A scientist performs an experiment in order to measure a certain physical quantity and takes 100 observations. He repeats the same experiment and takes 400 observations, by doing so

Options

A. The possible error remains same
B. The possible error is doubled
C. The possible error is halved
D. The possible error is reduced to one fourth

Answer

To solve the problem, we need to understand how the number of observations affects the mean error in an experiment. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Mean Error Formula
The mean error (Δx_m) is defined as:
\[ \Delta x_m = \frac{\sum \Delta x}{n} \]
where \( \sum \Delta x \) is the total error in the observations, and \( n \) is the number of observations.

### Step 2: Calculate Mean Error for 100 Observations
Initially, when the scientist takes 100 observations, we can denote the mean error as:
\[ \Delta x_{m1} = \frac{\sum \Delta x}{100} \]

### Step 3: Calculate Mean Error for 400 Observations
When the scientist repeats the experiment and takes 400 observations, the mean error becomes:
\[ \Delta x_{m2} = \frac{\sum \Delta x}{400} \]

### Step 4: Compare the Two Mean Errors
Since the total error \( \sum \Delta x \) remains constant (the possible error does not change), we can compare the two mean errors:
\[ \Delta x_{m2} = \frac{\sum \Delta x}{400} = \frac{1}{4} \cdot \frac{\sum \Delta x}{100} = \frac{1}{4} \Delta x_{m1} \]
This shows that the mean error when taking 400 observations is one-fourth of the mean error when taking 100 observations.

### Step 5: Conclusion
Thus, by increasing the number of observations from 100 to 400, the possible error is reduced to one-fourth of its original value.

### Final Answer
The correct option is that the possible error is reduced to \( \frac{1}{4} \).

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