# Frequency distributions

## Discrete versus continuous data

With cartons of eggs, we can count the number of cartons which have one, two, or three (etc.) broken eggs; there cannot be a fractional number of broken eggs in a carton. Such data is called discrete -- every datum takes on one of a few specified values. When measuring height, any real number is a possible value; am I 6'2", 6'1.9375", or 6'1.93814" tall? It is really more appropriate to describe my height as being in the interval 6'1.5" - 6'2.5" than as equal to 6'2". Data sets which can take on any value in a continuum are called continuous. With discrete distribution you can have, e.g, exactly two eggs, but with continuous distributions nobody is , e.g., exactly two metres tall. Hence with discrete distributions strict versus weak inequalities are important, but with continuous distributions it does not matter whether inequalities are strict or weak.

## Relative frequency distributions

We have constructed histograms so that the height of each bar is the number of data in each class. We can rescale our axis so that the total area of the histogram is equal to one, in which case the area of a rectangle will be the proportion of the data set which is in the class. This is easily generalized for any curve with the total area under the curve equal to one: The proportion of a data set in an interval is equal to the area under the curve above that interval. In the following graph we can calculate that the relative frequency of data in the interval (-.5, .5) is .375 (the area of the yellow region) by using the area formula for rectangles.

If the graph is not rectilinear, more sophisticated techniques are necessary, but it can be shown that the area of the shaded region in the following graph is 0.43.