The length of a rod as measured in an experiment was found to be 1.48 m, 2.46 m, 2.49 m, 2.50 m, and 2.48 m. Find the percentage error.

Correct Option: 0.40%

To find the percentage error, we first notice that the first value in the problem, 1.48 m, contains a typographical error and should be 2.48 m to align with the rest of the dataset and the standard calculation. Let's perform the calculation using the corrected dataset: 2.48 m, 2.46 m, 2.49 m, 2.50 m, and 2.48 m.

Step 1: Calculate the mean value of the measured length ($a_{\text{mean}}$)
The mean value is calculated by taking the average of all five measurements:

$a_{\text{mean}}=\frac{2.48+2.46+2.49+2.50+2.48}{5}$

$a_{\text{mean}}=\frac{12.41}{5}=2.482\text{m}$

Rounding to two decimal places (since the measurements are given to two decimal places), we get:

$a_{\text{mean}}\approx 2.48\text{m}$

Step 2: Calculate the absolute error for each measurement ($\Delta a_i=|a_i-a_{\text{mean}}|$)
We find the absolute difference between each measured value and the mean value (2.48 m):
$\Delta a_1=|2.48-2.48|=0.00\text{m}$
$\Delta a_2=|2.46-2.48|=0.02\text{m}$
$\Delta a_3=|2.49-2.48|=0.01\text{m}$
$\Delta a_4=|2.50-2.48|=0.02\text{m}$
$\Delta a_5=|2.48-2.48|=0.00\text{m}$

Step 3: Calculate the mean absolute error ($\Delta a_{\text{mean}}$)
We take the average of all individual absolute errors:

$\Delta a_{\text{mean}}=\frac{0.00+0.02+0.01+0.02+0.00}{5}$

$\Delta a_{\text{mean}}=\frac{0.05}{5}=0.01\text{m}$

Step 4: Calculate the percentage error
The percentage error is given by the formula:

$\text{Percentage Error}=\frac{\Delta a_{\text{mean}}}{a_{\text{mean}}}\times 100\%$

Substituting the values we calculated:

$\text{Percentage Error}=\frac{0.01}{2.48}\times 100\%$

$\text{Percentage Error}\approx 0.403\%\approx 0.40\%$

Thus, the percentage error is approximately 0.40%.