Question

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

• A. 0.24
• B. 0.06
• C. 0.18
• D. 0.08

Hint:

Relative error is a dimensionless quantity. If you are asked for percentage error, simply multiply the relative error by 100.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Relative error is the ratio of the mean absolute error to the mean value of the measurement.
Step 2: Key Formula or Approach:
1. Mean value \(\bar{L} = \frac{\sum L_i}{n}\).
2. Mean absolute error \(\Delta \bar{L} = \frac{\sum |L_i - \bar{L}|}{n}\).
3. Relative error \( = \frac{\Delta \bar{L}}{\bar{L}}\).
Step 3: Detailed Explanation:
1. \(\bar{L} = \frac{20.00 + 19.75 + 17.01 + 18.25}{4} = \frac{75.01}{4} \approx 18.75\,\text{cm}\).
2. Absolute errors: - \(|20.00 - 18.75| = 1.25\) - \(|19.75 - 18.75| = 1.00\) - \(|17.01 - 18.75| = 1.74\) - \(|18.25 - 18.75| = 0.50\) 3. \(\Delta \bar{L} = \frac{1.25 + 1.00 + 1.74 + 0.50}{4} = \frac{4.49}{4} \approx 1.12\).
4. Relative error \( = \frac{1.12}{18.75} \approx 0.0597 \approx 0.06\).
Step 4: Final Answer:
The relative error is 0.06.