## Absolute Error Relative Error

Absolute Error Relative error are two types of error with which every experimental scientist and mathematician should be familiar. Whenever we measure a quantity there is almost always a difference between our measurement and the actual value. We call this difference the error.

### Absolute Error

Absolute error is the amount of physical error in measurement. When estimating, one would notice deviation of measurements from the actual values. The deviation is called the error.
${\rm{Absolute Error}} = {\rm{Actual Value}} - {\rm{Estimated Value}}$

#### Example 1

If a line is estimated to have a length of 8 cm when the actual length is 7.8 cm, there is an absolute error of 0.2 cm.

#### Example 2

You measured a line and find it it to be 57 cm and you report the absolute error is the measurement as 57 cm +/- 1 cm. The absolute error is 1 cm by noting that absolute error is reported in the same units as the measurement.

### Relative Error

Error is often expressed as a percentage of the exact value, and in this case we use the size of the error, ignoring its sign. We therefore use the modulus (difference in positive) of the error. The significance of he error depends on the actual length involved.

• An error 0.1 cm is very significant if the actual length is only 1 cm.
• An error 0.1 cm may not be significant when the actual length is 10 m.

The significance of the error is seen when the error is compared with the actual value.
${\rm{Relative Error}} = \frac{{{\rm{Absolute Error}}}}{{{\rm{Actual Value}}}}$

#### Example 3

Mary measured a line as 48.4 m. If the actual length is 50 m, find her relative error.
${\rm{Relative Error}} = \frac{{50 - 48.4}}{{50}} = 0.032$

### Percentage Error

${\rm{Percentage Error}} = {\rm{Relative Error}} \times 100\%$

The error involved in measurement is due to

• inappropriateness of measuring instruments
• human error in reading or measuring
• inaccuracy of instruments

Errors can be minimised by using using the right instrument and taking a lot of care when making any measurement.