Surveying and error theory: how do random (accidental) errors accumulate as the number of independent observations/operations increases? Choose the most accepted proportionality used for precision estimates in civil engineering measurements.

Introduction / Context:
Random (accidental) errors arise from small, uncontrollable variations in observation conditions—sight clarity, hand steadiness, reading interpolation, and atmospheric flicker. In surveying and geospatial engineering, understanding how such errors accumulate is essential for planning redundancy, estimating precision, and choosing appropriate adjustment methods.

Given Data / Assumptions:

• Errors are random with zero mean and finite variance.
• Individual observations/operations are independent and similarly distributed.
• No dominant systematic (bias) or gross errors are present.

Concept / Approach:

For independent random errors, variances add. If each operation has standard deviation sigma, then over n operations, the resultant standard deviation is sigma_total = sqrt(n) * sigma. Hence the overall random error grows with the square root of the number of operations, not linearly with n. This square-root law underpins the familiar improvement in precision by averaging multiple observations (precision improves only as 1/sqrt(n)).

Step-by-Step Solution:

Verification / Alternative check:

Least-squares theory uses the same variance-addition principle; traverse closures and leveling misclosures follow root-n behavior when random direction/reading errors dominate.

Why Other Options Are Wrong:

Linear with n: implies systematic drift, not random accumulation. Reciprocal with n or cube root: do not reflect variance addition for independent errors. “None” ignores established statistics.

Common Pitfalls:

Confusing random with systematic errors; assuming averaging eliminates error completely; ignoring correlation which can violate sqrt(n) scaling.

Final Answer:

square root of the number of operations involved