Question

During the experiment, the refractive indices of the glass were observed to be $1.45,1.56,1.54,1.44,1.54$ and $1.53$. What is the relative error?

Answer 1

First of all find the mean value of refractive indices observed. Then, find the absolute error of each value observed by subtracting them from their mean value and taking only the absolute value. The relative error is calculated by dividing the mean value of absolute error and mean value of refractive indices observed.

Complete step by step solution:
Let the mean value of refractive indices observed in the experiment be $\overline n$ and let the refractive indices be ${n_1},{n_2},{n_3} \ldots + {n_6}$
The refractive indices which were observed are $1.45,1.56,1.54,1.44,1.54$ and $1.53$. Now, the average or mean value is calculated by formula –
$\overline n = \dfrac{{{n_1} + {n_2} + \ldots + {n_n}}}{{{n_t}}}$

where, ${n_t}$ is the total number of observations
Putting the values of refractive indices in the above formula –
$\overline n = \dfrac{{1.45 + 1.56 + 1.54 + 1.44 + 1.54 + 1.53}}{6} \\\ \overline n = \dfrac{{9.06}}{6} \\\ \overline n = 1.51 \\\$
Now, we have to calculate the absolute error in each measurement of refractive indices observed –
This is calculated by taking only the absolute value by subtracting observed refractive indices with their mean value –
$\Rightarrow \left| {observed - mean} \right|$
By using this formula for absolute value of each measurement, we get –
$\Rightarrow \left| {1.45 - 1.51} \right| = 0.06 \\\ \Rightarrow \left| {1.56 - 1.51} \right| = 0.05 \\\ \Rightarrow \left| {1.54 - 1.51} \right| = 0.03 \\\ \Rightarrow \left| {1.44 - 1.51} \right| = 0.07 \\\ \Rightarrow \left| {1.53 - 1.51} \right| = 0.03 \\\ \Rightarrow \left| {1.53 - 1.51} \right| = 0.02 \\\$
We got all the values of each absolute error of each measurement.
Now, calculating the mean value of absolute error we got above so, let the mean value of absolute error be $\Delta n$ -
$\Delta n = \dfrac{{0.06 + 0.05 + 0.03 + 0.07 + 0.03 + 0.02}}{6} \\\ \Delta n = 0.043 \\\$
Now, the relative error is calculated by dividing the mean value of absolute error and mean value of refractive indices observed which can be represented mathematically by –
$\Rightarrow \dfrac{{\Delta n}}{{\overline n }}$
Now, putting the values of mean absolute error and mean value of each measurement of refractive indices observed in the experiment –
$\Rightarrow \dfrac{{0.043}}{{1.51}} = 0.0285$

Hence, the relative error is $0.0285$.

Note: Relative error is a measure of the uncertainty of measurement compared to the size of the measurement. It is used for putting error into perspective. This error compares the measurement with an exact value.