6 + 2.8 cm2
W
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
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.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Absolute, fractional and percentage uncertainties
Fractional error is given by
fractional error
Fractional Error =
Absolute Error
Measured Value
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 = 1 / 11 = 0.09.
FYI
”Fractional” errors are usually expressed as
decimal numbers rather than fractions.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Absolute, fractional and percentage uncertainties
Percentage error is given by
percentage error
Absolute Error
Percentage Error = Measured Value · 100%
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%
FYI
Don’t forget to include the percent sign.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Absolute, fractional and percentage uncertainties
PRACTICE:
A student measures the voltage shown. What
are the absolute, fractional and percentage
uncertainties of his measurement? Find the
precision and the raw uncertainty.
SOLUTION:
Absolute uncertainty = 0.001 V.
Fractional uncertainty = 0.001/0.385 = 0.0026.
Percentage uncertainty = 0.0026(100%) = 0.26%.
Precision is 0.001 V.
Raw uncertainty is 0.001 V.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Absolute, fractional and percentage uncertainties
PRACTICE:
SOLUTION:
Find the average of the two measurements:
(49.8 + 50.2) / 2 = 50.0.
Find the range / 2 of the two measurements:
(50.2 – 49.8) / 2 = 0.2.
The measurement is 50.0  0.2 cm.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Propagating uncertainties through calculations
To find the uncertainty in a sum or difference you
just add the uncertainties of all the ingredients.
In formula form we have
uncertainty in sums and differences
If y = a  b then ∆y = ∆a + ∆b
FYI
Note that whether or not the calculation has a + or a -,
the uncertainties are ADDED.
Uncertainties NEVER REDUCE ONE ANOTHER.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Propagating uncertainties through calculations
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 12.56  0.07 meters above the
ground. How far is the bottom of the ladder
from the ground?
SOLUTION:
y = a – b = 12.56 - 9.51 = 3.05 m
∆y = ∆a + ∆b = 0.15 + 0.07 = 0.22 m
Thus the bottom is 3.05  0.22 m from the ground.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Propagating uncertainties through calculations
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
uncertainty in products and quotients
If y = a · b / c then ∆y / y = ∆a / a + ∆b / b + ∆c / c
FYI
Whether or not the calculation has a  or a ,
the uncertainties are ADDED.
You can’t add numbers having different units, so we
use fractional uncertainties for products and quotients.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Propagating uncertainties through calculations
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?
SOLUTION: Use rate = distance divided by time.
r = d / t = 64.7 / 8.65 = 7.48 m s-1
∆r / r = ∆d / d + ∆t / t = .5 / 64.7 + .05 / 8.65 = 0.0135
∆r / 7.48 = 0.0135 so that
∆r = 7.48( 0.0135 ) = 0.10 m s-1.
Thus, the car is traveling at 7.48  0.10 m s-1.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Propagating uncertainties through calculations
PRACTICE:
SOLUTION:
∆P / P = ∆I / I + ∆I / I + ∆R / R
∆P / P = 2% + 2% + 10% = 14%.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Propagating uncertainties through calculations
PRACTICE:
SOLUTION:
∆F / F = 0.2 / 10 = 0.02 = 2%.
∆m / m = 0.1 / 2 = 0.05 = 5%.
∆a / a = ∆F / F + ∆m / m = 2% + 5% = 7%.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Propagating uncertainties through calculations
PRACTICE:
SOLUTION:
∆r / r = 0.5 / 10 = 0.05 = 5%.
A = r2.
Then ∆A / A = ∆r / r + ∆r / r = 5% + 5% = 10%.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Uncertainty of gradient and intercepts
IB has a requirement that when you conduct an
experiment of your own design, you must have five
variations in your independent variable.
And for each variation of your independent variable
you must conduct three trials to gather the values of
the dependent variable.
The three values for each dependent variable will then
be averaged.
The
“good”
header
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
( responding )
Dependent
( manipulated )
Independent
3 Trials
Uncertainty of gradient and intercepts
This is a well designed table containing all of
the information and data points required by IB:
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Uncertainty of gradient and intercepts
In order to determine the uncertainty in the dependent
variable we reproduce the first two rows of the previous
table:
The uncertainty in the average height h was taken to
be half the largest range in the trial data, which is in
the row for n = 2 sheets: 53.4 - 49.6 = 2.0.
2
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Uncertainty of gradient and intercepts
The size of the error bar in the graph is then up two
and down two at each point in the graph:
Error bars
go up 2 and
down 2 at
each point.
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Uncertainty of gradient and intercepts
To determine the uncertainty in the gradient and
intercepts of a best fit line we look only at the first and
last error bars, as illustrated here:
bmin
bbest
bmax
Recall: y = mx + b
slope &
mbest  m = mbest  mmax - mmin
uncertainty
2
bbest  b = bbest 
bmax - bmin
2
intercept &
uncertainty
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Uncertainty of gradient and intercepts
The slope uncertainty calculation is shown here:
mmax - mmin
m =
2
-1.375 - -1.875
m =
2
m =  0.25
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Uncertainty of gradient and intercepts
The intercept uncertainty calculation is shown here:
bmax - bmin
b =
2
b = 53 - 57
2
b =  2
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Uncertainty of gradient and intercepts
Finally, the finished graph:
m = -1.6  0.3 cm sheet -1
b = 56  2 cm
Topic 1: Measurement and uncertainties
1.2 – Uncertainties and errors
Uncertainty of gradient and intercepts
EXAMPLE:
SOLUTION:
Look for the line to lie within all horizontal and vertical
error bars.
Only graph B satisfies this requirement.