 Three
Major Measures of Central
Tendency

 MODE
 is the most
frequent value in a data distribution
 The mode is suitable
for all types of data: NOMINAL through
RATIO
 In practice, the mode
is suitable only for variables with limited
values
 Visual display of
mode and bimodal distributions using smooth frequency
polygons
 MEDIAN
 is the value that
exactly divides an ordered frequency distribution
into equal halves
 Suitable for ORDINAL
variables and higher  NOT for NOMINAL
data
 Visual display of
data: in a symmetrical, unimodal distribution, the
mode and median are identical

 MEAN
 Commonly known as the
"average," the mean is the sum of all values divided by
the number of cases:
(This is your first
confrontation with higher mathematics.)
 In a symmetrical,
unimodal distribution, the mode, median, and mean
will be identical.

 In a unimodal
but skewed distribution, the median will
lie between the mode and the mean.
Skewed
distributions are defined by positions of their
TAILS

NEGATIVELY
skewed distributions have their tails to the
LEFT

POSITIVELY
skewed distributions have their tails to the
RIGHT



Negative
skew

Positive
skew

 Consider the example
of family income, which is a positively skewed
distribution
 A few very wealthy
families will skew the distribution to the right and
thus raise the mean,
 but a few very
wealthy people will have little effect on the
median.
 Thus, the median is a
preferred measure of central tendency for family
income.
But in general, the mean
is the most important measure of central tendency in
statistics  for a technical reason: the mean is the
number which has the smallest squared distance from all
other numbers in the distribution.
Let's study the means, medians, and modes you were asked to
compute for the assigned variables, by following the usual
procedure, going first to the Analyze Menu, then
choosing Frequencies, which produces this dialog box,
which shows that five variables were selected from list at
the left, moved to the right list, and that the "Display
frequency tables" box was NOT checkbecause these are
continuous variables and the frequency tables would have
little value.
:
The next step was
to click on the statistics box
above to go to this dialog box:
Checking the three
measures of Central TendencyMean, Median, and
Modeproduces this result:


% women population in 1989

% black population in 1990

% vote for G. W. Bush in
2000

% vote for Al Gore in 2000

% vote for Ralph Nader in
2000

N

Valid

51

51

51

51

51


Missing

0

0

0

0

0

Mean


48.07

10.636

49.697

46.04

3.039

Median


47.65

7.137

50.42

46.44

2.54

Mode


48.2

0.3^{a}

9^{a}

47.9

0

^{a }Multiple modes exist. The
smallest value is shown

How come the mean
vote for G.W. Bush, who did not win a plurality of the
popular vote, was higher than the mean vote for Al Gore,
who won the popular vote but lost the electoral
vote?
 This statistical
curiousity illustrates what's called the ecological
fallacy
  the danger of
attributing the result of data analysis at one level
to another level.
 We attributed the
result of a finding at the state level to the
national level.
Measures of central
tendency are useful, but statistics actually relies more
on the other type of summary statistics: measures of
dispersion (variation).
