If the difference between the mode and the median of certain observations is `54`, then the difference between the median and the mean is________

Options

A. `36`
B. `18`
C. `27`
D. `81`

Answer

To solve the problem step-by-step, we will use the information given about the mode, median, and mean.

### Step-by-Step Solution:

1. **Understand the Given Information**:
We know that the difference between the mode and the median is 54. This can be expressed as:
\[ \text{Mode} - \text{Median} = 54 \]

2. **Express Mode in Terms of Median**:
From the equation above, we can express the mode as:
\[ \text{Mode} = \text{Median} + 54 \]

3. **Use the Relationship Between Mode, Median, and Mean**:
There is a relationship in statistics that states:
\[ \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \]
We will use this relationship to find the mean.

4. **Substitute the Expression for Mode**:
Now, we can substitute the expression for the mode into the equation:
\[ \text{Median} + 54 = 3 \times \text{Median} - 2 \times \text{Mean} \]

5. **Rearrange the Equation**:
Rearranging gives us:
\[ 54 = 3 \times \text{Median} - \text{Median} - 2 \times \text{Mean} \]
Simplifying this, we have:
\[ 54 = 2 \times \text{Median} - 2 \times \text{Mean} \]

6. **Factor Out the Common Term**:
Factoring out the 2 from the left side gives:
\[ 54 = 2 \times (\text{Median} - \text{Mean}) \]

7. **Solve for the Difference Between Median and Mean**:
Dividing both sides by 2 gives:
\[ \text{Median} - \text{Mean} = \frac{54}{2} = 27 \]

### Final Answer:
The difference between the median and the mean is **27**.

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