Errors and Uncertainty
In science, every measurement will have
limitations on its accuracy. To indicate how
well a data point should be represented,
significant figures and uncertainties are used.
The history of science has many disastrous
examples of
where uncertainties in
measurements were not taken into account
WHY DO WE HAVE UNCERTAINTY ?
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1. Measuring Uncertainty – the lowest increment on the
measuring device determines the precision of the
instrument. You cannot have a precision greater than
the last digit marked on the instrument!!!
- ruler marked every mm d = 45.6 cm  0.1 cm
-thermometer marked every o T = 12 oC  1 oC
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Note that uncertainty is limited to the last place setting.
It is incorrect to write l = 18.5 cm  1.5 cm.
2.
Random Errors – Many experiments, when repeated, will yield
slightly different results each time. Random errors are produced
by unknown and unpredictable variations in the experimental
situation.
Ex – when shooting bullets, it is unlikely that they have the same exit speed
due to variations in the cartridge, fluctuations in gunpowder explosion etc…
Ex – when bouncing a ball, it is unlikely to rebound to the same height each
time bounce is repeated. This may be due to orientation of ball in air,
orientation of ball as it hits surface, uneven surface etc…
A simple technique to deal with random errors is to repeat the experiment
many times. Find the mean of the data values. Find the variation in the data by
taking
Variation = high - low
2
- data would be represented by mean  variation
Other sophisticated methods are used such as working out the standard
deviation but we will get into these methods as needed.
Small random errors means experiment has high precision.
3. Systematic Errors – are errors associated with a particular
instrument or experimental technique. Systematic errors are not
reduced by repeating measurements.
 -is measuring device perfectly accurate ( calibration)?
 - is device zeroed ?
 -is the experiment being performed in a wind or presence of air
currents?
 -is there a bias on behalf of the experimenter? ( right handed/left
handed)
 -is odometer on car accurate if tires have been changed ?
 most electrical equipment is assumed to have a 5% systematic
error
 Small systematic errors means experiment has high accuracy
The Mathematics of Uncertainty
Data value Absolute Uncertainty
Relative Uncertainty
x
x
x
x
Example:
Height: 16.8 cm
Absolute Uncertainty : 0.5 cm
 Note that absolute error has same units as measurement
Relative Uncertainty :
Relative Uncertainty has no units
Note that relative uncertainty can be
improved by measuring over a longer
time period or greater length.
0 . 5 cm
16 . 8 cm
 0 . 03  3 %
Addition / Subtraction of Measurements
When two measurements are combined, the uncertainty of each
should first be expressed in absolute form.
 The absolute uncertainty of the sum/difference is the sum of
the absolute uncertainties in the two numbers.
Ex: ( 4 m ± 1m) + ( 12 m ± 2 m) = 16 m ± 3 m
Ex:
( 952 ± 6) kg -
( 554 ± 10.) kg =
( 398 ± 16) kg
Multiplication / Division
Numbers should first be expressed as relative uncertainty
The relative uncertainty in the result is the sum of the
individual relative uncertainty.
Ex: ( 4.0 ± 1.0) mm * ( 12 ± 2) mm
( 4.0mm ± 0.25) * ( 12 mm ± 0.2)
( 48 mm2 ± 0.4 )
or ( 48 mm2 ± 40% )