1.
To appreciate that all physical readings
contain may contain errors and hence there
is uncertainty
2.
To be able to conduct basic uncertainty
calculations
3.
To be able to use key terms such as accurate,
precise and reliable correctly.
Introductory Guide: Pages 19-23
A significant historical example...
Anecdote! Reference needed
Kepler is credited with much of the early work on
establishing the orbital paths of the planets
Current views had a flat Earth with the planets
revolving around it
Kepler used careful observation and measurement to
establish that the planets in our solar system revolved
around the sun.
His results showed broadly circular orbits with a slightly
elliptical character
A lesser scientist would simply have said, the planets
move in circular orbits and the deviation between
observed behaviour and my nice new circular orbit law is
simply experimental error.
However, Kepler carried out a very detailed analysis of
the errors or uncertainty in his readings and found that
the elliptical nature of the orbits was too significant to
be explained by experimental error alone.
He declared the orbits to be elliptical!
When we take an experimental reading we hope that
is it near the true value (accurate)
However all of our readings will contain errors and so
there is a level of uncertainty or doubt for each value
Error is the difference between the measured value
and the ‘true value’ of the thing being
measured.
Uncertainty is a quantification of the doubt about the
measurement result.
Our results may contain systematic sources of error :
For example a micrometer which does not read zero at
zero (find this value and subtract from all readings)
A metre rule with the end worn away (use the middle
of the ruler)
Our results may contain many random sources of error
Human misjudgements (reaction time, parallax errors)
Environmental factors (change in temperature, wind)
Equipment limitations
It is always desirable to repeat readings when possible
(NB. practical exam time is effectively unlimited)
Once we have repeated readings we can comment
upon the reliability of the data.... (the range or
variance within the results)
Taking “3 repeat readings” and finding the mean
average has become popular. However, the number of
repeat readings should reflect the spread obtained
within the readings
Reliable data will be broadly repeatable in terms of
the values obtained
For a single readings, we are short of additional data
to comment upon the reliability of the results and
so....
The uncertainty is simply the precision of the
measuring device. Consider a typical meter rule with a
mm scale
Measuring something around foot long may return a
reading of say 30cm, we can specify the uncertainty as
 1mm *
Our reading would be 0.300  0.001 m
* Many people argue that the greatest uncertainty is
half a division... i.e. 0.5mm
Once multiple readings have been taken we have a much
better first hand idea of the uncertainties involved.... We
are considering the method as well as the precision of the
measuring devices. We can see it in the data.
For repeated readings the uncertainty is reflected in the
range of readings obtained.
For a simple analysis we may consider the uncertainty to
be half of the range in the results.
Consider the following resistance readings : 609; 666;
639; 661; 654; 628Ω
Our mean average value is 643Ω. The Largest value is
666Ω, while the smallest is 609Ω
The range within our readings is (666-609) = 57Ω
We can estimate our uncertainty to be 57/2Ω
And so we quote our measurement as
643  29Ω
Note a more thorough analysis can be employed which
uses “standard deviation”
So far what we have been specifying is an absolute
uncertainty. We have our reading plus or minus an
absolute value.
Fractional uncertainty = absolute uncertainty / value
This can be multiplied by 100% to achieve a
percentage.
From our last example :
Fractional uncertainty
= 29 / 643
= 0.045
Or 4.5%
Adding or subtracting quantities:
uncertainties
add absolute
Multiplying of dividing quantities:
uncertainties
add percentage
Raising to a power quantities: multiply percentage
uncertainty by the power
Constants in uncertainty calculations
For example if we have an estimate for the uncertainty in a
radius how is this reflected in the circumference calculation?
Circumference = 2π x radius
The Rules
Multiplying a number by a constant there are 2 rules depending
on which type of uncertainty you have :
Rule - Absolute Uncertainty:
c(A ± ΔA) = cA ± c(ΔA)
Consider: 1.5(2.0 ± 0.2) m = (3.0 ± 0.3) m
Absolute Uncertainty is multiplied by the constant.
Rule - Relative Uncertainty:
c(A ± εA) = cA ± εA
Consider: 1.5(2.0 m ± 1.0%) = (3.0 m ± 1.0%)
Relative Uncertainty is not multiplied by the constant.