Question

Repeated observations in an experiment gave the values 1.29, 1.33, 1.34, 1.35, 1.32, 1.36, 1.30 and 1.33. Calculate the mean value error, relative error and the percentage error.

Answer 1

Hint: In this question use the concept that the mean value will simply be the sum of all the observations divided by the total number of observations that is $\bar x = \dfrac{{\sum\limits_{i = 1}^n {{x_i}} }}{n}$. Moreover for absolute error calculate the modulus of difference of mean value and the individual readings. For relative error find the ratio of mean absolute error to mean and percentage error will simply be the multiplication of relative error and 100. This will help approaching all the parts of this given problem statement.


Complete step-by-step solution -

Given repeated observations in an experiment are,
1.29, 1.33, 1.34, 1.35, 1.32, 1.36, 1.30 and 1.33.
Let,
${x_1} = 1.29,{x_2} = 1.33,{x_3} = 1.34,{x_4} = 1.35,{x_5} = 1.32,{x_6} = 1.36,{x_7} = 1.30{\text{ and }}{x_8} = 1.33$
$\left( i \right)$ Mean value
Mean value is the sum of all the values divided by the number of values.
So as we see that there are 8 readings.
And the sum of the readings is
1.29 + 1.33 + 1.34 + 1.35 + 1.32 + 1.36 + 1.30 + 1.33 = 10.62
Let the mean of readings is x
$ \Rightarrow x = \dfrac{{10.62}}{8} = 1.3275$
$\left( {ii} \right)$ Absolute error
Absolute error is the modulus of difference of mean value and the individual readings.
Therefore,
$\Delta {x_1} = \left| {x - {x_1}} \right|$ = |1.3275 – 1.29| = 0.0375
$\Delta {x_2} = \left| {x - {x_2}} \right|$= |1.3275 – 1.33| =|-0.0025|= 0.0025
$\Delta {x_3} = \left| {x - {x_3}} \right|$ = |1.3275 – 1.34| = |-0.0125| = 0.0125
$\Delta {x_4} = \left| {x - {x_4}} \right|$ = |1.3275 – 1.35| = |-0.0225| = 0.0225
$\Delta {x_5} = \left| {x - {x_5}} \right|$ = |1.3275 – 1.32| = 0.0075
$\Delta {x_6} = \left| {x - {x_6}} \right|$ = |1.3275 – 1.36| = |-0.0325| = 0.0325
$\Delta {x_7} = \left| {x - {x_7}} \right|$ = |1.3275 – 1.30| = 0.0275
$\Delta {x_8} = \left| {x - {x_8}} \right|$ = |1.3275 – 1.33| = |-0.0025| = 0.0025
$\left( {iii} \right)$ Relative error
It is the $\left( \pm \right)$of the ratio of mean absolute error to mean.
So first find out the mean absolute error $\left( {\Delta x} \right)$
$ \Rightarrow \Delta x = \dfrac{{\Delta {x_1} + \Delta {x_2} + \Delta {x_3} + \Delta {x_4} + \Delta {x_5} + \Delta {x_6} + \Delta {x_7} + \Delta {x_8}}}{8}$
Now substitute all the values we have,
$ \Rightarrow \Delta x = \dfrac{{0.0375 + 0.0025 + 0.0125 + 0.0225 + 0.0075 + 0.0325 + 0.0275 + 0.0025}}{8}$
$ \Rightarrow \Delta x = 0.018125$
So the relative error is
$ \Rightarrow \Delta \bar x = \pm \dfrac{{\Delta x}}{x} = \pm \dfrac{{0.018125}}{{1.3275}} = \pm 0.01365$
$\left( {iv} \right)$ Percentage error
Percentage error is the multiplication of relative error and 100.
%error = 100$ \times \left( { \pm 0.01365} \right)$ = $ \pm 1.365$%
So this is the required answer.

Note – Errors can simply be defined as the difference between the actual values with that of the calculated value. Error can be caused due to any reasons that include human error, or even machine error. Errors can broadly be classified into 3 categories that are syntax error, logical error and run-time errors.