If the difference of mode and median is 36, then the difference of median and mean is :

Question

Accepted Answer

To solve the problem, we need to understand the relationship between the mode, median, and mean in a data set. The problem states that the difference between the mode and the median is 36. We are asked to find the difference between the median and the mean.

In statistics, there is a well-known empirical relationship between the mean, median, and mode for a moderately skewed distribution, which is given by:

$$ \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} $$

Given:
$$ \text{Mode} - \text{Median} = 36 $$

Using the empirical relationship, we can express the mode in terms of the median and mean:

$$ \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} $$

Substituting the given difference:

$$ (3 \times \text{Median} - 2 \times \text{Mean}) - \text{Median} = 36 $$

Simplifying the equation:

$$ 2 \times \text{Median} - 2 \times \text{Mean} = 36 $$

$$ 2 \times (\text{Median} - \text{Mean}) = 36 $$

Dividing both sides by 2:

$$ \text{Median} - \text{Mean} = 18 $$

Thus, the difference between the median and the mean is 18, which corresponds to Option 2.

Now, let's explain why each of the other options is incorrect:

1. **Option 1 (12):** This option suggests that the difference between the median and the mean is 12. However, based on the calculation using the empirical relationship, we derived that the difference is 18, not 12.

3. **Option 3 (16):** This option suggests that the difference is 16. Again, our calculation shows that the difference is 18, making this option incorrect.

4. **Option 4 (22):** This option suggests that the difference is 22. Our derived difference of 18 does not match this value, so this option is incorrect.

Therefore, based on the empirical relationship and the given information, Option 2 (18) is the correct answer.