The mean square deviation of a set of m observations `y_1, y_2…..y_m` about a point K is defined as `1/m sum_(i = 1)^(m) (y_i - k)^(2)`. The mean square deviation about -3 and 3 are 16 and 8 respectively, then standard deviation of this set of observation?

Options

A. `(sqrt23)/(3)`
B. `sqrt(7)`
C. `(sqrt41)/(3)`
D. `(sqrt(38))/(3)`

Explanation

To solve the problem, we need to find the standard deviation of the set of observations given the mean square deviations about two points, -3 and 3.

### Step-by-Step Solution:

1. **Understanding Mean Square Deviation**:
The mean square deviation (MSD) about a point \( K \) for a set of observations \( y_1, y_2, \ldots, y_m \) is given by:
\[ MSD(K) = \frac{1}{m} \sum_{i=1}^{m} (y_i - K)^2 \]
This formula represents the average of the squared differences between each observation and the point \( K \).

2. **Given Information**:
We are given:
- \( MSD(-3) = 16 \)
- \( MSD(3) = 8 \)

3. **Using the Formula**:
From the definitions:
\[ MSD(-3) = \frac{1}{m} \sum_{i=1}^{m} (y_i + 3)^2 = 16 \]
This implies:
\[ \sum_{i=1}^{m} (y_i + 3)^2 = 16m \]
Similarly,
\[ MSD(3) = \frac{1}{m} \sum_{i=1}^{m} (y_i - 3)^2 = 8 \]
This implies:
\[ \sum_{i=1}^{m} (y_i - 3)^2 = 8m \]

4. **Expanding the Squares**:
Now, we can expand both equations:
- For \( MSD(-3) \):
\[ \sum_{i=1}^{m} (y_i + 3)^2 = \sum_{i=1}^{m} (y_i^2 + 6y_i + 9) = \sum_{i=1}^{m} y_i^2 + 6\sum_{i=1}^{m} y_i + 9m = 16m \]
- For \( MSD(3) \):
\[ \sum_{i=1}^{m} (y_i - 3)^2 = \sum_{i=1}^{m} (y_i^2 - 6y_i + 9) = \sum_{i=1}^{m} y_i^2 - 6\sum_{i=1}^{m} y_i + 9m = 8m \]

5. **Setting Up the Equations**:
Let \( S = \sum_{i=1}^{m} y_i \) and \( Q = \sum_{i=1}^{m} y_i^2 \). We can rewrite the equations:
- From \( MSD(-3) \):
\[ Q + 6S + 9m = 16m \quad \Rightarrow \quad Q + 6S = 7m \quad \text{(1)} \]
- From \( MSD(3) \):
\[ Q - 6S + 9m = 8m \quad \Rightarrow \quad Q - 6S = -m \quad \text{(2)} \]

6. **Solving the System of Equations**:
Now we can solve equations (1) and (2):
- Adding (1) and (2):
\[ (Q + 6S) + (Q - 6S) = 7m - m \]
\[ 2Q = 6m \quad \Rightarrow \quad Q = 3m \quad \text{(3)} \]
- Substituting \( Q \) back into equation (1):
\[ 3m + 6S = 7m \quad \Rightarrow \quad 6S = 4m \quad \Rightarrow \quad S = \frac{2m}{3} \quad \text{(4)} \]

7. **Finding the Variance**:
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{1}{m} \sum_{i=1}^{m} (y_i^2) - \left(\frac{1}{m} \sum_{i=1}^{m} y_i\right)^2 = \frac{Q}{m} - \left(\frac{S}{m}\right)^2 \]
Substituting \( Q \) and \( S \) from (3) and (4):
\[ \sigma^2 = \frac{3m}{m} - \left(\frac{2m/3}{m}\right)^2 = 3 - \left(\frac{2}{3}\right)^2 = 3 - \frac{4}{9} = \frac{27}{9} - \frac{4}{9} = \frac{23}{9} \]

8. **Finding the Standard Deviation**:
The standard deviation \( \sigma \) is the square root of the variance:
\[ \sigma = \sqrt{\frac{23}{9}} = \frac{\sqrt{23}}{3} \]

### Final Answer:
The standard deviation of the set of observations is \( \frac{\sqrt{23}}{3} \).