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 find the standard deviation of the observations given the mean square deviations about the points -1 and 1, we can follow these steps:

### Step 1: Understand the Mean Square Deviation
The mean square deviation (MSD) about a point \( m \) is defined as:
\[ \sigma_m^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - m)^2 \]
where \( \sigma_m^2 \) is the mean square deviation about the point \( m \), \( n \) is the number of observations, and \( x_i \) are the observations.

### Step 2: Set Up the Equations
We are given:
- The mean square deviation about \( m_1 = -1 \) is 7:
\[ \sigma_{-1}^2 = 7 = \frac{1}{n} \sum_{i=1}^{n} (x_i + 1)^2 \]
- The mean square deviation about \( m_2 = 1 \) is 3:
\[ \sigma_{1}^2 = 3 = \frac{1}{n} \sum_{i=1}^{n} (x_i - 1)^2 \]

### Step 3: Expand the Equations
Using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \), we can expand both equations:

1. For \( m_1 = -1 \):
\[ 7 = \frac{1}{n} \left( \sum_{i=1}^{n} (x_i^2 + 2x_i + 1) \right) \]
This simplifies to:
\[ 7n = \sum_{i=1}^{n} x_i^2 + 2\sum_{i=1}^{n} x_i + n \]
Let \( S = \sum_{i=1}^{n} x_i \) and \( S_2 = \sum_{i=1}^{n} x_i^2 \):
\[ 7n = S_2 + 2S + n \]
Rearranging gives us:
\[ S_2 + 2S = 6n \quad \text{(Equation 1)} \]

2. For \( m_2 = 1 \):
\[ 3 = \frac{1}{n} \left( \sum_{i=1}^{n} (x_i^2 - 2x_i + 1) \right) \]
This simplifies to:
\[ 3n = \sum_{i=1}^{n} x_i^2 - 2\sum_{i=1}^{n} x_i + n \]
Rearranging gives us:
\[ S_2 - 2S = 2n \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( S_2 + 2S = 6n \)
2. \( S_2 - 2S = 2n \)

Subtract Equation 2 from Equation 1:
\[ (S_2 + 2S) - (S_2 - 2S) = 6n - 2n \]
This simplifies to:
\[ 4S = 4n \implies S = n \]

### Step 5: Substitute Back to Find \( S_2 \)
Now substitute \( S = n \) back into either equation, let's use Equation 1:
\[ S_2 + 2(n) = 6n \]
\[ S_2 + 2n = 6n \implies S_2 = 4n \]

### Step 6: Calculate the Variance
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{S_2}{n} - \left(\frac{S}{n}\right)^2 \]
Substituting \( S_2 = 4n \) and \( S = n \):
\[ \sigma^2 = \frac{4n}{n} - \left(\frac{n}{n}\right)^2 = 4 - 1 = 3 \]

### Step 7: Calculate the Standard Deviation
The standard deviation \( \sigma \) is the square root of the variance:
\[ \sigma = \sqrt{3} \]

### Final Answer
The standard deviation of the observations is:
\[ \sqrt{3} \]