How random error changes with the number of observations

Key concept: This question tests the square-root law for random error in the mean of repeated observations.

Step by step

Key relationship

For independent random observations, the random error in the arithmetic mean is inversely proportional to the square root of the number of observations.

So if the number of observations changes from $n_1$ to $n_2$, then the random error changes as

$\text{error ratio}=\sqrt{\frac{n_1}{n_2}}.$

Here, the mean of 100 observations has random error $x$. For 400 observations, the new error is

$x\sqrt{\frac{100}{400}}=x\sqrt{\frac14}=\frac12 x.$

Therefore the correct answer is

$\boxed{\frac12 x}$

which is option (D).

Why this happens

Averages become more stable when more measurements are included. Random positive and negative deviations tend to cancel out, so the uncertainty in the mean decreases as the sample size increases. But the decrease is not linear; it follows the square-root law.

That is why quadrupling the number of observations does not make the error one-fourth as large by a direct rule. Instead, it makes the error one-half as large, because $\sqrt{400}=2\sqrt{100}$.

Common formula to remember

If the standard error of the mean is involved, the same idea appears as

$\sigma_{\bar x}=\frac{\sigma}{\sqrt n}.$

This is the reason the answer scales like $1/\sqrt{n}$ rather than $1/n$.

Pitfall alert

A frequent mistake is to assume that if the number of observations is four times larger, the error must be one-fourth as large. That would be true for a direct inverse relationship, but random error in the mean follows a square-root law instead. Another mistake is to confuse random error with systematic error. Increasing the number of observations reduces random fluctuations, but it does not fix a consistent instrument bias. So the square-root rule applies only to random error in repeated independent measurements.

Try different conditions

If the number of observations increased from 100 to 900, the random error would become $x\sqrt{100/900}=x/3$. If the question instead asked for 25 observations, the error would be $x\sqrt{100/25}=2x$. A variant phrasing might ask for the standard error of the mean rather than random error; in that case, the same $1/\sqrt n$ relationship is used, so the logic stays the same.

Further reading

standard error, arithmetic mean, square-root law