326 The Basics of financial economeTrics

cumulative frequency distribution, this time using relative frequencies.
Hence, we have

=∑

Fxemfp() fi

i

k

1

In our example, Femfp($50) 18/30 0.6 60%===.
Note that the empirical cumulative frequency distribution can be evalu-
ated at any real x even though x need not be an observation. For any value x
between two successive observations x(i) and x(i+1), the empirical cumulative
frequency distribution as well as the empirical cumulative relative frequency
distribution remain at their respective levels at x(i); that is, they are of con-
stant level Fxempi()() and Fxemfp()()i , respectively. For example, consider the
empirical relative cumulative frequency distribution for the data shown in
Table A.1.
The computation of either form of empirical cumulative distribution
function is obviously not intuitive for categorical data unless we assign some
meaningless numerical proxy to each value such as “Sector A” = 1, “Sector
B” = 2, and so on.

continuous versus discrete Variables

When quantitative variables are such that the set of values—whether
observed or theoretically possible—includes intervals or the entire real num-
bers, then the variable is said to be a continuous variable. This is in contrast
to discrete variables, which assume values only from a finite or countable
set. Variables on a nominal scale cannot be considered in this context. And
because of the difficulties with interpreting the results, we will not attempt
to explain the issue of classes for rank data either.
When one counts the frequency of observed values of a continuous vari-
able, one notices that hardly any value occurs more than once.^3 Theoreti-
cally, with 100% chance, all observations will yield different values. Thus,
the method of counting the frequency of each value is not feasible. Instead,
the continuous set of values is divided into mutually exclusive intervals.
Then for each such interval, the number of values falling within that interval
can be counted again. In other words, one groups the data into classes for
which the frequencies can be computed. Classes should be such that their
respective lower and upper bounds are real numbers.

(^3) Naturally, the precision given by the number of digits rounded may result in higher
occurrences of certain values.