If random error in the arithmetic mean of 100 observations is x, then the randow error in the arithmetic mean of 500 observations would be

Options

A. 5x
B. `x//5`
C. 25x
D. `x//25`

Answer

To solve the problem, we need to understand how random errors in the arithmetic mean change with the number of observations. The random error in the arithmetic mean decreases as the number of observations increases.

### Step-by-Step Solution:

1. **Understanding Random Error**: The random error in the arithmetic mean is inversely proportional to the square root of the number of observations. This means that as we increase the number of observations, the random error decreases.

2. **Formula for Random Error**: The formula for the random error (E) in the arithmetic mean based on the number of observations (n) is given by:
\[ E \propto \frac{1}{\sqrt{n}} \]
This implies that if we have a random error \( x \) for \( n = 100 \), we can express it as:
\[ E_{100} = \frac{k}{\sqrt{100}} = \frac{k}{10} = x \]
where \( k \) is a constant.

3. **Finding the Random Error for 500 Observations**: Now, we need to find the random error when the number of observations is increased to 500. Using the same formula:
\[ E_{500} = \frac{k}{\sqrt{500}} \]

4. **Relating the Two Errors**: We can relate \( E_{500} \) to \( E_{100} \):
\[ E_{500} = \frac{k}{\sqrt{500}} = \frac{k}{\sqrt{100 \times 5}} = \frac{k}{10 \sqrt{5}} = \frac{x}{\sqrt{5}} \]

5. **Final Expression**: Thus, the random error in the arithmetic mean of 500 observations is:
\[ E_{500} = \frac{x}{\sqrt{5}} \]

### Conclusion:
The random error in the arithmetic mean of 500 observations would be \( \frac{x}{\sqrt{5}} \).