JEE PYQ: Units & Measurements - Question ID 79c08e5867f3 (JEE Main 2016)

Core Formula & Concept:

In experiments involving repeated measurements, the mean value of the observations is taken as the best estimate of the true physical quantity. The uncertainty in this mean is quantified by two key statistical measures:

• Mean ($\bar{x}$): $\bar{x} = \frac{1}{N} \sum_{i=1}^{N} x_i$ where $x_i$ are the individual measurements and $N$ is the number of observations.
• Standard Deviation ($\sigma$): $\sigma = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} (x_i - \bar{x})^2 }$ This measures the spread of the data around the mean.
• Standard Error of the Mean ($S_{\bar{x}}$): $S_{\bar{x}} = \frac{\sigma}{\sqrt{N}}$ This gives the uncertainty in the mean value due to random errors.

Additionally, the instrumental error must be considered. Since the clock has a minimum division of 1 s, the least count error is $\pm 0.5$ s per measurement. However, when averaging multiple measurements, this error reduces.

The total uncertainty in the reported mean is the combination of the standard error of the mean and the instrumental error. The larger of the two is typically taken as the dominant uncertainty.

--- Step-by-Step Derivation:

Step 1: Compute the Mean Time for 100 Oscillations

Given data set: $90$ s, $91$ s, $95$ s, $92$ s. $\bar{x} = \frac{90 + 91 + 95 + 92}{4} = \frac{368}{4} = 92 \text{ s}$

Step 2: Compute the Standard Deviation ($\sigma$)

Compute deviations from the mean: $$\begin{align*} (90 - 92) &= -2 \\ (91 - 92) &= -1 \\ (95 - 92) &= +3 \\ (92 - 92) &= 0 \\ \end{align*}$$ Square the deviations: $(-2)^2 = 4, \quad (-1)^2 = 1, \quad 3^2 = 9, \quad 0^2 = 0$ Sum of squared deviations: $4 + 1 + 9 + 0 = 14$ Standard deviation: $\sigma = \sqrt{ \frac{14}{4} } = \sqrt{3.5} \approx 1.87 \text{ s}$

Step 3: Compute the Standard Error of the Mean ($S_{\bar{x}}$)

$S_{\bar{x}} = \frac{\sigma}{\sqrt{N}} = \frac{1.87}{\sqrt{4}} = \frac{1.87}{2} = 0.935 \text{ s}$

Step 4: Estimate Instrumental Error

The clock has a least count of 1 s, so the maximum instrumental error per measurement is $\pm 0.5$ s. However, since we are averaging 4 measurements, the instrumental error in the mean reduces due to statistical averaging. The instrumental error in the mean is: $\frac{0.5}{\sqrt{4}} = \frac{0.5}{2} = 0.25 \text{ s}$

Step 5: Determine the Dominant Uncertainty

We have two sources of uncertainty: - Random error (standard error): $0.935$ s - Instrumental error: $0.25$ s The larger of the two is $0.935$ s, but this is still less than 1 s. However, in experimental reporting, uncertainties are typically rounded to one significant figure, and the precision of the instrument must be respected. But here's a crucial point: the standard deviation of the data is $1.87$ s, which is larger than the least count. This suggests that the spread in measurements is primarily due to random fluctuations, not just instrumental limitations. Moreover, the question asks for the reported mean time, and in such cases, the uncertainty is often taken as the maximum deviation from the mean or the standard deviation, rounded appropriately. Let’s compute the maximum absolute deviation: $\max(|90 - 92|, |91 - 92|, |95 - 92|, |92 - 92|) = \max(2, 1, 3, 0) = 3 \text{ s}$ However, using the maximum deviation is overly conservative. Instead, we use the standard deviation of the sample, which is $1.87$ s, and round it to $2$ s for simplicity and to match the precision of the instrument. Alternatively, the standard error is $0.935$ s, but since the least count is 1 s, we cannot report uncertainty below 1 s. However, the options suggest a larger uncertainty. The most reasonable approach is to consider the standard deviation of the measurements as the uncertainty, rounded to the nearest whole number, because the spread in data ($1.87$ s) is larger than the least count error. Thus, we report: $92 \pm 2 \text{ s}$ This matches option C. --- Common Traps & Exam Tip:

Trap 1: Confusing Standard Deviation with Standard Error
Many students compute the standard error ($S_{\bar{x}} = 0.935$ s) and conclude the uncertainty is $\pm 1$ s, ignoring the fact that the data spread is larger. The standard deviation ($\sigma = 1.87$ s) better reflects the actual variability in measurements.

Trap 2: Ignoring Instrumental Error
Some students forget to consider the least count error of the clock. However, in this case, the random error dominates, so the instrumental error is not the limiting factor.

Trap 3: Over-rounding or Under-rounding
Rounding $1.87$ s to $1$ s is incorrect because it underestimates the uncertainty. Rounding to $2$ s is appropriate and matches the options.

Exam Tip:
When multiple measurements are taken, the uncertainty in the mean is best represented by the standard deviation of the sample, especially when it exceeds the instrumental error. Always round the uncertainty to one significant figure and ensure it aligns with the precision of the measuring instrument.

In this case, the correct reported mean time is: $\boxed{92 \pm 2 \text{ s}}$ which corresponds to Option C.