Karl Pearson's coefficient of skewness of a distribution is `0*32`. Its s.d. is `6*5` and mean is `29*6`. The mode and median of the distribution are

Options

A. `27*52,28.91`
B. `26*91,27*23`
C. `25*67,26*34`
D. none of these

Answer

To find the mode and median of the distribution given Karl Pearson's coefficient of skewness, we can use the following formulas:

1. **Coefficient of Skewness (Pearson's)**:
\[ \text{Skewness} = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}} \]
Given:
- Skewness = 0.32
- Mean (\(\bar{x}\)) = 29.6
- Standard Deviation (SD) = 6.5

2. **Rearranging the formula to find Mode**:
\[ \text{Mode} = \text{Mean} - (\text{Skewness} \times \text{Standard Deviation}) \]

3. **Substituting the values**:
\[ \text{Mode} = 29.6 - (0.32 \times 6.5) \]

4. **Calculating the product**:
\[ 0.32 \times 6.5 = 2.08 \]

5. **Finding Mode**:
\[ \text{Mode} = 29.6 - 2.08 = 27.52 \]

Now, we need to find the median using the second formula for skewness:

6. **Using the second formula for skewness**:
\[ \text{Skewness} = \frac{3(\text{Mean} - \text{Median})}{\text{Standard Deviation}} \]

7. **Rearranging to find Median**:
\[ 3(\text{Mean} - \text{Median}) = \text{Skewness} \times \text{Standard Deviation} \]
\[ \text{Mean} - \text{Median} = \frac{\text{Skewness} \times \text{Standard Deviation}}{3} \]

8. **Substituting the values**:
\[ 29.6 - \text{Median} = \frac{0.32 \times 6.5}{3} \]

9. **Calculating the right side**:
\[ \frac{2.08}{3} \approx 0.6933 \]

10. **Finding Median**:
\[ 29.6 - \text{Median} = 0.6933 \]
\[ \text{Median} = 29.6 - 0.6933 \approx 28.9067 \approx 28.91 \]

### Final Answers:
- Mode = 27.52
- Median = 28.91