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Physics with a Foundation Year
Handling of experimental errors
Introduction
The subject of experimental error is covered at sufficient depth for this course in the three
FLAP modules P1.1, Introducing measurement, P1.2, Errors and uncertainty and
P1.3, Graphs and measurements. You are encouraged to study these over the next week or
two, but in order to begin laboratory work with the minimum of delay the key points are
summarized below. Italicized words in what follows are also Glossary entries and so you
can read more about these in the FLAP Glossary, viewable on the departmental intranet.
Whenever you do an experiment you take measurements and these measurements are limited
by equipment and by technique. The reliability of the measurements is necessarily limited,
however well you conduct the experiment and however sophisticated the equipment. These
limitations are called experimental errors and are unavoidable. They are quite different from
mistakes which are avoidable and should be avoided. In any experiment the final quoted
result is subject to error and it is as important to quote an estimate for the error expected as it
is to quote the final result. A final result quoted without an error estimate is almost useless
and will never be taken seriously by any other scientist. There are two types of error which
can arise. Random errors are equally likely to produce an overestimate or underestimate of a
measurement or result and systemmatic errors which give a bias in the measurement or
result. The precision of a measurement (how reproducible it is) is given by the random error,
whereas the accuracy of a measurement (how far from some true value it could be) is
affected by both the random and the systemmatic errors.
Random errors
Random errors indicate the reproducibility of a measurement and are estimated by repeating
the measurement a few times and by observing the scatter of the results around the mean of
the set of results. The mean is then taken as the best result for the measured quantity and an
estimate for the random error comes from the scatter about the mean. There are several ways
of estimating the random error from a set of measurements. The simplest method is no more
than a ‘rule of thumb’ and it is to look at the difference between the highest and lowest
values in the set and to take two thirds of this range as the full scatter, with half of this value
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quoted for the scatter about the mean. An example will make this clear. If a set of
measurements for a measured time has a highest value of 5.3 s, a lowest value of 4.7 s and a
mean of 5.0 s we would quote the best value as 5.0 (0.2) s. This method of quoting a
random error is quick and simple but it is subject to a criticism; the more measurements we
take for the set then the larger the scatter will become, since eventually we will surely obtain
values above the previous highest value and below the previous lowest value. A more
sophisticated treatment, which is not vulnerable to this criticism,. involves calculation of the
standard deviation of the set of measurements. This method returns a value for the random
error on each of a set of measurements without this value being affected by the size of the
set. To find a standard deviation we first find the mean and the deviation from the mean
(departure from this mean) of each value.
The mean deviation might be expected to
represent the scatter about the mean but if you ponder on this a moment you may realise that
the mean deviation is always zero, because, on average, there is as much positive as negative
deviation about a mean! If, instead, we first square these deviations (so, making them all
positive) and then find the mean squared deviation we will not have the problem of this
vanishing to zero and so will obtain a quantity which reflects the scatter of the average
reading from the mean. We can illustrate this calculation using our earlier example, but now
with all values given:
Time/s
Deviation /s (Deviation)2/s2
5.1
+ 0.1
0.01
5.3
+ 0.3
0,09
4.8
 0.2
0.04
4.7
 0.3
0,09
4.9
 0.1
0.01
5.2
+ 0.2
0.04
Mean is 5.00 Mean is 0 s
Mean is 0.047 s2
s
rms is 0.22 s
2
Notice that the mean is quoted to one extra significant figure, since it should be more precise
than the individual readings and we wish to avoid throwing away this potential advantage.
Notice also that the mean squared deviation has units of s2 not s and so cannot be taken as
the error on the values directly. For this we need to take the square root of the mean squared
deviation (written as rms, the root mean squared deviation). We then take 0.2 s (rounded to
the same number of significant figures as the individual values) as a representative scatter on
the individual readings. In this example we end up with exactly the same error on the
readings as we did with the ‘two thirds spread’ rule of thumb approach earlier – but this is
not always the case and the rms method is to be preferred where you have more than three or
four readings.
Notice particularly that both methods above give an estimate for the probable departure of an
individual reading from the mean of the set. They do not give the precision of the mean
value of the set of readings. As we have just stated it is expected that the mean of the set
should be more precise than the individual readings, because the random errors should
