Question

In a graphical representation of a frequency distribution, if the distance between mode and mean is k times the distance 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

Step 1: Recall the empirical relationship between mean, median, and mode.

In moderately skewed distributions, there is an approximate empirical relationship:

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

Step 2: Express the distances mathematically.

Let the mean be \( \mu \), the median be \( M \), and the mode be \( \text{Mo} \). The distance between the mode and mean is:

\[ |\text{Mo} - \mu|. \]

The distance between the median and mean is:

\[ |M - \mu|. \]

Step 3: Use the empirical formula to relate these distances.

From the empirical relationship \( \text{Mo} = 3M - 2\mu \), we can write:

\[ \text{Mo} - \mu = (3M - 2\mu) - \mu = 3M - 3\mu = 3(M - \mu). \]

Taking absolute values:

\[ |\text{Mo} - \mu| = 3 |M - \mu|. \]

Step 4: Compare the distances.

The problem states that the distance between the mode and mean is \( k \) times the distance between the median and mean. From the above equation:

\[ |\text{Mo} - \mu| = k |M - \mu|. \]

Comparing this with \( |\text{Mo} - \mu| = 3 |M - \mu| \), we find:

\[ k = 3. \]

Final Answer: The value of \( k \) is \( \mathbf{3} \), which corresponds to option \( \mathbf{(2)} \).