Question

The five successive measurements of the thickness of a glass plate using a screw

gauge are 1.56 mm, 1.54 mm, 1.57 mm, 1.53 mm and 1.52mm. Evaluate :

a) The mean value of thickness

b) Absolute error in each measurement

c) The mean absolute error

d) Relative error

e) Percentage error​

Answer 1

The five successive measurements of the thickness of a glass plate using screw gauge are 1.56mm, 1.54mm, 1.57mm, 1.53mm and 1.52mm.

A] The mean value of thickness :

If a₁ , a₂ , ... , aₙ are the reading of an experiment, then true value or mean value of the quantity is given by the arithmetic mean,

$\displaystyle\sf:\implies\:\bar{a}=\dfrac{1}{n}\sum\limits_{i=1}^{n}a_i$

$\sf:\implies\:\bar{a}=\dfrac{a_1+a_2+a_3+a_4+a_5}{5}$

$\sf:\implies\:\bar{a}=\dfrac{1.56+1.54+1.57+1.53+1.52}{5}$

$\sf:\implies\:\bar{a}=\dfrac{7.72}{5}$

$:\implies\:\underline{\boxed{\bf{\red{\bar{a}=1.544\:mm}}}}$

B] Absolute error :

$\sf{Absolute\:error=|True\:value - Measured\:value|}$

or $\sf{\Delta a_i=|\bar{a}-a_i}|$

$\sf:\implies\:\Delta a_1=|1.544-1.56|=0.016$

$\sf:\implies\:\Delta a_2=|1.544-1.54|=0.004$

$\sf:\implies\:\Delta a_3=|1.544-1.57|=0.026$

$\sf:\implies\:\Delta a_4=|1.544-1.53|=0.014$

$\sf:\implies\:\Delta a_5=|1.544-1.52|=0.024$

C] The mean absolute error :

$\displaystyle\sf:\implies\:\Delta\bar{a}=\dfrac{1}{n}\sum\limits_{i=1}^{n}\Delta a_i$

$\sf:\implies\:\Delta\bar{a}=\dfrac{\Delta a_1+\Delta a_2+\Delta a_3+\Delta a_4+\Delta a_5}{5}$

$\sf:\implies\:\Delta\bar{a}=\dfrac{0.016+0.004+0.026+0.014+0.024}{5}$

$\sf:\implies\:\Delta\bar{a}=\dfrac{0.084}{5}$

$:\implies\:\underline{\boxed{\bf{\gray{\Delta\bar{a}=0.0168}}}}$

D] Relative error :

$\sf:\implies\:Relative\:error=\dfrac{Final\:absolute\:error}{True\:value}$

$\sf:\implies\:\delta a=\dfrac{\Delta\bar{a}}{\bar{a}}$

$\sf:\implies\:\delta a=\dfrac{0.0168}{1.544}$

$:\implies\:\underline{\boxed{\bf{\orange{\delta a=0.0108}}}}$

E] Percentage error :

$\sf:\implies\: Percentage\:error=\delta a\times 100%$

$\sf:\implies\: Percentage\:error=0.0108\times100\%$

$:\implies\:\underline{\boxed{\bf{\pink{Percentage\:error=1.08\%}}}}$