Explanation:

Index numbers are a statistical measure that represents changes in a variable or a group of variables over time. They are used to compare and analyze data in a standardized way. Index numbers are free from a unit of measurement because they focus on relative changes rather than absolute values.

Relative Changes:
The correct answer to the question is option 'C', which states that index numbers show relative changes. Relative changes refer to the percentage change in a variable or a group of variables over time. Index numbers compare the current value of a variable to a base period value and express the change as a percentage. This allows for easy comparison and analysis of data, regardless of the unit of measurement.

Average Changes:
Option 'A' states that index numbers show average changes. While index numbers can be used to calculate average changes, this is not their primary purpose. Average changes refer to the arithmetic mean of a set of values, which is not the focus of index numbers.

Variations:
Option 'B' states that index numbers show variations. Variations refer to the fluctuations or differences in the values of a variable over time. While index numbers can reflect variations, their main purpose is to show relative changes rather than variations.

Conclusion:
In conclusion, index numbers are free from a unit of measurement because they show relative changes. This allows for standardized comparison and analysis of data, regardless of the specific units used to measure the variable. Index numbers are a valuable tool in statistics and economics to track and analyze changes over time.