average out to some extent in calculating the mean. It is also expected that if more readings
are taken then the more precise the mean should become. There are statistical arguments
based on the theory of random errors which come up with a result for how to deal with this.
In the Foundation course you are not expected to have studied these arguments but the results
are fairly straightforward to apply and you are recommended to quote these where
appropriate.
Providing you have more than three or four values being averaged then the standard error on
the mean is taken to be less than the standard deviation on the individual results by a
factor n  1 , where n is the total number of readings being averaged: For the example
above, n = 6 and so the standard error on the mean is 0.22/5 s =0.10 s. Finally, we quote
the best value result, with its precision as 5.00 ( 0.10) s.
Systemmatic errors
Systemmatic errors control the departure of a measured value or the mean of a set of repeated
measurements or a final result from the true value for the quantity. Systemmatic errors will
bias both individual and mean results, since they are present in each individual measurement.
Examples of such errors are calibration errors and procedural effects such as parallax errors
on reading a scale. Whenever an instrument is used for a measurement the reading is
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obtained from a scale or the value of a quantity is marked on the item (eg a resistor). Usually
the scale has divisions or digits or a marking code is used and the manufacturer should
ensure that these are true, and all marked values are correct, allowing accurate measurements
and deductions to be made. However, no item can be marked with its exact value and no
instrument scale is exact and so manufacturers declare the accuracy they will guarantee (and
charge you accordingly). When you quote your final result for a measurement you must
quote its expected accuracy and this requires you to incorporate both your precision estimate
(from random errors) and your systemmatic error estimate. Often, the systemmatic error is
the harder estimate since you are in the hands of manufacturers and unless you have access to
better equipment it is difficult to check the reliability of your instuments.
Combining errors
Whenever you do an experiment it is nearly always the case that you have to combine the
effects of several sources of error.
You will always have to combine random and
systemmatic errors at some point and usually there are several sources of each of these types
of error. How errors should be combined is a major topic and is discussed in detail in the
FLAP modules quoted. Here we give just a brief ssummary of the ideas.
The first thing to notice is that errors usually have units and if they are to be combined then
they must either have the same units or be expressed in a dimensionless way, as fractional or
percentage errors (in which case they have no units). The second thing to notice is that
where there are multiple error sources it matters a great deal whether they are independent or
dependent errors. Suppose there are ten measured quantities, each with a 1% error, which
contibute to an overall calculated value. It would be rather unlikely that these errors would
together conspire to produce an overall 10% error in the calculated value, unless these errors
were dependent – so that if one gave a high reading then all gave a high reading. Only an
extreme pessimist would quote 10% as an overall error from ten independent errors of 1%.
An extreme optimist might quote 0% as the overall error, arguing that 5 results could be high
and 5 low. A good scientist is a realist rather than an optimist or a pessimist and what is
required is a realistic estimate of the probable error. In this example it must fall somewhere
between 0% and 10%, but what procedure should be adopted to obtain this?
In P1.2 the key result on combining errors in measured quantities (A, B, C ,D …) which are
ABC
multiplied or divided to obtain a final result (X)
(eg X 
) is given as:
D
4
2
2
2
2
X
 A 
 B 
 C 
 D 
Probable error in the derived quantity
 
 
 
 
  ...
X
 A
 B 
 C 
 D 
Each term in this expression is an error (eg A) divided by the quantity itself (eg A), so
turning the result into a fractional error and allowing these to be summed. The summation
is not a simple sum but rather the sum of the squares which is then square-rooted to give the
probable error overall. This expression returns a value which lies between the two extemes
of the optimist and the pessimist.
In our example it gives the probable error of ten
independent 1% errors combined as 10 %  3.2% .
This same principle and expression can be used to combine independent random and
systemmatic errors into a final conclusion. On the other hand, if some of the errors are
suspected of being dependent then they should be added numerically, as normal, so that a
probable error from three 1% errors would be 3%.
One application of the expression above shows how to incorporate errors from measured
quantities which are raised to powers in the derived quantity. For example, if the quantity X
A 2 BC
is given by
) we note that since A2  A  A and D  D1 2  D1 2
X
D
Probable
error
2
in
2
the
2
derived
2
X
 2A 
 B 
 C 
 0.5D 
 
 
 
 
  ...
X
 A 
 B 
 C 
 D 
Other rules for combining errors are given in the Module summary for P1.2.
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quantity