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: Mode formula in terms of mean and median.
The mode of a distribution can be calculated using the empirical relationship: \[ \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \]
Step 2: Applying the values.
Given that the mean = 17 and the median = 18, we substitute these values into the formula: \[ \text{Mode} = 3 \times 18 - 2 \times 17 = 54 - 34 = 18.5 \]
Step 3: Conclusion.
Therefore, the mode of the distribution is \( \boxed{18.5} \). The correct answer is (3) 18.5.

Answer 2

This question relies on the empirical relationship between the mean, median, and mode of a moderately skewed frequency distribution, given by \( \text{Mode} = 3\,\text{Median} - 2\,\text{Mean} \). Substituting the given mean of 17 and median of 18 lets us check each option directly.

\[ \text{Mode} = 3(18) - 2(17) = 54 - 34 = 20 \]

1. 20: This matches the value obtained directly from the empirical formula above.
2. 17.5: This lies between the mean and the median, which is not where the empirical relationship places the mode; substituting it back would require \( 3(18)-2(17) \) to equal \( 17.5 \), which it does not.
3. 18.5: This is close to the median but does not satisfy \( 3\,\text{Median}-2\,\text{Mean}=54-34 \), which is exactly \( 20 \), not \( 18.5 \).
4. 19: Substituting into the same check, \( 54-34=20\ne19 \), so this does not satisfy the relationship either.

Only 20 is consistent with the standard mean-median-mode relationship applied to the given values.

Therefore, the correct answer is 20.