If mode of a series exceeds its mean by 12, then mode exceeds the median by :

Question

Accepted Answer

To determine why Option 3 is the correct answer, we need to understand the relationship between the mode, mean, and median in a data series. The question states that the mode exceeds the mean by 12. We are asked to find out by how much the mode exceeds the median.

In statistics, there is an empirical relationship between the mean, median, and mode for a moderately skewed distribution, which is often expressed as:

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

Given that the mode exceeds the mean by 12, we can express this as:

$$ \text{Mode} = \text{Mean} + 12 $$

Substituting this into the empirical relationship, we have:

$$ \text{Mean} + 12 = 3 \times \text{Median} - 2 \times \text{Mean} $$

Rearranging the terms, we get:

$$ 3 \times \text{Median} = 3 \times \text{Mean} + 12 $$

$$ \text{Median} = \text{Mean} + 4 $$

Now, we know:

$$ \text{Mode} = \text{Mean} + 12 $$
$$ \text{Median} = \text{Mean} + 4 $$

To find how much the mode exceeds the median, we subtract the median from the mode:

$$ \text{Mode} - \text{Median} = (\text{Mean} + 12) - (\text{Mean} + 4) $$

$$ \text{Mode} - \text{Median} = 12 - 4 = 8 $$

Thus, the mode exceeds the median by 8, which corresponds to Option 3.

**Explanation for Incorrect Options:**

1. **Option 1 (4):** This option suggests that the mode exceeds the median by 4. However, as shown in the calculation, the difference is actually 8, not 4.

2. **Option 2 (6):** This option suggests a difference of 6 between the mode and the median. Again, the calculation clearly shows that the difference is 8, not 6.

3. **Option 4 (10):** This option suggests a difference of 10. The calculation demonstrates that the actual difference is 8, making this option incorrect.

Therefore, based on the empirical relationship and the given condition, Option 3 is indeed the correct answer.