Question

The length of a highway was measured in 10 equal sections. If the precision of the individual section measurements is \(\sigma\), then what is the precision of the measured length of the highway?

• A. \(\sigma\sqrt{10}\)
• B. \(\sigma/\sqrt{10}\)
• C. \(10\sigma\)
• D. \(\sigma/10\)

Hint:

The highway length is a sum of 10 independent measurements, not an average, so use variance propagation for a sum: variances add, giving precision proportional to the square root of the number of sections.

Answer 1

The Correct Option is A

Step 1: Set up the total length as a sum of independent measurements.
Let the highway length be \(L = l_1 + l_2 + \cdots + l_{10}\), where each \(l_i\) is the independently measured length of one of the 10 equal sections, and each section is measured with the same precision (standard deviation) \(\sigma\).
Step 2: Apply the law of propagation of variance.
For a function \(L = f(l_1, l_2, \ldots, l_{10})\) of independent random variables, the variance propagation law gives \(\sigma_L^2 = \sum_{i=1}^{10} \left(\dfrac{\partial L}{\partial l_i}\right)^2 \sigma_i^2\). Here every partial derivative \(\dfrac{\partial L}{\partial l_i} = 1\), since \(L\) is a simple sum, and every \(\sigma_i = \sigma\).
Step 3: Substitute and simplify.
\[ \sigma_L^2 = \sum_{i=1}^{10} (1)^2 \sigma^2 = 10\sigma^2 \]
Step 4: Take the square root.
\[ \sigma_L = \sqrt{10\sigma^2} = \sigma\sqrt{10} \]
Step 5: Explain why the other options are wrong.
Option (C), \(10\sigma\), would be correct only if the errors were fully correlated (added directly without any cancellation), which is not the case for independent random errors; it overstates the true precision. Options (B) and (D), \(\sigma/\sqrt{10}\) and \(\sigma/10\), describe the precision of the mean (average) of 10 measurements of the same quantity, not the precision of their sum, so they do not apply here.
Step 6: Conclude.
The precision of the total measured length is \(\sigma\sqrt{10}\), reflecting the fact that random errors partially cancel when independent measurements are combined, but not as much as they would for an average.
\[ \sigma_L = \boxed{\sigma\sqrt{10}} \]