### Uncertainty & Errors in Measurement

```Uncertainty & Errors in
Measurement
Waterfall by M.C. Escher
Keywords
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Uncertainty
Precision
Accuracy
Systematic errors
Random errors
Repeatable
Reproducible
Outliers
Measurements = Errors

Measurements are done directly by humans or
with the help of

Humans are behind the development of
instruments, thus there will always be
associated with all
instrumentation, no matter how
precise that instrument is.
Uncertainty
When a physical quantity is taken, the
uncertainty should be stated.
Example
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
Exercise
A reward is given for a missing diamond,
which has a reported mass of 9.92 +/0.05g.You find a diamond and measure its
mass as 10.1 +/- 0.2g. Could this be the
missing diamond?
Significant Figures
(1)
(2)
____ significant figures in 62cm3
____ significant figures in 100.00 g.
The 0s are significant in (2) What is the uncertainty range?
Measurements Sig. Fig.
Measurements
Sig. Fig.
1000s
Unspecified
0.45 mol dm-3
2
1 x 103s
1
4.5 x 10-1s mol dm-3
2
1.0 x 103s
2
4.50 x 10-1s mol dm-3
3
1.00 x 103s
3
4.500 x 10-1s mol dm-3
4
1.000 x 103s
4
4.5000 x 10-1s mol dm-3
5
Random (Precision) Errors

An error that can
based on
individual interpretation.
 Often, the error is the result of
mistakes or errors.
 Random error is not ______ and can fluctuate
up or down. The smaller your random error is,
Random Errors are caused by
instrument.
 The effects of changes in the surroundings
such as temperature variations and air
currents.
 Insufficient data.
 The observer misinterpreting the reading.

Minimizing Random Errors
By repeating measurements.
 If the same person duplicates the
experiment with the same results, the
results are repeatable.
 If several persons duplicate the results,
they are reproducible.

19.9 , 20.2 , 20.0, 20.0, 20.1, 19.9, 20.3, 19.9, 20.2, 22.3
(a)
What is the mean temperature?
The temperature is reported as
as it has a range of
Systematic Errors
An error that has a fixed margin, thus
producing a result that differs from the
true value by a fixed amount.
 These errors occur as a result of poor
experimental design or procedure.
 They cannot be reduced by repeating the
experiment.

20.0 , 20.3 , 20.1, 20.1, 20.2, 20.0, 20.4, 20.0, 20.3
19.9 , 20.2 , 20.0, 20.0, 20.1, 19.9, 20.3, 19.9, 20.2
All the values are ____________.
(a) What is the mean temperature?
The temperature is reported as
Examples of Systematic Errors
Measuring the volume of water from the
top of the meniscus rather than the
bottom will lead to volumes which are
too ________.
 Heat losses in an exothermic reaction will
 Overshooting the volume of a liquid
delivered in a titration will lead to
volumes which are too ______ .

Minimizing Systematic Errors
Control the variables in your lab.
 Design a “perfect” procedure ( not ever
realistic)

Errors
Systematic errors
Apparatus
are taken
Random errors
being high or low from
1 measurement to the
next
Accuracy
Precision
• How close to the
accepted(true)
• The reproducibility
• Reproducibility does
not guarantee
accuracy. It could
simply mean you have
a very determinate
systematic error.
If all the temperature reading is 200C but
the true reading is 190C .
This gives us a precise but inaccurate
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
Exercise
Putting it together
Example
The accurate pH for pure water is 7.00 at 250C.
Scenario I
You consistently obtain a pH reading of
6.45 +/- 0.05
Accuracy:
Precision:
Scenario II
You consistently obtain a pH reading of
8 +/-2
Accuracy:
Precision:
subtraction
When adding and subtracting, the final result
should be reported to the same number of
decimal places as the least no. of decimal
places.
Example:
(a) 35.52 + 10.3 (b) 3.56 – 0.021
Calculations involving multiplication
& division
When adding and subtracting, the final result
should be reported to the same number of
significant figures as the least no. of
significant figures .
Example:
5 .2 7
(a) 6.26 x 5.8
(b)
12
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 ____________
Multiple math operations
Example:  5.254 + 0.0016 
34.6
 2.231× 10
-3
Uncertainties in calculated results
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
Absolute & Percentage Uncertainty
Consider measuring 25.0cm3 with a
pipette that measures to +/- 0.1 cm3.
We write
Absolute Uncertainty
3
25.0  0.1cm
0.1
Percentage Uncertainty
 100%  0.4%
25.0
The uncertainties are themselves approximate and
are generally not reported to more than
1 significant fgure.
Percentage Uncertainty &
Percentage Error
P ercentage uncertainty =
absolute uncertainty
 100%
m easured value
Percentage error =
accepted value-experim ental value
accepted value
 100%
the absolute uncertainties
Example
0
0

0
.0
5
C
Initial temperature = 34.50 C 
0
0

0
.0
5
C
Final temperature = 45.21 C 
Change in temperature, ΔH
When multiplying or dividing measurement, add the
percentage uncertainties
Example
g
Given that mass = 9.24 g   0 .0 0 5 and
3
3

0
.0
5
cm

volume = 14.1 cm 
What is the density?
Example
Calculate the following:
(a)  5.2  0.1m   10.2  0.5 m 
(b)  5 .2  0 .1m   1 0 .2  0 .5 m 
Example:
3

0
.0
2
cm
 subtract the
When using a burette 
, you
initial volume from the final volume. The volume
delivered is
Final vol = 38.46
Initial vol = 12.15
What is total volume delivered?
Example
The concentration of a solution of hydrochloric
acid =
moldm
1 .0 0 -3 and
0 .0 5the volume =
cm3 .
1 0 .0  0 .1
Calculate the number of moles and give the
absolute uncertainty.
When multiplying or dividing by a pure number,
multiply or divide the uncertainty by that
number
Example
 4 .9 5 ± 0 .0 5  × 1 0
Powers :
 When raising to the nth power, multiply the
percentage uncertainty by n.
 When extracting the nth root, divide the
percentage uncertainty by n.
Example
 4 .3 ± 0 .5 cm 
3
Averaging :
repeated measurements can lead to an average value
for a calculated quantity.
Example
Average ΔH
=[+100kJmol-1(  10%)+110kJmol-1 ( 10%)+
108kJmol-1 ( 10%)]  3
= 106kJmol-1 ( 10%)]
Calculations
No. of decimal places
Multiply & Divide
No. of significant
figures
No. with the fewest sig
fig used determines
the sig fig to be used
Assume that the liquid in the thermometer
is calibrated by taking the melting point at
00C and boiling point at 1000C (1.01kPa).
will be biased.
Instruments have measuring scale identified
and also the tolerance.
Manufacturers claim that the thermometer
reads from -100C to 1100C with
uncertainty +/- 0.20C.
Upon trust, we can reasonably state the
room temperature is
20.10C +/- 0.20C.
Graphical Technique
y-axis : values of dependent variable
 x-axis : values of independent variables

Plotting Graphs

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
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.
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