Describing categorical and discrete populations
Since we have defined the probability of any value to be its proportion in the population from which we are sampling, graphical displays of these population proportions also describe the probabilites.
Bar charts were used earlier to describe the distribution of values in finite data sets, but a bar chart whose vertical axis is labeled with proportions (not counts) can be used in the same way to describe an infinite population.
Bar charts and the law of large numbers
An alternative interpretation of these bar charts comes from the law of large numbers. If we imagine repeating the data collection to increase the sample size indefinitely, the law of large numbers states that the sample proportions in the different categories will eventually stabilise at the underlying population proportions (probabilities). The sample bar chart will therefore stabilise at the above bar chart of the probabilities.
The diagram below shows the bar chart of a random sample of 20 values from a discrete infinite population.
Take a few samples to observe the variability in the shape of the bar chart.
Now increase the sample size to 200 and take a few more samples. The shape of the bar chart becomes more stable. As the sample size is increased further, the bar chart becomes less variable and our description of the infinite population is the limiting bar chart (describing an infinite sample from the population).
The 'infinite-sample' barchart gives probabilities that describe the population distribution.