The sum of deviations of n observations about 25 is 25 and sum of deviations of the same n observations about 35 is -25 . The mean of observations is

Options

A. 25
B. 30
C. 35
D. 40

Answer

To find the mean of the observations based on the given information, we can follow these steps:

### Step 1: Understanding the Given Information
We are given two pieces of information about the sum of deviations of n observations:
1. The sum of deviations about 25 is 25.
2. The sum of deviations about 35 is -25.

### Step 2: Setting Up the Equations
Using the notation \( \sigma \) for the sum of observations \( x_i \):
- The first equation can be expressed as:
\[ \sigma (x_i - 25) = 25 \]
This simplifies to:
\[ \sigma x_i - 25n = 25 \quad \text{(1)} \]

- The second equation can be expressed as:
\[ \sigma (x_i - 35) = -25 \]
This simplifies to:
\[ \sigma x_i - 35n = -25 \quad \text{(2)} \]

### Step 3: Rearranging the Equations
From equation (1):
\[ \sigma x_i = 25 + 25n \quad \text{(3)} \]

From equation (2):
\[ \sigma x_i = -25 + 35n \quad \text{(4)} \]

### Step 4: Equating the Two Expressions for \( \sigma x_i \)
Set equations (3) and (4) equal to each other:
\[ 25 + 25n = -25 + 35n \]

### Step 5: Solving for \( n \)
Rearranging the equation gives:
\[ 25 + 25 = 35n - 25n \]
\[ 50 = 10n \]
\[ n = 5 \]

### Step 6: Finding \( \sigma x_i \)
Substituting \( n = 5 \) back into equation (3):
\[ \sigma x_i = 25 + 25 \times 5 \]
\[ \sigma x_i = 25 + 125 = 150 \]

### Step 7: Calculating the Mean
The mean \( \bar{x} \) is given by:
\[ \bar{x} = \frac{\sigma x_i}{n} = \frac{150}{5} = 30 \]

### Final Answer
The mean of the observations is \( \boxed{30} \).