Question

An angle was measured with a standard error of 5". How many observations a surveyor needs to take in order to obtain a standard error of 1" for the mean value of this angle?

• A. 5
• B. 1
• C. 10
• D. 25

Hint:

Averaging $n$ independent measurements reduces random error by $\sqrt{n}$. To improve precision by a factor $k$, you need $n=k^2$ observations.

Answer 1

The Correct Option is D

Standard error (SE) of the mean: If each observation has standard deviation $\sigma$, then for $n$ i.i.d.\ observations, the SE of the sample mean is $\sigma_{\bar{x}}=\dfrac{\sigma}{\sqrt{n}}$.
Here the single‐observation standard error is $5''$ (acts as $\sigma$). We require $\sigma_{\bar{x}}=1''$. Hence \[ \frac{5''}{\sqrt{n}}=1'' \;⇒\; \sqrt{n}=5 \;⇒\; n=25. \] Therefore, the surveyor must take 25 observations.
\[ \boxed{25} \]