The random error in the arithmetic mean of 100 observations is x; then random error in the arithmetic mean of 400 observations would be

The correct option is x/2.

Step-by-Step Explanation:
In statistics, the random error (often characterized by the standard error of the mean) in the arithmetic mean of a set of observations is inversely proportional to the square root of the number of observations, $N$.
This relationship can be mathematically expressed as:
$\mathrm{Error}\propto \frac{1}{\sqrt{N}}$

Let $E_1$ be the random error for $N_1=100$ observations, and $E_2$ be the random error for $N_2=400$ observations.
According to the problem statement, $E_1=x$.

Using the proportionality relation, we can write the ratio of the two errors as:
$\frac{E_2}{E_1}=\sqrt{\frac{N_1}{N_2}}$

Substitute the given values into the equation:
$\frac{E_2}{x}=\sqrt{\frac{100}{400}}$
$\frac{E_2}{x}=\sqrt{\frac{1}{4}}$
$\frac{E_2}{x}=\frac{1}{2}$

Solving for $E_2$, we get:
$E_2=\frac{x}{2}$

Therefore, the random error in the arithmetic mean of 400 observations would be $\frac{x}{2}$ (or x/2).