# 3.02 Precision and relative uncertainty

Consider the following two measurements

1m +/- 1cm and 1cm +/- 1mm.

When uncertainty values are written this way they are referred to as absolute uncertainty values. We can clearly see that the absolute uncertainty of the second value is the smaller of the two.

Now look at the same measurements when the uncertainty is expressed as a percentage of the nominal value (ie the value at the mid point of the uncertainty interval).

1m +/- 1% and 1cm +/-10%

When uncertainties are expressed as a percentage of the nominal value they are referred to as relative uncertainty values. We can clearly see that the relative uncertainty of the first value is the now the smaller of the two.

The precision of a number is determined by the relative uncertainty and therefore the first of these two numbers is more precise

# Accuracy and relative error

Consider the following errors that have been determined for the two measurements below

2m +/- 1% that was found to have an error of 5cm
10cm +/- 1% that was found to have an error of 1cm

The absolute size of the second error is smaller (1cm) is smaller than the first (5cm)
However if we express the errors as a percentage of the nominal value this gives

2m +/- 1% with an error of 2.5%
10cm +/- 1% with an error of 10%

The accuracy of a number is determined by the relative error. Therefore the first of the two measurements is the more accurate!

How accurate and precise does a measurement need to be?
It all depends on the purpose for which the measurement is made. A precision to within +/- 1 minute may be acceptable if say you are measuring the time you have cooked some food in an oven. However if you were measuring the time taken for an athlete to run 100m you would need to be able to measure the time to within 1/100th of a second.
So if we say a measurement is accurate and precise what we are really saying is that the accuracy and precision are good enough for that particular application.