Calculating the Mode Using the Empirical Relation

This question asks us to find the mode of a statistical distribution when we are given its mean and median. We are specifically instructed to use the empirical relation, which provides a useful approximation for moderately skewed distributions.

Understanding the Empirical Relation

The empirical relation connects the three main measures of central tendency: the mean, median, and mode. For distributions that are not heavily skewed (i.e., moderately skewed), the following approximate relationship holds:

\( \text{Mean} - \text{Mode} \approx 3 \times (\text{Mean} - \text{Median}) \)

We can rearrange this formula to solve for the mode:

\( \text{Mode} \approx \text{Mean} - 3 \times (\text{Mean} - \text{Median}) \)

Alternatively, another common form of the rearranged formula is:

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

We will use the second form for our calculation as it is often more direct.

Applying the Formula to the Given Data

From the question, we have the following information:

• Mean = 8.73
• Median = 11

Now, we substitute these values into the empirical relation formula for the mode:

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

Step-by-Step Calculation

1. Substitute values:

   \( \text{Mode} \approx 3 \times (11) - 2 \times (8.73) \)

2. Perform multiplication:

   \( \text{Mode} \approx 33 - (2 \times 8.73) \)

   \( 2 \times 8.73 = 17.46 \)

   So, the equation becomes:

   \( \text{Mode} \approx 33 - 17.46 \)

3. Perform subtraction:

   \( \text{Mode} \approx 15.54 \)

Conclusion

Using the empirical relation with the given mean of 8.73 and median of 11, the calculated mode of the distribution is approximately 15.54.