JEE PYQ: Units & Measurements - Question ID c2b16168014c (JEE Main 2020)

Core Formula & Concept:

In experimental physics, when multiple measurements of a physical quantity are taken, the best estimate of the true value is given by the arithmetic mean of the readings. However, due to random errors, the measurements scatter around this mean. The uncertainty in the mean is quantified using the standard deviation of the mean (also called the standard error), which is computed as:

$\sigma_{\text{mean}} = \frac{\sigma}{\sqrt{N}}$ where:

• $\sigma$ is the standard deviation of the individual readings,
• $N$ is the number of readings.

The final reported value must be rounded to reflect the precision of the measurement. The mean and the uncertainty must be rounded to the same decimal place, and the uncertainty is typically rounded to one significant figure unless the leading digit is 1, in which case two significant figures may be retained.

Step-by-Step Derivation:

Step 1: Compute the arithmetic mean

Given readings: $5.50\, \text{mm},\ 5.55\, \text{mm},\ 5.45\, \text{mm},\ 5.65\, \text{mm}$. The average is already provided as $\bar{d} = 5.5375\, \text{mm}$.

Step 2: Compute the standard deviation of the mean

Given $\sigma = 0.07395\, \text{mm}$ and $N = 4$, the standard deviation of the mean is: $\sigma_{\text{mean}} = \frac{\sigma}{\sqrt{N}} = \frac{0.07395}{\sqrt{4}} = \frac{0.07395}{2} = 0.036975\, \text{mm}$

Step 3: Round the uncertainty

The uncertainty $\sigma_{\text{mean}} = 0.036975\, \text{mm}$ must be rounded to one significant figure (since the leading digit is 3, not 1): $\sigma_{\text{mean}} \approx 0.04\, \text{mm}$ However, the question asks for the standard deviation of the data (not the mean) to be used in reporting. The standard deviation of the data is $\sigma = 0.07395\, \text{mm}$. When reporting the average diameter, the uncertainty is typically taken as the standard deviation of the data (not the mean) unless specified otherwise. This is a common convention in introductory experiments. Rounding $\sigma = 0.07395\, \text{mm}$ to one significant figure gives: $\sigma \approx 0.07\, \text{mm}$

Step 4: Round the mean to match the uncertainty

The mean $\bar{d} = 5.5375\, \text{mm}$ must be rounded to the same decimal place as the uncertainty $0.07\, \text{mm}$, which is the hundredths place. Thus: $\bar{d} \approx 5.54\, \text{mm}$

Step 5: Final reported value

Combining the rounded mean and uncertainty: $d = (5.54 \pm 0.07)\, \text{mm}$ This matches option A. Common Traps & Exam Tip:

Trap 1: Students often confuse the standard deviation of the data ($\sigma$) with the standard deviation of the mean ($\sigma_{\text{mean}}$). The question explicitly provides the standard deviation of the data, so the uncertainty in the average should be based on this value (rounded appropriately), not the standard error.

Trap 2: Over-precision in reporting. The mean and uncertainty must be rounded to the same decimal place. Reporting $5.5375 \pm 0.07395$ is incorrect because it suggests false precision.

Trap 3: Misinterpreting the question’s intent. The question asks for the average diameter to be recorded, implying the use of the standard deviation of the data (not the mean) for uncertainty, following common experimental conventions.

Exam Tip: Always round the uncertainty to one significant figure (unless the leading digit is 1, where two may be used) and round the mean to the same decimal place as the uncertainty.