# Absolute Error, Mean Absolute Error, Relative Error (Or Percentage Error)

a) Absolute error: Let a physical quantity A be measured n times and let A₁, A₂, … An be the results of these measurements. If Am is the arithmetic mean of these measurements, then

$${{A}_{m}}=\frac{{{A}_{1}}+{{A}_{2}}+…+{{A}_{i}}+…+{{A}_{n}}}{n}$$

Or $${{A}_{m}}=\frac{\sum\limits_{i=1}^{n}{{{A}_{i}}}}{n}$$ … (1)

Since the true value of the quantity is now known, we usually take Am to be the true value.

The magnitude of the deviation of the result of any measurement of the quantity from the arithmetic mean is called the absolute error of the measurement. Obviously, the absolute errors in the measurement of the quantity A are

ΔA₁ = |Am – A₁|

ΔA₂ = |Am – A₂|

…..   …..  …..  ….

ΔAᵢ = |Am – Aᵢ|

ΔAn = |Am – An|

ΔA₁, ΔA₂, …..ΔAn represent the magnitudes and should always be taken as positive.

b) Mean absolute error: It is the arithmetic mean of all the absolute errors.

Clearly, Amean (mean absolute error)

$$\Delta {{A}_{mean}}=\frac{\Delta {{A}_{1}}+\Delta {{A}_{2}}+…+\Delta {{A}_{i}}+…+\Delta {{A}_{n}}}{n}$$

Or $$\Delta {{A}_{m}}=\frac{\sum\limits_{i=1}^{n}{\Delta {{A}_{i}}}}{n}$$ … (2)

ΔAmean is also called the final absolute error.

Thus, the final result of measuring physical quantity A is given in the form

A = Am ± ΔAmean … (3)

This result implies that any single measurement of the quantity A is likely to be such that

Δm – ΔAmean ≤ A ≤ Am + ΔAmean … (4)

c) Relative error or percentage error: The accuracy of measuring physical quantities is compared by calculating the relative error or the percentage error.

Relative error is the ratio of the mean absolute error to the mean value of the quantity being measured.

Thus, relative error = ΔAmean/Am … (5)

Relative error expressed in percentage is known as the percentage error.

Thus, percentage error = ΔAmean/Am x 100% … (6)