Whenever you measure something, you can’t measure it
exactly, there’s always some amount of *error* in your measurement. The
error is pretty simple to define, it’s just the difference between what you
measure and what it should be exactly:

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Measurement error = measured value – correct / true value

If we’re taking about measuring something to be 300 metres
long with a precision of 1 metre, that means the actual value could be as large
as 300.4999…metres, or as small as 299.500…1 metres. Basically, when we say a
precision of 1 metre, we mean the measurement is correct *to the nearest
metre*.

So say I measured something to be 299.8 metres long, and it was actually 300 metres long. My measurement error would be calculated like this:

_{}

### Absolute error

The absolute error is just the absolute value of the
error. So in the last calculation, the error was *negative* 0.2 metres.
The absolute error is just the positive version of this:

_{}

The two vertical lines either side of the ‘0.2 m’ mean ‘take the absolute of’.

### Greatest possible error

The largest error you can make doing a measurement is exactly one half of the precision of that measurement. For instance, when we measured 300 metres to a precision of 1 metre, we could have been off by up to 0.5 metres larger or smaller:

Greatest Possible
Error = _{} of
the precision

The mathematical way of writing down this greatest possible
error for an actual measurement is to use a *plus minus* sign (_{}):

_{}

### Relative error

The relative error gives you an idea of how large the error
is, given how big the thing you’re measuring is. If you’re measuring the
distance from the Earth to the moon, an error of 10 cm doesn’t really matter
very much. However, if you’re measuring how much space you have in a room to
put furniture in, an error of 10 cm is quite *significant*, and may result
in you having a bad day when the expensive furniture you bought doesn’t fit in!

To work out the relative error, you just need to divide the error you have by the true value of the thing you’re measuring.

_{}

### Percentage error

The percentage error is just the relative error expressed as a percentage. So say I get a relative error of ‘0.5’. To convert this into a percentage I just multiply by 100 to get a percentage error of 50%.

_{}

For instance, let’s work out two relative and percentage errors:

· A 10 cm error measuring the distance to the moon (approximately 400,000 km)

· A 10 cm error measuring the length of a room 5 metres long

For the moon:

_{}

_{}

For the room:

_{}

_{}

The relative and percentage errors give you an idea of the significance of the error – for the moon measurement, the very, very small relative and percentage errors tell us that the error isn’t really significant. For the room measurement however, we get a 2 % error – enough to mess things up if we’re trying to put a piece of furniture in as big as what we think the room is.