Proportions from a histogram
In the histogram of a finite data set, the area above any class equals the proportion of values in the class.
Proportion and area
The diagram below shows the histogram of a population of 50 values. Each value is represented by a rectangle with the same area.
Drag with the mouse over some of the histogram classes to highlight them. The proportion of values in the selected classes equals the area above these classes.
Probabilities from a probability density function
Since the probability density function (pdf) describing a continuous distribution is a type of histogram, it satisfies the same property.
Probability and area under the pdf
The probability that the value of a continuous random variable, \(X\), is between two values, \(P(a \lt X \lt b)\), equals the area between \(a\) and \(b\) under the distribution's pdf.
If a histogram of the distribution is drawn with equal class widths, the probability for any class is the same as the class rectangle's proportion of the total histogram area (since the total of the class probabilities is one).
Histogram classes are mutually exclusive, so we can add the probabilities of adjacent classes to find probabilities for wider ranges of values. The total area for these adjacent classes is therefore also proportional to their probability.
This also holds when histogram class widths are reduced and, therefore, in the limit for the distribution's pdf.
The diagram below illustrates the result.
Probability and area
In the diagram below, again drag with the mouse over the diagram to highlight an interval of values. The probability of getting a value from the interval is equal to the area above that interval.