Question

The mean of n observations is \(\bar{x}\). If one observation \(x_{n+1}\) is added and the new mean of n+1 observations was found to be same as previous. What was the value of \(x_{n+1}\)?

• A. 0
• B. 1
• C. n
• D. \(\bar{x}\)

Hint:

Adding an observation equal to the mean leaves the mean completely unchanged ($x_{n+1} = \bar{x}$).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The arithmetic mean remains unchanged upon adding a new observation if and only if the new observation is exactly equal to the existing mean.
Key Formula or Approach:
\[ \bar{x}_{n+1} = \frac{\sum_{i=1}^n x_i + x_{n+1}}{n + 1} \]

Step 2: Detailed Explanation:
Given that the sum of the initial \(n\) observations is:
\[ \sum_{i=1}^n x_i = n \bar{x} \]
Adding the new observation \(x_{n+1}\), the new mean is:
\[ \bar{x}_{n+1} = \frac{n \bar{x} + x_{n+1}}{n + 1} \]
Since the new mean equals the original mean \(\bar{x}\):
\[ \frac{n \bar{x} + x_{n+1}}{n + 1} = \bar{x} \]
\[ n \bar{x} + x_{n+1} = (n + 1)\bar{x} = n \bar{x} + \bar{x} \]
Subtracting \(n \bar{x}\) from both sides:
\[ x_{n+1} = \bar{x} \]

Step 3: Final Answer:
Hence, the value of \(x_{n+1}\) is \(\bar{x}\), corresponding to option (D).