Uncertainty & Errors in
Measurement
Waterfall by M.C. Escher
Objectives
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Difference between random errors
(uncertainties) and systematic errors
Difference between precision and accuracy
Repeatable
Reproducible
Outliers
Calculations involving addition &
subtraction
When adding and subtracting quantities, the
absolute uncertainties are added.
Example:
(a) Mass of 1st zinc = 1.21g ± 0.01g
Mass of 2nd zinc = 0.56g ± 0.01g
Total mass of the 2 pieces of zinc =
(b) Final burette reading = 38.46 cm3 ± 0.05 cm3
Initial burette reading = 12.15 cm3 ± 0.05 cm3
Volume titrated =
WS
Calculations involving multiplication
& division
When multiplying or dividing quantities, then the
percent (or fractional) uncertainties are added.
Example:
Molarity of NaOH(aq) = 0.20 M (± 0.05 M)
Percentage uncertainty =
Volume of NaOH(aq) = 25.00 cm3 (± 0.10 cm3)
Percentage uncertainty =
Therefore, the no. of moles of NaOH =
May convert % uncertainty back to absolute uncertainty.
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Example
When the temperature of 0.125kg of water is
increased by 7.20C. Find the heat required.
Heat required
= mass of water x specific heat capacity x
temperature rise
= 0.125 kg x 4.18 kJ kg-1 0C-1 x 7.20C
=
Since the temperature recorded only has 2 sig fig,
the answer should be written as ____________
WS
Multiple math operations
Example:  5.254 + 0.0016 
34.6
 2.231×10-3
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Quoting values with uncertainties
Measured value ±
uncertainty
Value you should
quote
253.4 ± 0.3
253.56 ± 0.1
0.06200 ± 0.0001
261.4 ± 8
261.4 ± 20
261.4 ± 100
The uncertainty is usually quoted to one significant figure.
Your measurement should be stated so that the
significant is in the last significant figure.
Errors (uncertainties) in raw data
When a physical quantity is taken, the
uncertainty should be stated
These uncertainties may be estimated by
 from the smallest division from a scale
 from the last significant figure in a digital
measurement
 from
data provided by the manufacturer
.
Digital Instruments
If the balance is accurate to
+/- 0.001g, the measurement is 45.310g
If the balance is accurate to +/- 0.01g, the
measurement is 45.31g
Uncertainty for digital instrument :
+/- the smallest division
Analogue Instruments
A burette of value 34.1cm3 becomes
34.10cm3 (±0.05cm3)
Note: the volume is cited to 2 decimal
places so as to be consistent with the
uncertainty.
Uncertainty for analogue instrument:
half of the smallest division.
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Higher levels of uncertainty is normally
indicated by an instrument manufacturer.
WS:Practice
Errors
Systematic errors
Apparatus
Way in which readings
are taken
Random errors
Equal chance of reading
being high or low from
1 measurement to the
next
Random Errors
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Arise from the imprecision of measurements
and lead to readings being above or below the
‘true’ value.
Random Errors are caused by
The readability of the measuring
instrument.
 The effects of changes in the surroundings
such as temperature variations and air
currents.
 Insufficient data.
 The observer misinterpreting the reading.
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Minimizing Random Errors
By
 using more precise measuring equipment
 repeating measurements so that te
random errors cancel out.
Systematic Errors
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Arise from a problem in the experiment
set-up that results in the measured values
deviating from the ‘true’ value in the same
direction, that is always higher or always
lower.
Examples of Systematic Errors
Miscalibration of a measuring device.
 Measuring the volume of water from the
top of the meniscus rather than the
bottom will lead to volumes which are
too ________.
 Overshooting the volume of a liquid
delivered in a titration will lead to
volumes which are too ______ .
 Poor insulation in calorimetry
experiments
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Minimizing Systematic Errors
Control the variables in your lab.
