11.1  UNCERTANTIES AND ERRORS IN MEASUREMENT AND RESULTS
FOCUS CONCEPT  All measurement has a limit of precision and accuracy, and this must be taken into account when evaluating
experimental results.
INTRO
 Error analysis is the study and evaluation of uncertainty in measurement.
 In science, the word “error” does not mean “mistake” or “blunder”. “Error” in a scientific measurement means the inevitable
uncertainty that attends all measurements. It is the job of the scientist to keep these errors to a minimum. Assessing the magnitude of
these errors is necessary in order to determine the reliability of a final result.
 “Experimental error" (aka: experimental uncertainty) refers to variability in results due to limitations in the experimental design
and measuring DEVICE; it's the reason scientists perform multiple trials of any given experiment, and report their results as a
statistical average with a plus or minus deviation included.
 In this course, although we will often average measurements, we'll rarely do more advanced statistical analysis.
 In the “Conclusion and Evaluation section of your formal lab reports, do list any observed reasons that you feel may have
contributed to errors in your experiment, including:
specific problems with the equipment,
difficulties in reading the equipment; or,
limitations in the design of the experiment.
 Don't just guess about why your experiment might have gone wrong - only mention specific sources of error that you have a
legitimate reason to believe affected your results, and explain how those sources of error affected your results, e.g. "The average final
velocity of the pendulum was 15.6% smaller than it was at the top, which we presume is at least partly caused by loss of energy due to
air friction as the pendulum swung down."
*SIGNIFICANT DIGITS IN MEASUREMENTS
(http://youtu.be/F9R8johYCww)
COUNTED QUANTITIES
MEASURED QUANTITIES
- _____________exact
- ______ uncertainty
- _____________ number of
significant digits
e.g 22 people, 3 eggs, 12 pencils
- _______________exact
- last digit in a measurement is always ____________
(an estimate)
e.g. Consider a mass reading on three different scales.
Bathroom scale 
56  1 kg
(___ sd)
Physician’s office  55.8  0.1 kg
(___ sd)
Science lab 
55.778  0.001 kg (___ sd)
*In any measurement, the digits that are reliably known are called __________________ digits. These include the digits known for
certain and the single last digit that is _____________________. (↑ decimal places = ↑ precision)
Rules for Determining the Number of Significant Digits Associated with a Measurement
RULE
MEASUREMENT
(EXAMPLE)
1. All NON-ZERO numbers ARE significant.
1,292 mm
2. “CAPTIVE” ZEROS (i.e. zeroes placed between other digits) ARE significant.
400,006 cm
3. LEADING ZEROS are NEVER significant.
e.g. 0.0000089 kg
= 8.9 x 10-6 kg
4. TRAILING ZEROS:
a) ARE significant if they indicate a measurement (i.e. the zeros are to the right of a decimal).
b) Zeroes at the end of a number are significant only if they are indicated to be so using
scientific notation. (e.g. 1000 seconds has an unspecified # of sig digs)
- 5.800 km
- 900.0 mL,
- 703.0 N
- 1 x 103 s
- 1.0 x 103 s
- 0.45 mol dm-3
