Empirical Formula: Median Exceeds Mean

We are given the relationship between the median and mode:

Median = Mode - 61.54

This can be rewritten as:

Mode = Median + 61.54

Applying the Empirical Formula

The empirical relationship between mean, median, and mode for moderately skewed distributions is:

$Mode = 3 \times Median - 2 \times Mean$

Calculating Median - Mean Difference

Substitute the expression for the mode from the given information into the empirical formula:

$Median + 61.54 = 3 \times Median - 2 \times Mean$

Now, rearrange the equation to find the difference between the median and the mean (Median - Mean):

1. Move the terms involving Mean to one side and others to the other side:

   $2 \times Mean = 3 \times Median - Median - 61.54$

2. Simplify the equation:

   $2 \times Mean = 2 \times Median - 61.54$

3. Isolate the difference (Median - Mean):

   $2 \times Median - 2 \times Mean = 61.54$

   $2 \times (Median - Mean) = 61.54$

4. Solve for (Median - Mean):

   $Median - Mean = \frac{61.54}{2}$

   $Median - Mean = 30.77$

Therefore, the median of the data exceeds its mean by 30.77.