Descriptive Statistics 327
Moreover, whether the class bounds are elements of the classes or not
must be specified. The class bounds of a class have to be bounds of the
respective adjacent classes as well, such that the classes seamlessly cover the
entire data. The width should be the same for all classes. However, if there
are areas where the data are very intensely dense in contrast to areas of
lesser density, then the class width can vary according to significant changes
in value density. In certain cases, most of the data are relatively evenly scat-
tered within some range while there are extreme values that are located in
isolated areas on either end of the data array. Then, it is sometimes advis-
able to specify no lower bound to the lowest class and no upper bound to
the uppermost class. Classes of this sort are called open classes. Moreover,
one should consider the precision of the data as they are given. If values are
rounded to the first decimal place but there is the chance that the exact value
might vary within half a decimal about the value given, class bounds have
to consider this lack of certainty by admitting half a decimal on either end
of the class.
cumulative Frequency distributions
In contrast to the empirical cumulative frequency distributions, in this sec-
tion we will introduce functions that convey basically the same information,
that is, the frequency distribution, but rely on a few more assumptions. These
cumulative frequency distributions introduced here, however, should not be
confused with the theoretical definitions given in probability theory in the next
appendix, even though one will clearly notice that the notion is akin to both.
The absolute cumulative frequency at each class bound states how many
observations have been counted up to this particular class bound. However,
we do not exactly know how the data are distributed within the classes. On
the other hand, when relative frequencies are used, the cumulative relative
frequency distribution states the overall proportion of all values up to a
certain lower or upper bound of some class.
So far, things are not much different from the definition of the empirical
cumulative frequency distribution and empirical cumulative relative fre-
quency distribution. At each bound, the empirical cumulative frequency
distribution and cumulative frequency coincide. However, an additional
assumption is made regarding the distribution of the values between bounds
of each class when computing the cumulative frequency distribution. The
data are thought of as being continuously distributed and equally spread
between the particular bounds. Hence, both forms of the cumulative fre-
quency distributions increase in a linear fashion between the two class
bounds. So for both forms of cumulative distribution functions, one can
compute the accumulated frequencies at values inside of classes.