D

The random error in the arithmetic mean decreases as the sample size increases. The formula for the random error in the mean is given by \( \frac{s}{\sqrt{n}} \), where \(s\) is the standard deviation and \(n\) is the sample size.

Given that the random error for 100 observations is \(x\), we have:\
\( x = \frac{s}{\sqrt{100}} = \frac{s}{10} \)

Now, for 400 observations:
\( n = 400 \)
So, the random error becomes:
\( new\ random\ error = \frac{s}{\sqrt{400}} = \frac{s}{20} \)
Now, we relate this to \(x\):
\( \text{Random error for } 400\ observations = \frac{1}{2} \times x\)

Thus, the answer is \( \frac{1}{2} x \), which corresponds to Option D.