For an asymmetrical distribution, if the median is 7 and the mean is 10, then the mode is :

Question

Accepted Answer

In an asymmetrical distribution, the relationship between the mean, median, and mode can be described using the empirical rule known as the "Pearson's first coefficient of skewness." This rule states that the skewness can be approximated by the formula:

$$ \text{Skewness} = \frac{3(\text{Mean} - \text{Median})}{\text{Standard Deviation}} $$

However, a more straightforward relationship often used is:

$$ \text{Mode} \approx 3(\text{Median}) - 2(\text{Mean}) $$

Given in the question:
- Median = 7
- Mean = 10

Using the formula for the mode:

$$ \text{Mode} \approx 3(7) - 2(10) $$
$$ \text{Mode} \approx 21 - 20 $$
$$ \text{Mode} \approx 1 $$

Thus, the mode is approximately 1, which corresponds to Option 4.

**Explanation for why Option 4 is correct:**
- The calculation using the formula $ \text{Mode} \approx 3(\text{Median}) - 2(\text{Mean}) $ yields a result of 1, which matches Option 4. This supports the given correct answer.

**Explanation for why the other options are incorrect:**
1. **Option 1 (11):** This option is incorrect because the calculated mode using the formula is 1, not 11. The value 11 would imply a different relationship between the mean and median, which does not fit the given data.
   
2. **Option 2 (4):** This option is incorrect because the calculated mode is 1. A mode of 4 would suggest a different skewness and distribution shape that does not align with the given mean and median.

3. **Option 3 (3):** This option is incorrect because the calculated mode is 1. A mode of 3 would also imply a different distribution pattern that is not supported by the given mean and median.

In conclusion, based on the relationship between the mean, median, and mode in an asymmetrical distribution, Option 4 is the correct answer, as it aligns with the calculated mode of 1.