6.4: Probability Histograms

A probability histogram is a visual representation of a probability distribution. Similar a typical histogram, the probability histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents. The vertical axis is labeled with probability. Each rectangular bar in the histogram is 1 unit wide, which suggests that the area under each bar equals the probability, P(x), where x is 1, 2, 3, and so on. The concept that the area is equal to the probabilities is useful in statistics. The histogram (like the stemplot) can give the shape of the data, the center, and the spread of the data.

Further, the mean, variance, and standard deviation can be calculated and visualized in the probability histogram. The mean is calculated using the equation:

The variance is calculated using the formula:

The standard deviation can be obtained by finding the square root of the variance.

This content is adapted from Openstax, Introductory Statistics, Section 2.2 Histograms

Probability histograms provide visual insights into the centering and spread of probability distributions that are difficult to comprehend in tabular form.

Consider a case of the number of seats occupied in a carpool. 

Based on daily observation, the probability that any number of seats between one to five is occupied is calculated. Here, the occupied seat count is the random variable.

Plotting a histogram with the number of seats occupied on the X-axis and corresponding probabilities on the Y-axis creates a probability histogram.

Each of these rectangular bars is one unit wide. This means that the area of each rectangle also represents the probability of each outcome.

To get more insights into the data, mean and variance can be calculated using the distribution table. 

To find the mean, multiply x with the corresponding probability, and add them up. 

Similarly, multiply the square of the x minus population mean by the probability, and add them up to obtain the variance.

The square root of the variance gives the standard deviation.