Relative Errors

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Although absolute errors are useful, they do no necessarily give an indication of the importance of an error. An error 1cm is very significant in a measurement of 2cm, but virtually irrelevant in a measurement of 10m. For this reason, the more useful way to express the error, is as a relative error.

The relative error is defined as

For example, calculate the relative error of the calculation

The calculated value is 25m and the absolute error is 1.5m. Therefore the relative error is

Notice that there are no units on a relative error.

For a second example, find the relative error of

The calculated value is 600m and the absolute error is 9m. The relative error is therefore

You should notice that in this case, although the absolute error is much larger,( 9m against 1.5m,) the relative error is only a quarter of the rel. error in the first case. You could therefore say that the second result is more accurate than the first.

Percentage Error

Relative errors are sometimes expressed as percentages. To convert a relative error into a percentage error, you just multiply by 100, so the percentage errors in the above examples were 6% and 1.5% respectively.

Example

The dimensions of a field are measured to be 800m by 300m

The measuring wheel has a relative error of 0.02.

Calculate

1	The absolute error in the length measurements
2	The absolute error in the area of the field
3	The relative error in the area of the field
4	The percentage error in the area of the field

1 To calculate the absolute errors, we just need to reverse the formula

2 Calculated area is 240000m²

To find the greatest possible area, we must calculate

So the absolute error is 9696m².

3 The relative error will be found by dividing the absolute error by 240000m².

4 To find the percentage error, multiply the last answer by 100, so the per. error is 4.04%

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