The mean square deviation of a set of observation `x_(1), x_(2)……x_(n)` about a point m is defined as `(1)/(n)Sigma_(i=1)^(n)(x_(i)-m)^(2)`. If the mean square deviation about `-1 and 1` of a set of observation are 7 and 3 respectively. The standard deviation of those observations is

Options

A. `sqrt2`
B. 2
C. 5
D. `sqrt3`

Answer

To solve the problem, we need to find the standard deviation of a set of observations given the mean square deviations about the points -1 and 1.

### Step 1: Set up the equations for mean square deviation

The mean square deviation about a point \( m \) is given by:

\[ \text{MSD}(m) = \frac{1}{n} \sum_{i=1}^{n} (x_i - m)^2 \]

Given that the mean square deviation about -1 is 7, we can write:

\[ \frac{1}{n} \sum_{i=1}^{n} (x_i + 1)^2 = 7 \]

Multiplying both sides by \( n \):

\[ \sum_{i=1}^{n} (x_i + 1)^2 = 7n \]

Expanding the left side:

\[ \sum_{i=1}^{n} (x_i^2 + 2x_i + 1) = 7n \]

This simplifies to:

\[ \sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i + n = 7n \]

Rearranging gives us:

\[ \sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i = 6n \tag{1} \]

### Step 2: Set up the second equation for mean square deviation

Next, we consider the mean square deviation about 1, which is given as 3:

\[ \frac{1}{n} \sum_{i=1}^{n} (x_i - 1)^2 = 3 \]

Multiplying both sides by \( n \):

\[ \sum_{i=1}^{n} (x_i - 1)^2 = 3n \]

Expanding the left side:

\[ \sum_{i=1}^{n} (x_i^2 - 2x_i + 1) = 3n \]

This simplifies to:

\[ \sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i + n = 3n \]

Rearranging gives us:

\[ \sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i = 2n \tag{2} \]

### Step 3: Solve the equations

Now we have two equations:

1. \( \sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i = 6n \)
2. \( \sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i = 2n \)

Let's add these two equations:

\[ \left( \sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i \right) + \left( \sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i \right) = 6n + 2n \]

This simplifies to:

\[ 2\sum_{i=1}^{n} x_i^2 = 8n \]

Dividing by 2:

\[ \sum_{i=1}^{n} x_i^2 = 4n \tag{3} \]

Now, let's subtract equation (2) from equation (1):

\[ \left( \sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i \right) - \left( \sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i \right) = 6n - 2n \]

This simplifies to:

\[ 4\sum_{i=1}^{n} x_i = 4n \]

Dividing by 4:

\[ \sum_{i=1}^{n} x_i = n \tag{4} \]

### Step 4: Calculate the standard deviation

The formula for the standard deviation \( \sigma \) is given by:

\[ \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} x_i^2 - \left( \frac{1}{n} \sum_{i=1}^{n} x_i \right)^2} \]

Substituting the values from equations (3) and (4):

\[ \sigma = \sqrt{\frac{1}{n} (4n) - \left( \frac{1}{n} (n) \right)^2} \]

This simplifies to:

\[ \sigma = \sqrt{4 - 1} = \sqrt{3} \]

Thus, the standard deviation of the observations is:

\[ \boxed{\sqrt{3}} \]