- 4.5 x 10-1 mol dm-3
- 1.00 x 103 s
- 4.50 x 10-1 mol dm-3
-1.000 x 103 s
- 4.500 x 10-1 mol dm-3
SIG
DIGS
PRACTICE ACTIVTY 1:
*Determine the number of significant digits in the following numbers:
NUMBER
SIG. DIGS
NUMBER
0.02
0.0005
0.020
0.1020
501
10 001
501.0
3.50 x 104
3
5.000 x 10
2 x 109
5 000.0
8.4572 x1023
6 052.00
8040
10 800 000
3.01 x 1021
SIG. DIGS
NUMBER
80405
0.0300
699.5
2.000 x102
0.90100
90 100
4.7 x 10-8
0.000 410
SIG. DIGS.
PRACTICE ACTIVITY 2: Round the following to the indicated number of significant digits.
*When performing calculations that involve a series of mathematical steps, don’t round off numbers until you are ready to state the
final answer…use all the decimal places reported by your calculator re: intermediate calculations.
a) 2150 (2 s.d.) -___________
b) 0.0256 (2 s.d.) - _________
c) 0.0346 (2 s.d.) - __________
d) 0.0050129 (3 s.d.) - _________
e) 1.994 (3 s.d.) - _________
f) 2149.99 (3 s.d.) - ________
*SIGNIFICANT FIGURES IN CALCULATIONS
ADDITION AND SUBSTRACTION
MULIPLICATION AND DIVISION
RULE: Round the answer to the LEAST number of
places in the _________________ portion of any number
in the problem.
RULE: The LEAST number of __________________ figures in any
number of the problem determines the number of significant figures
in the answer.
e.g. Add the following measurements:
123.0 cm + 12.40 cm + 5.380 cm
e.g. A cyclist travels 4.00 x 103 m on a racetrack in 292.4 s. Calculate
the average speed of the cyclist.
Solution:
Solution: Average speed (v) is distance (d) divided by time (t):
123.0
cm
12.40 cm
5.380 cm
cm
v=
*Thus, ______________ cm should be rounded to
_____________ cm since 123.0 cm has only one decimal
place.
4.00 x 103 m
d
=
=
t
292.4 s
*Thus, the average speed should be reported to _____ significant
digits – ____________ m/s.
PRACTICE: Perform the following operations expressing the answer in the correct number of significant digits.
a) 1.35  2.267 =
b) 1 035  42 =
c) 12.01 + 35.2 + 6 =
d) 55.46 – 28.9 =
e) 0.021  3.2  100.1 =
f) 1.278  103  1.4267  102 =
g) 0.2129 + 0.002 + 0.03 =
h) 101.4 + 25 + 201 =
i) 1.0 + 2.04 + 5.03 =
j) 2.5 x 1.1111 =
k) 8.314 x 2.5x10-2 =
l) 35.45 x 2.25 =
m) 150  4 =
n) 505 – 450.25 =
o) 1.252  0.115  0.012 =
p) 0.15 + 1.15 + 2.051 =
q) 41.11 + 20.5 + 18.333 =
*EXPERIMENTAL ERRORS
Random vs. Systematic Error (http://youtu.be/x5Euj2d39kI)
RANDOM ERRORS
SYSTEMATIC ERRORS
 cause a series of measurements to fluctuate above ________
below a ______________
_______ – sided, predictable, unavoidable, quantifiable error.
 Repeated measurements _________ reveal this type of
uncertainty.
 cause a series of measurements to be EITHER too high
_______too low (i.e. either all higher than an accepted value
OR all lower than an accepted value)
 ONE – sided, quantifiable error.
 Repeated measurements WILL ________ reveal this type of
uncertainty regardless of the number of trials performed.
 Usually difficult to detect unless the expected/actual value is
known.
 Can sometimes be remedied/ameliorated/lessened by careful
experimental design.
*E.g. A sample of metal is weighed 5 times on the same scale.
TRIAL
1
2
3
4
5
MASS / g
2.86
2.87
2.85
2.84
2.88
mean = __________
