3.7 Measures of the Center of a Distribution 45* A continuous distribution has an infi nite mean if = xf ( x ) dx = ∞ , where f ( x ) is the probability
density function, and the integral is taken over all x where f ( x ) > 0.symmetric or close to symmetric, the mean and median are nearly
equal, and the mean has statistical properties that favor it over the
median. But for skewed distributions or data with one or more gross
outliers, the median is usually the better choice.
For discrete data, the mode is the most frequently occurring value.
Sometimes, there can be more than one mode. For continuous data, the
mode of the distribution is the highest peak of the probability density
function. If the density has two or more peaks of equal height that is
the highest, then these peaks are all modes. Sometimes, authors will be
less strict and refer to all the peaks of the probability density function
to be modes. Such distributions are called multimodal, and in the case
of two peaks, bimodal.
The normal distribution and other symmetric distributions (such as
Student ’ s t distribution) have one mode (called unimodal distributions),
and in that case, the mode = median = mean. So the choice of the
measure to use depends on a statistical property called effi ciency. There
are also symmetric distributions that do not have a fi nite mean. * The
Cauchy distribution is an example of a unimodal symmetric distribution
that does not have a fi nite mean. For the Cauchy, the median and mode
both exist and are equal.
Now let ’ s give a formal defi nition for the mode. The mode of a
sample is the most frequently occurring value. It will not be unique if
two or more values tie for the highest frequency of occurrence.
Probability distributions with one mode are called unimodal.
Distributions with two or more peaks are called multimodal. Strictly
speaking, a distribution only has two or more distinct modes if the
peaks have equal maximum height in the density (probability distribu-
tion for a continuous distribution) or probability mass function (name
for the frequency distribution for a discrete distribution). However,
when not strict, Figure 3.7 b is called bimodal even though the peaks
do not have the same height.
Figure 3.7 shows the distinction between a unimodal and a bimodal
density function.
Had the two peaks had the same height, then the bimodal distribu-
tion would have two distinct modes. As it is, it only has one mode. But
we still call it bimodal to distinguish it from the unimodal distribution.