Traverse bearings—probable error: The probable error of the adjusted bearing at the middle of a series of observations is best expressed as which of the following (r = probable error of a single observed bearing, n = number of observations)?

Introduction / Context:

When repeated observations are combined to obtain an adjusted bearing (or any repeated measurement), random errors tend to cancel out. The spread of the mean (or adjusted value) decreases with the number of observations, a fundamental result from error theory used in surveying adjustments.

Given Data / Assumptions:

• Each repeated observation of the bearing has the same precision (identical r).
• Errors are random, independent, and follow a normal distribution.
• We are interested in the probable error of the adjusted bearing (essentially the mean).

Concept / Approach:

For independent, equally precise observations, the dispersion of the mean is related to that of a single observation by the factor 1/√n. Thus the probable error (or standard error, up to a constant factor) of the average is r/√n. This is why surveyors repeat angle measurements and bearings in multiple sets; the precision of the adjusted value improves as the square root of the number of sets.

Step-by-Step Solution:

Verification / Alternative check:

Compute the sample variance of repeated sets and compare against the expected 1/√n improvement; practical field data track this trend unless dominated by systematics.

Why Other Options Are Wrong:

• r√n: would imply worse precision with more observations, contrary to statistics.
• r independent of n or n/r: ignore the averaging benefit.
• r/(n − 1): relates to an unbiased variance estimator, not probable error of the mean.

Common Pitfalls:

• Assuming more observations always help: only true if errors are random and independent; systematics must still be controlled.

Final Answer:

r / √n