Biostatistics

114 CHAPTER 4 PROBABILITY DISTRIBUTIONS
To help us understand the nature of the distribution of a continuous random variable,
let us consider the data presented in Table 1.4.1 and Figure 2.3.2. In the table we have 189
values of the random variable, age. The histogram of Figure 2.3.2 was constructed by
locating specified points on a line representing the measurement of interest and erecting a
series of rectangles, whose widths were the distances between two specified points on the
line, and whose heights represented the number of values of the variable falling between
the two specified points. The intervals defined by any two consecutive specified points we
called class intervals. As was noted in Chapter 2, subareas of the histogram correspond to
the frequencies of occurrence of values of the variable between the horizontal scale
boundaries of these subareas. This provides a way whereby the relative frequency of
occurrence of values between any two specified points can be calculated: merely determine
the proportion of the histogram’s total area falling between the specified points. This can be
done more conveniently by consulting the relative frequency or cumulative relative
frequency columns of Table 2.3.2.
Imagine now the situation where the number of values of our random variable is very
large and the width of our class intervals is made very small. The resulting histogram could
look like that shown in Figure 4.5.1.
If we were to connect the midpoints of the cells of the histogram in Figure 4.5.1 to
form a frequency polygon, clearly we would have a much smoother figure than the
frequency polygon of Figure 2.3.4.
In general, as the number of observations, n, approaches infinity, and the width of the
class intervals approaches zero, the frequency polygon approaches a smooth curve such as
is shown in Figure 4.5.2. Such smooth curves are used to represent graphically the
distributions of continuous random variables. This has some important consequences when
we deal with probability distributions. First, the total area under the curve is equal to one, as
was true with the histogram, and the relative frequency of occurrence of values between
any two points on the x-axis is equal to the total area bounded by the curve, the x-axis,
and perpendicular lines erected at the two points on the x-axis. See Figure 4.5.3. The
f (x)
FIGURE 4.5.1 A histogram resulting from a large number of values
and small class intervals.
x