Understanding the Relationship Between Mode, Median, and Mean

In statistics, the mode, median, and mean are measures of central tendency. They represent different aspects of the center of a dataset. For a symmetrical distribution, the mode, median, and mean are all equal. However, for skewed distributions, they differ, and there's an empirical relationship connecting them.

The question provides information about the difference between the mode and the median and asks for the difference between the median and the mean. We can use an empirical formula that relates these three measures for moderately skewed distributions. This formula is sometimes known as Kelly's empirical formula or simply the empirical relationship.

Applying the Empirical Relationship Formula

The empirical formula relating mode, median, and mean is:

$\text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean}$

This formula is an approximation but is widely used in practice when dealing with moderately skewed data where the exact relationship is unknown or difficult to determine.

Calculating the Difference Between Median and Mean

We are given that the difference between the mode and the median is 36:

$\text{Mode} - \text{Median} = 36$

We need to find the value of $\text{Median} - \text{Mean}$.

Let's rearrange the empirical formula to involve the differences given in the problem. We can substitute the approximate value of Mode from the formula into the given equation:

$(3 \times \text{Median} - 2 \times \text{Mean}) - \text{Median} \approx 36$

Simplify the left side of the equation:

$(3 \times \text{Median} - \text{Median}) - 2 \times \text{Mean} \approx 36$

$2 \times \text{Median} - 2 \times \text{Mean} \approx 36$

Factor out 2 from the left side:

$2 \times (\text{Median} - \text{Mean}) \approx 36$

Now, solve for the difference between median and mean:

$\text{Median} - \text{Mean} \approx \frac{36}{2}$

$\text{Median} - \text{Mean} \approx 18$}

Summary of Calculation Steps

Here are the steps followed:

1. Recall the empirical relationship: $\text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean}$.
2. Identify the given information: $\text{Mode} - \text{Median} = 36$.
3. Substitute the empirical formula for Mode into the given equation: $(3 \times \text{Median} - 2 \times \text{Mean}) - \text{Median} \approx 36$.
4. Simplify the equation: $2 \times \text{Median} - 2 \times \text{Mean} \approx 36$.
5. Factor out 2: $2 \times (\text{Median} - \text{Mean}) \approx 36$.
6. Solve for $\text{Median} - \text{Mean}$: $\text{Median} - \text{Mean} \approx \frac{36}{2} = 18$.

Based on the empirical relationship, if the difference between mode and median is 36, then the difference between median and mean is approximately 18.

Checking the Options

The calculated difference between median and mean is 18. Let's compare this with the given options:

The calculated value of 18 matches Option 2.