The mean square deviation of a set of n observation `x_(1), x_(2), ..... x_(n)` about a point c is defined as `(1)/(n) sum_(i=1)^(n) (x_(i) -c)^(2)`.
The mean square deviations about – 2 and 2 are 18 and 10 respectively, the standard deviation of this set of observations is

Options

A. 3
B. 2
C. 1
D. None of these

Answer

To find the standard deviation of the set of observations given the mean square deviations about -2 and 2, we can use the following steps:

### Step 1: Understand the Mean Square Deviation
The mean square deviation (MSD) about a point \( c \) is given by:
\[ \text{MSD}(c) = \frac{1}{n} \sum_{i=1}^{n} (x_i - c)^2 \]
where \( n \) is the number of observations.

### Step 2: Set Up the Equations
From the problem, we know:
- The mean square deviation about -2 is 18:
\[ \frac{1}{n} \sum_{i=1}^{n} (x_i + 2)^2 = 18 \]
Multiplying both sides by \( n \):
\[ \sum_{i=1}^{n} (x_i + 2)^2 = 18n \]

- The mean square deviation about 2 is 10:
\[ \frac{1}{n} \sum_{i=1}^{n} (x_i - 2)^2 = 10 \]
Multiplying both sides by \( n \):
\[ \sum_{i=1}^{n} (x_i - 2)^2 = 10n \]

### Step 3: Expand the Equations
Now we will expand both equations:
1. For \( \sum_{i=1}^{n} (x_i + 2)^2 \):
\[ \sum_{i=1}^{n} (x_i^2 + 4x_i + 4) = \sum_{i=1}^{n} x_i^2 + 4\sum_{i=1}^{n} x_i + 4n = 18n \]

2. For \( \sum_{i=1}^{n} (x_i - 2)^2 \):
\[ \sum_{i=1}^{n} (x_i^2 - 4x_i + 4) = \sum_{i=1}^{n} x_i^2 - 4\sum_{i=1}^{n} x_i + 4n = 10n \]

### Step 4: Set Up the System of Equations
Let:
- \( S = \sum_{i=1}^{n} x_i^2 \)
- \( T = \sum_{i=1}^{n} x_i \)

From the first equation:
\[ S + 4T + 4n = 18n \quad \Rightarrow \quad S + 4T = 14n \quad \text{(1)} \]

From the second equation:
\[ S - 4T + 4n = 10n \quad \Rightarrow \quad S - 4T = 6n \quad \text{(2)} \]

### Step 5: Solve the System of Equations
Now, we have two equations:
1. \( S + 4T = 14n \)
2. \( S - 4T = 6n \)

Subtracting equation (2) from equation (1):
\[ (S + 4T) - (S - 4T) = 14n - 6n \]
This simplifies to:
\[ 8T = 8n \quad \Rightarrow \quad T = n \]

Substituting \( T = n \) back into equation (1):
\[ S + 4n = 14n \quad \Rightarrow \quad S = 10n \]

### Step 6: Calculate the Variance and Standard Deviation
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} x_i^2 - \left(\frac{1}{n} \sum_{i=1}^{n} x_i\right)^2 \]
Substituting \( S \) and \( T \):
\[ \sigma^2 = \frac{S}{n} - \left(\frac{T}{n}\right)^2 = \frac{10n}{n} - \left(\frac{n}{n}\right)^2 = 10 - 1 = 9 \]

Thus, the standard deviation \( \sigma \) is:
\[ \sigma = \sqrt{9} = 3 \]

### Final Answer
The standard deviation of the set of observations is \( 3 \).

---