# What is Absolute Error, Relative Error and Percentage Error?

Posted in Dimensional Formulae | Email This Post |**Absolute Error Formula**

Absolute error is defined as the magnitude of difference between the actual and the individual values of any quantity in question.

Say we measure any given quantity for n number of times and a_{1}, a_{2} , a_{3} …..a_{n} are the individual values then

Arithmetic mean a_{m} = [a_{1}+a_{2}+a_{3}+ …..a_{n}]/n

a_{m}= [Σ_{i=1}^{i=n }a_{i}]/n

Now absolute error formula as per definition =

Δa_{1}= a_{m} – a_{1}

Δa_{2}= a_{m} – a_{2}

………………….

**Δa _{n}= a_{m} – a_{n}**

**Mean Absolute Error**= Δa_{mean}= [Σ_{i=1}^{i=n }|Δa_{i}|]/n

Note: While calculating absolute mean value, we dont consider the +- sign in its value.

**Relative Error or fractional error**

It is defined as the ration of mean absolute error to the mean value of the measured quantity

δa =mean absolute value/mean value = Δa_{mean}/a_{m}

**Percentage Error**

It is the relative error measured in percentage. So

Percentage Error =mean absolute value/mean value X 100= Δa_{mean}/a_{m}X100

**An example showing how to calculate all these errors is solved below**

The density of a material during a lab test is 1.29, 1.33, 1.34, 1.35, 1.32, 1.36 1.30 and 1.33

So we have 8 different values here so n=8

Mean value of density u= [1.29+1.33+1.34+1.35+1.32+1.36+1.30+1.33] / 8 = 1.3275 = 1.33 (rounded off)

Now we have to calculate absolute error for each of these 8 values

Δu_{1} = 1.33 – 1.29 = 0.04

Δu_{2} = 1.33 – 1.33= 0.00

Δu_{3} = 1.33 – 1.34= -0.01

Δu_{4} = 1.33 – 1.35= -0.02

Δu_{5} = 1.33 – 1.32= 0.01

Δu_{6} = 1.33 – 1.36= -0.03

Δu_{7} = 1.33 – 1.30= 0.3

Δu_{8} = 1.33 – 1.33= 0.00

Now remember we don’t take +- signs in calculating Mean absolute value

So mean absolute value = [0.04+0.00+0.01+0.02+0.01+0.03+0.03+0.00]/8 = 0.0175 = 0.02 (rounded off)

Relative error = +- 0.02/1.33 =+- 0.015 = +- 0.02

Percentage error = +- 0.015*100 = +- 1.5%

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