What is Absolute Error, Relative Error and Percentage Error?

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 …..a_n are the individual values then

Arithmetic mean a_m = [a₁+a₂+a₃+ …..a_n]/n

a_m= [Σ_i=1ⁱ⁼ⁿ a_i]/n

Now absolute error formula as per definition =
Δa₁= a_m – a₁
Δa₂= a_m – a₂
………………….
Δa_n= a_m – a_n

Mean Absolute Error= Δa_mean= [Σ_i=1ⁱ⁼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_mX100

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%