Question

If the difference between mode and mean of a data is k times the difference between median and mean, then the value of k is

• A. 2
• B. 3
• C. 1
• D. Cannot be determined

Answer 1

The Correct Option is B

To solve this problem, we will use the empirical relationship between mean, median, and mode typically observed in a moderately skewed distribution. 
This relationship is given by: Mode = 3 \(\times\) Median - 2 \(\times\) Mean

Let's denote:

• M_o = Mode
• M_e = Median
• X̄ = Mean

According to the problem, the difference between mode and mean is k times the difference between median and mean:
(M_o - X̄) = k(M_e - X̄)
Substitute the empirical relationship into the equation:
(3M_e - 2X̄ - X̄) = k(M_e - X̄)
3M_e - 3X̄ = kM_e - kX̄
Rearranging terms gives us:
3M_e - kM_e = 3X̄ - kX̄
M_e(3 - k) = X̄(3 - k)
Since M_e and X̄ are not equal in a skewed distribution, we cannot cancel (3 - k). 
Therefore, if 3 - k = 0, k = 3.
Thus, the value of k is 3.

Answer 2

Given: The difference between mode and mean of a data is \( k \) times the difference between median and mean.

Step 1: Recall the Empirical Formula 

The empirical relationship between mode, mean, and median is:

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

Rearranging this equation:

\[ \text{Mode} - \text{Mean} = 3 \times (\text{Median} - \text{Mean}) \]

Comparing with the given condition:

\[ k = 3 \]

Final Answer: \( \mathbf{3} \)