A data consists of n observations `x_(1), x_(2), ..., x_(n). If Sigma_(i=1)^(n) (x_(i) + 1)^(2) = 9n and Sigma_(i=1)^(n) (x_(i) - 1)^(2) = 5n`, then the standard deviation of this data is

Options

A. 5
B. `sqrt(5)`
C. `sqrt(7)`
D. 2

Answer

To solve the problem, we need to analyze the given equations and derive the standard deviation step by step.

### Step-by-Step Solution:

1. **Understanding the Given Equations:**
We have two equations:
\[ \sum_{i=1}^{n} (x_i + 1)^2 = 9n \]
\[ \sum_{i=1}^{n} (x_i - 1)^2 = 5n \]

2. **Expanding the First Equation:**
Expanding \((x_i + 1)^2\):
\[ (x_i + 1)^2 = x_i^2 + 2x_i + 1 \]
Therefore,
\[ \sum_{i=1}^{n} (x_i + 1)^2 = \sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i + n = 9n \]
Let \(\sum_{i=1}^{n} x_i^2 = A\) and \(\sum_{i=1}^{n} x_i = B\). Then we have:
\[ A + 2B + n = 9n \implies A + 2B = 8n \quad (1) \]

3. **Expanding the Second Equation:**
Expanding \((x_i - 1)^2\):
\[ (x_i - 1)^2 = x_i^2 - 2x_i + 1 \]
Therefore,
\[ \sum_{i=1}^{n} (x_i - 1)^2 = \sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i + n = 5n \]
Using \(A\) and \(B\):
\[ A - 2B + n = 5n \implies A - 2B = 4n \quad (2) \]

4. **Solving the System of Equations:**
Now we have two equations:
\[ A + 2B = 8n \quad (1) \]
\[ A - 2B = 4n \quad (2) \]
Adding (1) and (2):
\[ (A + 2B) + (A - 2B) = 8n + 4n \]
This simplifies to:
\[ 2A = 12n \implies A = 6n \]

Now substituting \(A = 6n\) into equation (1):
\[ 6n + 2B = 8n \implies 2B = 2n \implies B = n \]

5. **Finding the Mean:**
The mean \(\bar{x}\) is given by:
\[ \bar{x} = \frac{B}{n} = \frac{n}{n} = 1 \]

6. **Calculating the Standard Deviation:**
The formula for the standard deviation \(\sigma\) is:
\[ \sigma = \sqrt{\frac{A}{n} - \left(\frac{B}{n}\right)^2} \]
Substituting the values:
\[ \sigma = \sqrt{\frac{6n}{n} - \left(\frac{n}{n}\right)^2} = \sqrt{6 - 1} = \sqrt{5} \]

### Final Answer:
The standard deviation of the data is \(\sqrt{5}\).