Question 37: Units , Dimensions and Error Analysis

Question

Four persons measure the length of a rod as $20.00 \mathrm{~cm}, 19.75 \mathrm{~cm}, 17.01 \mathrm{~cm}$ and 18.25 cm . The relative error in the measurement of average length of the rod is :

Accepted Answer

The average (mean) of the measured values is the true value for our error calculation.
$$ \bar{x}=\frac{x_1+x_2+x_3+x_4}{4} $$
Substituting the given values :
$$ \bar{x}=\frac{20.00+19.75+17.01+18.25}{4}=\frac{75.01}{4} $$
$$ \overline{\mathrm{x}}=18.7525 \approx 18.75 \mathrm{~cm} $$
The absolute error is the magnitude of the difference between each individual measurement and the mean value.
$$ \left|\Delta \mathrm{x}_{\mathrm{i}}\right|=\left|\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right| $$
$$\Rightarrow $$ $\left|\Delta \mathrm{x}_1\right|=|20.00-18.75|=1.25$
$$\Rightarrow $$ $\left|\Delta \mathrm{x}_2\right|=|19.75-18.75|=1.00$
$$\Rightarrow $$ $\left|\Delta x_3\right|=|17.01-18.75|=|-1.74|=1.74$
$$\Rightarrow $$ $$ \left|\Delta \mathrm{x}_4\right|=|18.25-18.75|=|-0.50|=0.50 $$
Then the average of these absolute errors.
$$ \Delta \overline{\mathrm{x}}=\frac{\sum\left|\Delta \mathrm{x}_{\mathrm{i}}\right|}{4} $$
$$\Rightarrow $$ $\Delta \bar{x}=\frac{1.25+1.00+1.74+0.50}{4}=\frac{4.494}{4}$
$$\Rightarrow $$ $$ \Delta \overline{\mathrm{x}}=1.1225 \approx 1.12 \mathrm{~cm} $$
The relative error is defined as the ratio of the mean absolute error to the mean value.
Relative Error $=\frac{\Delta \overline{\mathrm{x}}}{\overline{\mathrm{x}}}$
Relative Error $=\frac{1.1225}{18.7525}$
Relative Error $\approx 0.0598$
Rounding to two decimal places,
Relative Error $\approx 0.06$
Therefore, the relative error in the measurement is 0.06 . Hence, the correct option is (C).

Answer

0.18

Answer

0.24

Answer

0.06

Answer

0.08

Units , Dimensions and Error Analysis

Four persons measure the length of a rod as 20.00 \mathrm{~cm}, 19.75 \mathrm{~cm}, 17.01 \mathrm{~cm} and 18.25 cm . The relative error in the measurement of average length of the rod is :

Choose the correct answer: