Errors and Uncertainties

Measurement of any quantity involves some uncertainty or error. Note that we are not talking about actual mistakes such as reading the number off a scale wrongly. Mistakes can be avoided by concentration and good experimental technique. However, no measurement is perfect – no matter how good your experimental technique or your instrument, there is always some associated uncertainty. For this reason it is better to use the term experimental uncertainty, rather than experimental error, though they mean the same thing. Whenever you measure a quantity, you must quote an associated plus or minus value, e.g. 45 ± 1 mm. There are two kinds of uncertainty: systematic and random.

Systematic Uncertainties

These arise because of faults in our equipment or in our experimental technique. They lead to a result which is always skewed in a particular direction. For example, if a voltmeter has a “zero error” which means that it always measures too low by 0.1 Volts, we will always have an uncertainty of –0.1 V, in addition to any random uncertainty. It is not always possible to quantify systematic uncertainties or take them into account. We should try to eliminate them or at least reduce them by using good equipment and good technique.

Random Uncertainties

These are due to statistical fluctuations and lead to a result which is spread around some mean value. They cannot be avoided, but they can be taken into account using standard techniques. They may be due to the limitations of an instrument or uncontrollable conditions. For example, an unstable power supply may make a voltage reading fluctuate randomly around a certain value.

Precision and Accuracy

These words do not mean the same thing. A result is accurate if it is close to the ‘true value’. A result is precise if its associated uncertainty is small. For example, if the actual mass of an orange is 205 g:

210 ± 5 g is neither accurate nor precise

210 ± 1 g is precise but not accurate

205 ± 5 g is accurate but not precise

205 ± 1 g is both accurate and precise

Estimating Uncertainties

There is no hard and fast rule for estimating uncertainties – it is your estimate, and as long as it is ‘sensible’, then it is not open to challenge. For example, one person might measure the length of a piece of string to be 44.7 ± 0.1 cm, while another might measure it as 44.7 ± 0.2 cm. These are both sensible estimates of uncertainty. However, someone who estimates the uncertainty to be ± 1 cm or ± 0.001 cm is not making a sensible judgment.

Sometimes a good way to estimate uncertainty is to take several measurements. For example, suppose you measure the current through a wire 5 times and get the values 5.3, 5.5, 5.1, 5.2 and 5.4. The mean of these values is 5.3 and the ‘spread’ is 0.4, so a sensible result would be 5.3 ± 0.2 A. (Note: see significant digits below).

We have a different problem if repeating the measurement always gives the same value. This does not mean that the uncertainty is zero!! The uncertainty is always at least one half the smallest division of the instrument you are using. For example, if you are using a ruler with mm divisions, the uncertainty is always at least 0.5 mm (and may be more).

Significant Digits

Uncertainties are estimates and should be given to one significant digit only. The measurement should be given to the same level of significance (i.e. the same order of magnitude or same number of decimal places). For example:

CorrectWrong

15.2 ± 0.415.154 ± 0.4

15,000 ± 1,00015,430 ± 1,000

543 + 5543.25 ± 5

(5.4 ± 0.3) x 107(5.40 + 0.3) x 107

[Note: it is good practice to give values and uncertainties to the same power of 10, i.e. do not write 5.4 x 107 ± 3 x 106].

Absolute and Relative Uncertainties

Suppose we measure a value, x, to be 3.2 ± 0.1. The absolute uncertainty, ∆x, is 0.1. However, the uncertainty can also be expressed as a fraction or percentage of the value, ∆x/x. This is known as the relative, fractional or percentage uncertainty. In this case it would be 0.1/3.2 ≈ 0.03 or 3% (to one significant digit, remember!)

Combining Uncertainties

Suppose we measure the quantities a and b with their associated absolute uncertainties, ∆a and ∆b. We need to find the value y, which is a combination of a and b. We use the following rules:

When adding or subtracting quantities, we add the absolute uncertainties.
When multiplying or dividing quantities, we add the relative uncertainties.

In equation form:

If y = a ± b then ∆y = ∆a + ∆b

If or then

Remember that the final uncertainty should be given to one significant digit only. These equations are in your data booklet. We will practise using them in the first few practicals.

Dominant and Negligible Uncertainties

In an experiment there will often be a dominant uncertainty which mainly determines the overall uncertainty. This is the source of uncertainty that you should focus on in your evaluation. When combining uncertainties, you can safely neglect small uncertainties (a rule of thumb is to neglect uncertainties of less than 1 % (when rounded up). You must say that you are neglecting an uncertainty, and why, when you write a lab report.

Graphs

When you plot points on a graph, you must indicate the associated uncertainty with an error bar (the term ‘uncertainty bar’ has not come into common usage). We will look at working out uncertainties from graphs at a later date.

BMc / NIST / Science / 14.8.03