Question

In a frequency distribution, the mean and median are 17 and 18 respectively. Then the mode of the distribution is

• A. 20
• B. 17.5
• C. 18.5
• D. 19

Hint:

The mode can be found using the empirical formula \( \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \) for symmetric distributions.

Answer 1

The Correct Option is A

Step 1: The empirical relationship between mean, median and mode can also be written as \( \text{Mean} - \text{Mode} = 3(\text{Mean}-\text{Median}) \).
Step 2: Substituting mean \( =17 \) and median \( =18 \): \( 17-\text{Mode} = 3(17-18) = 3(-1) = -3 \).
Step 3: Solving, \( \text{Mode} = 17-(-3) = 17+3 = 20 \).
\[ \boxed{\text{Mode} = 20} \]

Answer 2

A useful way to think about this without directly quoting the mode formula is to note that, in a moderately skewed distribution, the mode sits on the opposite side of the median from the mean, at twice the distance between the mean and the median. That is, \( \text{Median} - \text{Mode} = 2(\text{Mean} - \text{Median}) \).

Substituting mean \( =17 \) and median \( =18 \): \( 18 - \text{Mode} = 2(17-18) = 2(-1) = -2 \), so \( \text{Mode} = 18-(-2) = 20 \).

1. 20: Matches the value derived from this spacing relationship exactly.
2. 17.5: Substituting back, \( 18-17.5=0.5 \), which does not equal \( 2(17-18)=-2 \), so this option fails.
3. 18.5: Substituting back, \( 18-18.5=-0.5 \), again not equal to \( -2 \), so this fails too.
4. 19: Substituting back, \( 18-19=-1 \), still not equal to \( -2 \), so this option is also inconsistent.

Only a mode of 20 is consistent with the mean and median being spaced in this characteristic 1:2 ratio around it.

Therefore, the correct answer is 20.