Ram measures the period of oscillation of a simple pendulum. In successive measurements, the reading are 2.63 s, 2.56 s, 2.42 s, 2.71 s and 2.80 s. The mean absolute error and percentage error are respectively:

Question

Accepted Answer

To determine why Option 1 is the correct answer, we need to calculate the mean absolute error and the percentage error based on the given measurements of the pendulum's oscillation period.

1. **Calculate the Mean Period:**
   The given measurements are 2.63 s, 2.56 s, 2.42 s, 2.71 s, and 2.80 s. The mean period $ T $ is calculated as follows:
   $$
   T = \frac{2.63 + 2.56 + 2.42 + 2.71 + 2.80}{5} = \frac{13.12}{5} = 2.624 \, \text{s}
   $$

2. **Calculate the Absolute Errors:**
   The absolute errors for each measurement are:
   $$
   |2.63 - 2.624| = 0.006 \, \text{s}
   $$
   $$
   |2.56 - 2.624| = 0.064 \, \text{s}
   $$
   $$
   |2.42 - 2.624| = 0.204 \, \text{s}
   $$
   $$
   |2.71 - 2.624| = 0.086 \, \text{s}
   $$
   $$
   |2.80 - 2.624| = 0.176 \, \text{s}
   $$

3. **Calculate the Mean Absolute Error:**
   The mean absolute error is the average of these absolute errors:
   $$
   \text{Mean Absolute Error} = \frac{0.006 + 0.064 + 0.204 + 0.086 + 0.176}{5} = \frac{0.536}{5} = 0.1072 \, \text{s}
   $$
   Rounding 0.1072 to two decimal places gives 0.11 s.

4. **Calculate the Percentage Error:**
   The percentage error is calculated using the formula:
   $$
   \text{Percentage Error} = \left(\frac{\text{Mean Absolute Error}}{\text{Mean Period}}\right) \times 100 = \left(\frac{0.11}{2.624}\right) \times 100 \approx 4.19\%
   $$
   Rounding 4.19% to the nearest whole number gives 4%.

Thus, the mean absolute error is approximately 0.11 s, and the percentage error is approximately 4%, which matches the values given in Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (0.068 s and 4%):** The mean absolute error calculated as 0.11 s does not match the 0.068 s stated in this option. Therefore, this option is incorrect.

- **Option 3 (0.068 s and 2.6%):** Both the mean absolute error and the percentage error do not match the calculated values of 0.11 s and 4%, respectively. Hence, this option is incorrect.

- **Option 4 (0.11 s and 2.6%):** While the mean absolute error of 0.11 s is correct, the percentage error of 2.6% does not match the calculated 4%. Therefore, this option is incorrect.

In conclusion, Option 1 is the correct answer as it accurately reflects the calculated mean absolute error and percentage error.