mmax = __________
mmin = __________
absolute uncertainty =  ________
mass = _________________
 absolute uncertainty = range = random error, quantified
*The five different masses are due to random errors. What
causes random errors? *Cannot always be identified – possible
sources:
- ________ currents
- ____________ fluctuations in balance
- human _______________ variance (for analogue devices) –
e.g. reading a burette
- human ______________ time
- judging a colour change
*E.g. Let’s say the actual mass of the metal in the adjacent
example is 2.92 g. The 5 measurements, therefore, are ALL
_____________ than the actual value – the measurements are
_________________ due to _______________ error.
*What causes systematic errors? Usually/Examples:
(i) faulty/broken or ____________________ equipment:
- ________________ running two seconds too fast
- _____________________reading 2° higher than actual
- measuring length using stretched _______________
- ______ meter not calibrated
(ii) improper use of measuring instrument / poor experimental
technique
- forget to “_______” electronic balance
- reading a ______________ incorrectly / parallax error
- reading scale incorrectly
- unaccounted __________ loss in a calorimetry experiment
- adding too much titrant in titration experiment
- using an inappropriate indicator in a titration experiment
Precision vs. Accuracy
PRECISION
 how __________ a series of measurements are to each other
*Which measurement is more precise?....has less random error?
a) 2.85  0.01 g or 2.85  0.02 g
b) 2.850  0.001 g or 2.85  0.01 g
- (↑ decimal places = ↑ precision = ↑$ )
 usually quoted as a  value = range = _____________ error
 ↓ random error = ↑ ______________
 ↑ # of ___________ = ↓ random error = ↓ standard deviation
ACCURACY
 how close a result or measurement is to the ____________,
accepted value
 Accuracy / systematic error quantified using:
PRACTICE: Consider the following ideal gas constant measurements:
(Accepted = 8.314 J mol-1 K-1)
 0.03 J mol K
Group 2  8.513  0.006 J mol K
Group 1  8.34
-1
-1
-1
-1
Which of the measurements is: a) more accurate? b) more precise? c) Calculate
the relative(percent) error associated with each measurement.
SUMMARY
DEFINITION
AFFECTED BY
EVALUATE BY
LOOKING AT
HOW IMPROVE
PRECISION
- how close a series of
measurements are to each other
- ______________errors
(caused by air currents,
voltage fluctuations etc.)
- use more precise
measuring instrument
(↑ $)
- perform multiple trials
ACCURACY
- how close a result or
measurement is to the actual,
published value
- ______________ errors
(caused by faulty
equipment / improper use
of equipment)
- uncertainty range
(i.e. absolute error /
magnitude of 
value)
- magnitude of ____
error
- calibrate / use
equipment properly
- compare measurement
readings with more
reliable instrument
*UNCERTAINTY / ERRORS IN MEASUREMENT (SUMMARY)
“Uncertainty in measurements  (http://youtu.be/2JRwdeaX_w0)
“Absolute vs % Uncertainty  (http://youtu.be/kfsoxLuC570)
ABSOLUTE
UNCERTAINTY
- the actual uncertainty in the value /
measurement
e.g. 13.10  0.01 g
A.U =  _______
PERCENTAGE (RELATIVE)
UNCERTAINTY
absol . uncert .
% uncert . 
x 100
measurement
e.g. 13.10

