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Prefixes and Uncertainties
Prefixes and Uncertainties
The use of standard form and / or prefixes such as milli will be
encouraged throughout the course. Care must be taken when dealing with
significant figures. Data should be carried through in calculations and the
final rounding carried out at the end. Remember that you should only
quote your final answer to the same significance as the least significant
piece of data. SQA however will allow you to quote 2 too many or 1 too
few significant figures.
1 Units, prefixes and scientific notation
Units and prefixes
 SI units should be used with all the physical quantities. Prefixes
should be used where appropriate.
 These include pico (p), nano(n), micro(
giga(G) and tera(T).
Significant figures
 When carrying out calculations and using relationships to solve
problems, it is important to give answers to an appropriate number
of significant figures. This means that the final answer can have no
more significant figures than the value with least number of
significant figures used in the calculation.
Scientific notation
 Candidates should be familiar with the use of scientific notation
and this may be used as appropriate when large and small numbers
are used in calculations.
2 Uncertainties
Random and systematic uncertainty
 All measurements of physical quantities are liable to uncertainty,
which should be expressed in absolute or percentage form.
 Random uncertainties occur when an experiment is repeated and
slight variations occur.
 Scale reading uncertainty is a measure of how well an instrument
scale can be read.
 Random uncertainties can be reduced by taking repeated
measurements.
Prefixes and Uncertainties
Systematic uncertainties occur when readings taken are either all
too small or all too large. They can arise due to measurement
techniques or experimental design.
Uncertainties and data analysis
 The mean of a set of readings is the best estimate of a ‗true‘
value of the quantity being measured.
 When systematic uncertainties are present, the mean value of
measurements will be offset.
 When mean values are used, the approximate random uncertainty
should be calculated.
 When an experiment is being undertaken and more than one
physical quantity is measured, the quantity with the largest
percentage uncertainty should be identified and this may often be
used as a good
 estimate of the percentage uncertainty in the final numerical
result of an experiment.
 The numerical result of an experiment should be expressed in the
form final value ± uncertainty

These learning outcomes will be covered throughout the course when
doing experimental work.
Prefixes and Uncertainties
Uncertainties
The measurement of any Physical Quantity is subject to an
uncertainty.
There are a number of factors which may contribute to this
uncertainty and we need to be able to identify the effects of each.
Reading Uncertainty
Whatever instrument we use to take a measurement, we can estimate
the uncertainty in the reading.
Analogue Scales (rulers, protractors etc.) can be read to ±½ of the
smallest division. E.g. a metre stick graduated in centimetres can be
read to ±½ cm.
Digital Scales (digital multimeter, stop watch etc.)
Can be read to  the value of the last digit on the display. For example
if a digital voltmeter reads 5.76V then the uncertainty in the reading
is ± 0.01V. We can say with confidence the actual value is between
5.75V and 5.77V.
Systematic Effects
These are caused by faults in the apparatus or in poor experimental
technique.
For example, if a voltmeter reads 0.2V when it should read 0V then all
readings taken with that instrument will be too high by 0.2V.
Random Uncertainties
When we do experiments it is good practice to take a number of
readings for each measurement.
This often results in a scatter of values. We can deal with these by
calculating the mean value and the random uncertainty for the mean.
Calculation of Mean -Simply add up all the values you obtain and divide
by the number of readings taken.
Mean 
Sum of Values
Number of Readings
Calculation of Random Uncertainty (RUncert) –Subtract the minimum
value from the maximum value and divide by the number of readings
taken.
RUncert 
Maximum Value - Minimum Value
Number of Measuremen ts Made
Prefixes and Uncertainties
Example –A group of students measure the width of a sheet of A4
paper;
Person Making Reading
Value Obtained (mm)
Alice
209
Bob
210
Claire
210
Damien
211
Ethel
212
Fred
210
Gemma
209
Horace
208
Irene
210
Jeremy
211
Calculation of Mean
Mean 
209  210  210  211  212  210  209  208  210  211
10
= 210mm
Calculation of Random Uncertainty
RUncert 
Maximum Value - Minimum Value
Number of Measuremen ts Made
Uncert 
212  208
10
= 0.4mm
So we quote our final answer as 210  0.4mm
Percentage Uncertainty
It is sometimes useful to quote an uncertainty as a percentage of the
mean value. We calculate percentage uncertainties using;
%Uncertainty 
Random Uncertainty
100
Mean
So in the example above;
0.4
%Uncertaint y 
100  0.2%
210
Prefixes and Uncertainties
So we can now quote the value as;
210  0.4mm or as 210mm  0.2%
Percentage Uncertainties are particularly useful when we need to
combine uncertainties.
We may measure two physical quantities and calculate a third.
As a rule, we take the largest % uncertainty in the data and simply
quote this as the % uncertainty in the calculated value.
Example; The voltage across a resistor is measured as 12V ± 0.5V The
current through the resistor is measured as 2A ± 0.5A.
Calculate the value of the resistor and quote the uncertainty in this
value.
0.5
100  4.2%
12
0.5
%Uncertaint y in Current 
100  25%
2
%Uncertaint y in Voltage 
Resistance is found using Ohm’s Law;
R
V 12

 24
I 0.5
So we quote the final answer as;
24 ± 6 Ω
(25% is the largest of the %uncertainties in the data, 25% of 24 is
6Ω)