Descriptive Statistics 323
The set of data then form what is called time series data. In contrast, one
could pick a particular time period of interest such as the first quarter
of the current year and observe the dividend payments of all companies
comprising the Standard & Poor’s 500 index. By doing so, one would
obtain cross-sectional data of the universe of stocks in the S&P 500
index at that particular time.
Frequency distributions
One of the most important aspects of dealing with data is that they are effec-
tively organized and transformed in order to convey the essential informa-
tion contained in them. This processing of the original data helps to display
the inherent meaning in a way that is more accessible to intuition.
Relative Frequency Suppose that we are interested in a particular variable
that can assume a set of either finite or infinitely many values. These values
may be qualitative or quantitative in nature. In either case, the initial step
when obtaining a data sample for some variable is to sort the values of each
observation and then to determine the frequency distribution of the data
set. This is done simply by counting the number of observations for each
possible value of the variable. This is referred to as the absolute frequency.
Alternatively, if the variable can assume values on all or part of the real line,
the frequency can be determined by counting the number of observations
that fall into nonoverlapping intervals partitioning the real line.
In our illustration, we begin with qualitative data first and then move on
to the quantitative aspects. For example, suppose we want to compare the
industry composition of the component stocks in the Dow Jones Industrial
Average (DJIA), an index comprised of 30 U.S. stocks, the Dow Jones Global
total 50 Index (DJGTI), and the S&P 500. A problem arises because the num-
ber of stocks contained in the three indices is not the same. Hence, we can-
not compare the respective absolute frequencies. Instead, we have to resort
to something that creates comparability of the two data sets. This is done
by expressing the number of observations of a particular value as the propor-
tion of the total number of observations in a specific data set. That means we
have to compute the relative frequency.
Let’s denote the (absolute) frequency by a and, in particular, by ai for the
ith value of the variable. Formally, the relative frequency fi of the ith value
is then defined by
(^) f =a
n
i
i