A data consists of n observations : `x_(1), x_(2),……, x_(n)`. If `Sigma_(i=1)^(n)(x_(i)+1)^(2)=11n and Sigma_(i=1)^(n)(x_(i)-1)^(2)=7n`, then the variance of this data is

Options

A. 5
B. 8
C. 6
D. 7

Answer

To solve the problem, we need to find the variance of the given data based on the two equations provided. Let's go through the solution step by step.

### Step 1: Expand the first equation
We start with the first equation:
\[ \sum_{i=1}^{n} (x_i + 1)^2 = 11n \]
Expanding the left side:
\[ \sum_{i=1}^{n} (x_i^2 + 2x_i + 1) = \sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i + \sum_{i=1}^{n} 1 \]
Since \(\sum_{i=1}^{n} 1 = n\), we can rewrite the equation as:
\[ \sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i + n = 11n \]
This simplifies to:
\[ \sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i = 10n \tag{1} \]

### Step 2: Expand the second equation
Now, we consider the second equation:
\[ \sum_{i=1}^{n} (x_i - 1)^2 = 7n \]
Expanding this:
\[ \sum_{i=1}^{n} (x_i^2 - 2x_i + 1) = \sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i + \sum_{i=1}^{n} 1 \]
Again, substituting \(\sum_{i=1}^{n} 1 = n\), we have:
\[ \sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i + n = 7n \]
This simplifies to:
\[ \sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i = 6n \tag{2} \]

### Step 3: Subtract the two equations
Now, we will subtract equation (2) from equation (1):
\[ \left(\sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i\right) - \left(\sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i\right) = 10n - 6n \]
This leads to:
\[ 4\sum_{i=1}^{n} x_i = 4n \]
Dividing both sides by 4 gives:
\[ \sum_{i=1}^{n} x_i = n \]
Thus, we find:
\[ \frac{\sum_{i=1}^{n} x_i}{n} = 1 \tag{3} \]
This indicates that the mean of the data is 1.

### Step 4: Add the two equations
Next, we will add equations (1) and (2):
\[ \left(\sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i\right) + \left(\sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i\right) = 10n + 6n \]
This simplifies to:
\[ 2\sum_{i=1}^{n} x_i^2 = 16n \]
Dividing by 2 gives:
\[ \sum_{i=1}^{n} x_i^2 = 8n \tag{4} \]

### Step 5: Calculate the variance
The formula for variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - \left(\frac{\sum_{i=1}^{n} x_i}{n}\right)^2 \]
Substituting the values from equations (3) and (4):
\[ \sigma^2 = \frac{8n}{n} - (1)^2 = 8 - 1 = 7 \]

### Final Answer
Thus, the variance of the data is:
\[ \boxed{7} \]