If the difference between mean and mode is 63, then the difference between mean and median is

Options

A. 189
B. 21
C. `31.5`
D. `48.5`

Answer

To solve the problem step by step, we start with the information given in the question:

1. **Given Information**: We know that the difference between the mean and mode is 63.
\[ \text{Mean} - \text{Mode} = 63 \]

2. **Using the relationship between Mean, Median, and Mode**: We use the empirical relationship that relates mean, median, and mode:
\[ \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \]

3. **Substituting Mode in the equation**: From the first equation, we can express the mode in terms of the mean:
\[ \text{Mode} = \text{Mean} - 63 \]

4. **Setting the two expressions for Mode equal**: We can now set the two expressions for mode equal to each other:
\[ \text{Mean} - 63 = 3 \times \text{Median} - 2 \times \text{Mean} \]

5. **Rearranging the equation**: Now, we rearrange the equation to isolate the terms involving the mean and median:
\[ \text{Mean} + 2 \times \text{Mean} = 3 \times \text{Median} + 63 \]
\[ 3 \times \text{Mean} = 3 \times \text{Median} + 63 \]

6. **Dividing by 3**: We can simplify this equation by dividing everything by 3:
\[ \text{Mean} = \text{Median} + 21 \]

7. **Finding the difference between Mean and Median**: The difference between the mean and median is:
\[ \text{Mean} - \text{Median} = 21 \]

Thus, the difference between the mean and median is **21**.