Histograms and probability density functions

The distribution of values in a infinite categorical or discrete population can be displayed in the same way as a sample or finite population — with a bar chart. Finite samples of continuous numerical values are often displayed using histograms and these can also be used as graphical displays of infinite populations.

However we noted before that the exact shape of a sample histogram depends on the choice of classes that were used to draw it. Class width is usually reduced as much as possible to retain a fairly smooth histogram shape. For an infinite population, this reduction in class width can be taken to its extreme, resulting in a smooth histogram called a probability density function. This is often abbreviated to a pdf.

Probability density functions are still essentially histograms and share all properties of histograms.

The law of large numbers and histograms

Take a few samples to observe the variability in the shape of the histogram of samples of size 50.

Increase the sample size to 500, then 5000, and take more samples. As expected from the law of large numbers, the proportion in each class becomes less variable.

With the larger sample size, the classes can be made narrower without giving the histogram a jagged appearance. Make the classes Narrower until the histograms start to appear jagged.

Increase the sample size to 50,000 and note that the class width can be made still narrower.

With large samples, the shape of the histogram is approaching a smooth curve.

Finally change the sample size to Infinite and note that the histogram can now be made arbitrarily narrow, resulting in a smooth curve.

The limiting 'infinite sample' smooth histogram is the probability density function of the population.