If the median of a data is 61.54 less than its mode, then the median of the data exceeds its mean by _____. (Use the empirical formula to find the answer)

Correct Answer: The correct option is 30.77.

Step-by-step Explanation:

We are given the relationship between the mode and the median of a statistical dataset, and we need to find how much the median exceeds the mean using the empirical relationship.

Step 1: Write down the given information.

The median is 61.54 less than the mode. This can be expressed as:

$\text{Mode}-\text{Median}=61.54$

Or equivalently:

$\text{Mode}=\text{Median}+61.54$

Step 2: Recall the Empirical Formula.

The empirical relationship between Mean, Median, and Mode is given by:

$\text{Mode}=3\left(\text{Median}\right)-2\left(\text{Mean}\right)$

Step 3: Substitute the expression for Mode into the empirical formula.

Replacing $\text{Mode}$ with $\text{Median}+61.54$ in the formula:

$\text{Median}+61.54=3\left(\text{Median}\right)-2\left(\text{Mean}\right)$

Step 4: Rearrange the terms to solve for (Median - Mean).

Move $2\left(\text{Mean}\right)$ to the left side and $\text{Median}$ to the right side:

$2\left(\text{Mean}\right)+61.54=3\left(\text{Median}\right)-\text{Median}$

$2\left(\text{Mean}\right)+61.54=2\left(\text{Median}\right)$

Subtract $2\left(\text{Mean}\right)$ from both sides:

$2\left(\text{Median}\right)-2\left(\text{Mean}\right)=61.54$

Factor out 2:

$2(\text{Median}-\text{Mean})=61.54$

Divide both sides by 2:

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

$\text{Median}-\text{Mean}=30.77$

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