Mathematical Error

At the time of writing this, the 'population clock' run by the USA states that there are 316,356,429 people living in the USA. However, many sources will say that the population is 300,000,000. This means that these sources are approximating the value.

To find the error in this approximation, we do:

$300000000-316356429=-16356429$

Mei define an error

'When an exact value x is approximated by X, the error ε is given by: ε = X - x

Notice that if the approximation is too big the error is positive, and if the approximation is too small the error is negative.'

To put it simply, the error is a measure of the distance between and estimate of the value, and the actual value.

As well as this, we have what is called the absolute error, which is effectively the magnitude of the error.

Mei define it as this:

'When an exact value x is approximated by X, the absolute error is defined as the modulus of the error.

absolute error = |ε| = |X - x|'

The modulus of the errors means its magnitude; how big it is. The positive value of that number.

Although knowing the errors and absolute errors of estimates is very good, it is hard to compare them. Say I'm comparing two measurement techniques.One looks at the sizes of atoms, and the other looks at the sizes of galaxies. The error is automatically going to be greater with the measurements of the galaxies as it is just so much larger than an atom, which isn't a very fair comparison. This is when we use something called the relative error.

Mei define the relative error as:

'A useful measure of error is the relative error. This is defined by:

${\text{relative error}}={\frac {\text{error}}{\text{exact value}}}={\frac {\epsilon }{x}}$'

The relative error is measuring the ratio of the error to the exact value, making it a better representation of the error's you're dealing with.

There like before, we have a different version of the relative error;Absolute relative error

Mei define it as:

'When an exact value x is approximated by X, the absolute relative error is the modulus of the relative error, and is defined as:

${\text{absolute relative error}}=\left|{\frac {\text{error}}{\text{exact value}}}\right|=\left|{\frac {\epsilon }{x}}\right|=\left|{\frac {X-x}{x}}\right|$'

Again, it is the ratio between error and exact value, although this time the value is always positive.

Finally, Percent Error is the absolute relative error times 100. For example, if the absolute relative error is 0.25, the percent error is 25%.