If mean and mode of a distribution are 74 and 29 respectively, then median of the distribution is :

Question

Accepted Answer

To determine the median of a distribution when the mean and mode are given, we can use the empirical relationship between the mean, median, and mode. This relationship is often expressed as:

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

Given:
- Mean = 74
- Mode = 29

We need to find the median. Let's substitute the given values into the formula:

$$ 74 - 29 = 3(74 - \text{Median}) $$

$$ 45 = 3(74 - \text{Median}) $$

Now, divide both sides by 3 to solve for the median:

$$ 15 = 74 - \text{Median} $$

Rearrange the equation to find the median:

$$ \text{Median} = 74 - 15 $$

$$ \text{Median} = 59 $$

Therefore, the median of the distribution is 59, which corresponds to Option 2.

**Explanation for why each option is incorrect except Option 2:**

1. **Option 1 (49):** 
   - If the median were 49, substituting back into the empirical formula would not satisfy the equation. The calculation would not balance, indicating that 49 cannot be the correct median.

2. **Option 3 (62):**
   - Similarly, if the median were 62, substituting into the formula would result in an imbalance. The difference between the mean and the calculated median would not align with the empirical relationship, making 62 incorrect.

3. **Option 4 (63):**
   - Using 63 as the median in the formula would also lead to an incorrect balance. The empirical relationship would not hold true, confirming that 63 is not the correct median.

In conclusion, based on the empirical relationship and the given values, the median is correctly calculated as 59, which is Option 2. This option satisfies the relationship between mean, median, and mode, making it the correct answer.