Question

Define absolute error, mean absolute error, relative error and mean percentage error.

Answer 1

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%

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Answer 2

Answer:

In other words, the absolute error is the magnitude of the difference between the exact value and the approximation. The relative error is the absolute error divided by the magnitude of the exact value. The percent error is the relative error expressed in terms of per 100.

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