Discrete Frequency Distribution Calculator

In statistical analysis and machine learning, understanding the frequency distribution of observed data is a foundational skill. When working with discrete variables—variables that take on distinct, countable values—we often need to compute the relative frequency of each unique value, which provides an empirical estimate of the underlying probability distribution.

Given a collection of integer observations drawn from a discrete source, your task is to compute the discrete frequency distribution. For each unique value in the dataset, calculate its relative frequency—the proportion of times that value appears relative to the total number of observations.

Mathematical Formulation:

For a dataset containing N total observations, if a particular value x appears count(x) times, its relative frequency is:

$$\text{relative_frequency}(x) = \frac{\text{count}(x)}{N}$$

The sum of all relative frequencies will always equal 1.0, representing the complete distribution of the data.

Output Requirements:

• Return a list of tuples where each tuple contains (value, relative_frequency)
• The tuples must be sorted in ascending order by value
• If the input list is empty, return an empty list

Why This Matters: Understanding frequency distributions is essential in machine learning for tasks such as:

• Analyzing feature distributions during exploratory data analysis
• Computing prior probabilities in Bayesian classifiers
• Detecting class imbalances in classification datasets
• Building histograms and density estimates for visualization