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 a1, a2 , a3 …..an are the individual values then
Arithmetic mean am = [a1+a2+a3+ …..an]/n
am= [Σi=1i=n ai]/n
Now absolute error formula as per definition =
Δa1= am – a1
Δa2= am – a2
………………….
Δan= am – an
Mean Absolute Error= Δamean= [Σi=1i=n |Δai|]/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 = Δamean/am
Percentage Error
It is the relative error measured in percentage. So
Percentage Error =mean absolute value/mean value X 100= Δamean/amX100
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
Δu1 = 1.33 – 1.29 = 0.04
Δu2 = 1.33 – 1.33= 0.00
Δu3 = 1.33 – 1.34= -0.01
Δu4 = 1.33 – 1.35= -0.02
Δu5 = 1.33 – 1.32= 0.01
Δu6 = 1.33 – 1.36= -0.03
Δu7 = 1.33 – 1.30= 0.3
Δu8 = 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%