 Design a “perfect” procedure ( not ever
realistic)
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Percentage Uncertainty &
Percentage Error
Systematic error can be identified by comparison
with accepted literature values.
accepted value-experimental value
Percentage error =
100%
accepted value
absolute uncertainty
Percentage uncertainty =
100%
measured value
Practice Qn
(ii)
Density =
Percentage uncertainty of
Mass
Volume
(i)
Density
(a)
(b)
(i)
(c) Percentage error
Comment on the error
The percentage error (4.5%) is greater
than the percentage uncertainty (2.9%)
 The literature value does not fall within
the range 0.63 +/- 0.02 g/ml.
 Since random error is estimated by the
uncertainty and it is smaller than the
percentage error, systematic errors are at
work making the measured data
inaccurate.
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Data from Preparation of a Standard Solution
0.001g
( Electronic Balance is accurate to
)
Mass of anhydrous Na2CO3 = 1.104 g  0.001g
Titration
( Burette is accurate to 0.05cm)3
Initial Volume
Final Volume
Volume of Acid
60.00
53.50
6.50
53.50
47.00
6.50
47.00
40.00
7.00
Average :
6.70
0.05cm3
0.05cm3
0.10cm3
0.5cm
( Measuring cylinder is accurate to
) 3
3
3
3
3
6.70
cm

0.10
cm
of
Na
CO
is
titrated
with
10.0cm  0.5cm
2
3
HCl.
Percentage uncertainties due to measurements
0.001
100%  0.0906%
1.104
Mass of Na2CO3 =
Volume of HCl =
0.05
100%  0.7463%
6.70
Volume of Na2CO3 =
0.5
100%  5%
10.0
Total percentage uncertainty
 0.0906%  0.7463%  5%  5.837%
How do we quote the value in the report?
Molarity of HCl from experiment
1.104 1000 10 2 1000
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 
 0.3109
=
106 100 1000 1 6.70
Absolute uncertainty of molarity of HCl
5.837

 0.3109  0.02  one significant figure 
100
Therefore the concentration of HCl is
 0.31  0.02moldm3
Comparing % error & % random uncertainty
Since the percentage error (55.45%) is
greater than the percentage random
uncertainty (5.837%), it is suggested that
the experiment involves some systematic
errors.
How trustworthy is your reading?
Accuracy
• How close a
measured value
is to the
correct value.
Precision
• The
reproducibility
of your reading.
• How many
significant figures
there are in a
measurement.
Example
A mercury thermometer could measure
the normal boiling temperature of water
as 99.50C (±0.50C) whereas
A data probe recorded it as 98.150C
(±0.050C) .
Which is more accurate? more precise?
If all the temperature reading is 200C but
the true reading is 190C .
This gives us a precise but inaccurate
reading.
If you have consistently obtained a reading
of 200C in five trials. This could mean
that your thermometer has a large
systematic error.
systematic error
accuracy
random error
precision
systematic error
accuracy
random error
precision
Calculations
Add & Subtract
No. of decimal places
Multiply & Divide
No. of significant
figures
No. with the fewest sig
fig used determines
the sig fig to be used
for the answer.
Graphical Technique
y-axis : values of dependent variable
 x-axis : values of independent variables
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Plotting Graphs
Give the graph a title.
Label the axes with both quantities and
units.
 Use sensible linear scales – no uneven
jumps.
 Plot all the points correctly.
 A line of best fit should be drawn clearly. It
does not have to pass all the points but
should show the general trend.
 Identify the points which do not agree with
the general trend.
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Line of Best Equation
Change in volume of a fixed gas heated at a constant pressure
74.0
Temperature (0 C) Volume of Gas (cm3)
60.0
30.0
63.0
40.0
64.0
50.0
67.0
60.0
68.0
70.0
72.0
72.0
70.0
Volume (cm3)
20.0
68.0
66.0
64.0
62.0
60.0
58.0
0.0
10.0
20.0
30.0
40.0
50.0
0
temperature ( C)
60.0
70.0
80.0
Graphs can be useful to us in predicting
values.
 Interpolation – determining an unknown
value within the limits of the values
already measured.
 Extrapolation – requires extending the
graph to determine an unknown value
that lies outside the range of the values
measured.
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