Units and Dimensions - Absolute & Relative Errors

Description:

To decrease the errors in measurement, arithmetic mean (A.M.) is calculated to estimate the most correct value.

If ‘n’ measurements were taken as,

a₁, a₂, a₃, .....a_n

Then arithmetic mean is defined as,

a_mean = a₁ + a₂ + a₃ + ..... + a_n/n = n∑i=1a_i/n

The A.M. is much closer to the true value.

The magnitude (absolute value without the sign) of difference between A.M. and the individual measurements is called the Absolute Error. Therefore,

Δa₁ = |a₁ - a_mean|

Δa₂ = |a₂ - a_mean|

⋮

Δa_n = |a_n - a_mean|

Finally, the mean absolute error is defined as −

Δa_mean =

Δa₁ + Δa₂ + Δa₃ + ....+ Δa_n/n = n∑i=1Δa_i/n

The absolute error is expressed with the measurement in the following way −

a = a_mean ± Δa_mean

Example − Following measurements are made of an iron rod in the laboratory

10 cm, 12 cm, 9 cm, 8 cm, 13 cm

Find the mean absolute error.

Method − First we calculate the A.M. as,

a_mean = 10 + 12 + 9 + 8 + 13/5 = 52/5 = 10.4 cm

Now calculating absolute error for each measurement, Δa_n = |a_n − a_mean|

|10.4 - 10| = 0.4cm

|10.4 - 12| = 1.6 cm

|10.4 - 9| = 1.4 cm

|10.4 - 8| = 2.4 cm

|10.4 - 13| = 2.6 cm

Now, mean absolute error = 0.4 + 1.6 + 1.4 + 2.4 + 2.6/5 = 1.68 cm

Hence, the measurement can be expressed as −

length = 10.4 ± 1.68 cm

Relative and Percentage Error

Relative Error is defined as −

Relative Error = Δa_mean/a_mean

Percentage Error is defined as −

Percentage Error = Relative Error × 100

Relative and percentage errors are unit-less.

Example −

Person ‘A’ designs 1 m ruler with mean absolute error of 1 cm.

Relative error_A = 0.01 m/1 m = 0.01

Person ‘B’ designs 10 km road with mean absolute error of 1 cm.

Relative Error_B = 0.01 m/1000 m = 0.00001

Clearly, Person ‘B’ is much better at his job compared to Person ‘A’. This aspect can only be revealed by Relative error. (and not by Absolute error)