If the difference between the mean and the mode of certain observations is 93, then the difference between the mean and the median is ______

Answer

To solve the problem step by step, we will use the relationship between the mean, median, and mode.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We are given that the difference between the mean (M) and the mode (Mo) is 93. This can be expressed as:
\[ M - Mo = 93 \quad \text{(Equation 1)} \]

2. **Using the Relationship Between Mean, Median, and Mode**:
We can use the formula that relates mean, median, and mode:
\[ Mo = 3 \times \text{Median} - 2 \times M \quad \text{(Equation 2)} \]

3. **Substituting Equation 1 into Equation 2**:
From Equation 1, we can express the mode as:
\[ Mo = M - 93 \]
Now, we can substitute this expression for the mode into Equation 2:
\[ M - 93 = 3 \times \text{Median} - 2 \times M \]

4. **Rearranging the Equation**:
Let's rearrange the equation:
\[ M - 93 + 2M = 3 \times \text{Median} \]
This simplifies to:
\[ 3M - 93 = 3 \times \text{Median} \]

5. **Dividing by 3**:
To isolate the median, we divide the entire equation by 3:
\[ M - 31 = \text{Median} \]

6. **Finding the Difference Between Mean and Median**:
Now, we can find the difference between the mean and the median:
\[ M - \text{Median} = M - (M - 31) = 31 \]

### Final Answer:
The difference between the mean and the median is **31**.