PERCENTAGE ERROR
0.01 g
% uncert. =
Measurement  13.10 g

________
*Identify two ways by which the % uncertainty can be reduced. 1. ↑ ____________ size; 2. ↑ _____________ of measuring device
Analogue vs. Digital Uncertainty
ANALOGUE
DIGITAL
- If not available, use ½ of _________ count (i.e. smallest
subdivision / gradation on measuring instrument)
- If not available, use the value associated with the smallest
____________ value.
*EVALUATING PRECISION USING % UNCERTAINTY
*Consider the following:
SCALE
MEASUREMENT

Electronic Balance
0.25
Triple Beam
Balance
100.00
0.005 g

0.05 g
% UNCERTAINTY
(Relative uncert.)
Which scale is more
precise? Explain.
 ____%
(0.25 g  ____% )
 ______ %
(100.00 g  ____ %)
*REPORTING UNCERTANTIES (in “Conclusion” of formal lab)
RULE 1  report uncertainties to ______ SIG. FIG. (almost always)
RULE 2  experimental result and uncertainty must have SAME _______________ SIG. FIG.
EXAMPLES
INCORRECT
CORRECT
d = 5.67

a = 1261.29
q = 2.34

0.0415 g cm-3

199.43 cm/s2
.000225675 J
d = 18.8 g cm-3

9.63 %
(large uncert.)
(very small uncert.)
Which measurement is
more precise?
Explain.
*PROPAGATION OF UNCERTAINTIES IN CALCULATED RESULTS
(http://youtu.be/EvQOaPNhdxI)
Propagating uncertainty  taking uncertainty in measurements and carrying them through to the calculated result
 Shows the impact of the uncertainties on the final result.
ADDITION AND SUBTRACTION
e.g.
Initial burette reading (Vi) =
Final burette reading (Vf) =
3.10
36.45
MULTIPLICATION AND DIVISION


e.g. density of tap water – sample data
3
0.10 cm
0.10 cm3
d=
Volume added (ΔV = Vf -Vi ) = __________  __________
*Volmax = ________________________
9.90  0.005 g
10.1  0.1 cm3
R.Umass =________ R.Uvolume = _________ Σ R.U =__________
*Calculation of density with unrouded R.U sum:
*Volmin = ________________________

* (Volmax - Volmin) ÷ 2 = ______________
*Next, Convert to absolute uncertainty (round to ONE sig dig)
and make sure density value has same rightmost sig dig.

RULE: When ADDING or SUBTRACTING measurements,
ADD THE _________________ UNCERTAINTIES attached to
each measurement.
RULE: When MULTIPLYING or DIVIDING measurements,
ADD THE ___________________ (RELATIVE)
UNCERTAINTIES attached to each measurement.
*DISCUSSING ERRORS AND UNCERTANTIES IN FORMAL LAB RESULTS
(http://youtu.be/a3L1H0aflyM)
*Your discussion of experimental errors, in a formal lab report, should focus on the influence of random and systematic errors on
your results. Note that almost all measurements are subject to both systematic and random uncertainties. Refrain from referring to
"human error". Examples of so-called human error include misreading a ruler, adding the wrong reagent to a reaction mixture, mistiming the reaction, miscalculations, or any kind of mistake / blunder. Scientists would never report the results of an experiment
affected by human error - instead, they repeat the experiment more carefully.
PRACTICE: Evaluate the influence of random and systematic error on the following results.
LAB  Determine molar mass of methane. (Accepted value = 16.04 g mol-1)
SAMPLE RESULTS
 0.5 g mol
13.2  0.5 g mol-1
15  5 g mol-1
10  4 g mol-1
15.1
RANDOM ERROR
SYSTEMATIC ERROR
-1
*When the final uncertainty arising from random errors is calculated, this can then be compared with the % error. If the % error is
larger than the total uncertainty, then random error alone cannot explain the discrepancy and systematic errors must be involved.
FIXING RANDOM ERROR
(how get better precision)
1. Identify largest contributor of ______ uncertainty.
e.g.
d=
153 g  6%
25 cm3  0.3%
*Aim would be to decrease the % uncertainty with
respect to ____________ since it is a much larger (6%
vs. 0.3%) source of random error.
FIXING SYSTEMATIC ERROR
(how get better accuracy)
1. a) “Seek and destroy” – find and then eliminate source of
systematic error. (E.g. Are all the measuring devices working
properly? Are the displayed values accurate? Have I used the
equipment properly? Have I committed any avoidable procedural
errors? - forgot to “zero” the balance when obtaining a mass, for
example.)
b) Re-________________– Have all devices been calibrated properly?
2. Use ____________ sample size.
e.g.
0 .1 g
(100) = 10%
1g
0 .1 g
(100) = 5%
(ii) % uncert =
2 g
2. Evaluate the validity of any ___________________ that were
made. (e.g. room temperature/pressure, density/specific heat capacity
of water)
(i) % uncert =
3. Perform more _____________ – will statistically
decrease random error.
4. Change procedure  use more ____________
measuring instrument(s), for example
*ASSIGNMENT  Complete problem set handout.
3. Re-evaluate ____________  Look for places where something
you measured/calculated might have been different from the true
value.
*N.B. Repeating experiment will not eliminate systematic error.