THIRD
YEAR LABORATORY
EXPERIMENTAL
TECHNIQUE AND ERRORS
The
following notes present a few observations, hints and
rules of thumb on experimental technique and the treatment
of errors. They do not claim to be a formal, exhaustive treatment of the
subject.
Experimental
Technique
Initially, think about what the aim of the experiment
you are about to do is; what it is you're trying to find
out from your measurements.
Make sure you understand the function and limitations
of the apparatus used in the experiment. Be aware of the dominant sources of error in the experiment,
both random and systematic.
The accuracy of the final result is limited by
the accuracy of the worst measured quantity, so try to
minimize the dominant errors rather than increasing the
accuracy of readings with small errors.
Some systematic errors can be eliminated (e.g.
parallax, backlash) while random errors can be decreased
by taking more readings etc.
If possible, perform a preliminary experiment (a
rehearsal) to determine minima/maxima, interesting regions
etc. Do not
blindly take readings spread evenly out over the range
of the independent variable.
Less points are needed where the measured quantity
changes slowly, more where the changes are rapid, or some
quantity (e.g. full width at half maximum) needs to be
derived from the measurements.
Record your readings in tables to organise them.
Where practicable take more than one reading as
a check. Write
down raw readings without first performing
mental arithmetic such as correction for zero error etc.
Write down all your observations, relevant thoughts
and suspicions as the experiment progresses.
Watch out for uncontrolled factors which may be
affecting the experiment e.g. other radioactive sources
near a nuclear counting experiment etc.
Graph as you go, this makes 'strange' readings
easier to spot and check.
Choose the variables graphed so that straight line
graphs are obtained wherever possible. Choose your graph scales to make plotting and interpolation
simple and to spread the experimental points out over
most of the graph page.
Label the axes and give units for the plotted variables.
Use a
or a + to mark the experimental points, this avoids
confusion with spots, dust specks etc. Errors on graphed points should be shown as error bars, in
both directions if necessary.
Finally, ask yourself if your calculated results
are reasonable.
Check your arithmetic by doing calculations with
rounded off numbers.
Errors
Due to the limitations of apparatus and the statistical
nature of certain physical processes, errors (uncertainties
as distinct from mistakes), both systematic and random,
are associated with physical measurements.
Systematic errors affect the accuracy, random the
precision of measurements.
The calculation of the error in the final result
of an experiment is not just an exercise in arithmetic,
it is necessary in order to make a scientific
statement about the reliability of the result. Remember, however, that it is better to make a realistic estimate
of the errors, rather than to get lost in the mathematical
formalism.
The overall precision of a measured quantity does
not only depend on the uncertainty in reading a scale,
other factors, e.g. a personal decision as to when a pointer
is first seen to move, may be involved. Repeating readings gives an overall measure of
random (but not systematic) errors, making it unnecessary
to assess individual contributions.
If the number of readings of the measured quantity
is small, or the spread of readings is not random but
is determined by e.g. the limitations of the measuring
instrument, the traditional estimate of error is the 'maximum
likely error' and error propagation involves summing differentials.
Thus, if z is a function of measurable quantities
a, b, c... i.e. z = F(a,b,c...), then the error,
z, is given by:
where
a, b,
c... are the maximum likely errors in the measured
quantities. In
particular cases this formula reduces as follows (k, r
are error free constants) :
If a larger number, n, of randomly distributed
experimental readings, xi, is taken, the spread
of results is best described by the standard deviation,
, where:
The standard deviation should be independent of
n for n large (typically >40).
The standard error on the mean,
sm
(i.e. the error assigned to the set of measurements)
is then given by:
which
decreases as n increases.
There is 68% probability that the true value of
the mean lies within
sm
of x
and 95% probability that it lies between
x±
2s
m.
If Z is a function of the independently measurable
quantities A, B, C... i.e. Z = f(A,B,C...) then, in general,
the standard deviation, sZ,
is given by :
where sA,
sB, sC...
are the standard deviations of the individual quantities.
Since sums of squares are involved, errors less
than about 1/3 of the dominant error
may normally be ignored.
In some particular cases the above general form
reduces as follows (k,r are error free constants) :
if
Z = kA ± B then
sZ2
= (k sA)2
+ sB2
if
Z = kAB or kA/B then ( sZ/Z)2
= ( sA/A)2
+ ( sB/B)2
if
Z = kAr then sZ/Z
= r sA/A
*
* *
When fitting a straight line to a set of experimental
points by eye (a calculated least squares fit, where possible,
is preferable) an estimate of the error in the gradient
and intercepts may be obtained by drawing in lines of
minimum and maximum slope to pass through the error bars
on the experimental points.
A transparent ruler helps here.
To find gradients, two points, separated by the
maximum possible distance, should be used in order to
minimise errors due to reading the graph scales.
Errors in final results are normally quoted to
one significant figure (sometimes two, if the first figure
is a 1); the result should then be rounded off appropriately.
The use of scientific notation helps to prevent
confusion about the number of significant figures.
Computer or calculator generated results, e.g.
A = 0.03456789 ± 0.00245678, should always be appropriately
rounded off, i.e. in our case:
A
= (3.5 ± 0.2) x 10-2.
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