Calculating Mode using Empirical Relation

The problem asks for the mode of a data set given its mean and median.

Given Data:

• Mean ($ \bar{x} $) = 20
• Median (M) = 30

Empirical Relation:

For a moderately skewed distribution, the empirical relationship between mean, median, and mode is given by:

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

Or, using notation:

Mode $ = 3M - 2\bar{x} $

Calculation:

Substitute the given values into the formula:

Mode $ = (3 \times 30) - (2 \times 20) $

Mode $ = 90 - 40 $

Mode $ = 50 $

Result:

The mode of the data is 50.