```                              CENTRAL TENDENCY

I.  Measures of central tendency summarize a distribution by describing a typical
or average score.
A.  Measures of central tendency are efficient, describing an entire
distribution with a single value or score.
B.  In numerical distributions, measures of central tendency describe where
on a number line the distribution is located and can give some idea of
the general shape of the distribution.
II.  Measures of central tendency.
A.  The mode.
1.  The mode is simply the most commonly occurring value or score.
2.  Distributions may have more than one mode.
3.  The mode may be determined for variables measured at any level.
B.  The median.
1.  The median is the score or value in a distribution at which 1/2 of
the observations fall below it and 1/2 fall above it.  It is the
mid-point of a distribution.
2.  The median can be calculated for variables measured at the ordinal
or interval level.  However, unless an ordinal variable 7 or more
attributes, the median is usually not meaningful and is seldom
used.
3.  Determining the median.
a.  Order and number the observations in the distribution from
high to low.
b.  If the total number of observations in the distribution is
an odd number, add 1 to the total number of observations and
divide the result by 2.  The value of the observation with
that number is the median.
c.  If the total number of observations in the distribution is
an even number, divide the total number of observations by 2,
add the value of the observation with that number to the
value of the next observation then divide by 2.  The result
is the median.
d.  The median can be a fractional value even if all the
observations in the data set are whole numbers.
C.  The mean.
1.  The mean is the sum of all observations in a distribution divided
by the total number ofobservations.  It is the average score.
2.  Technically, the mean is only appropriate for continuous numerical
variables, however, it is frequently used with discrete numerical
variables as well.  In fact, it is sometimes calculated for ordinal
data, although this is not as widely accepted.
3.  The mean is calculated by adding all the observations in a data set
together, then dividing by the total number of observations.
4.  While the mean is usually considered the best measure of central
tendency for numerical variables, the median, and the mode, can be
informative.  In some cases, they may even be better measures than
the mean.
a.  Skewed distributions.
b.  Bimodal distributions.

III.  Reporting measures of central tendency.
A.  When reporting central tendency, generally, you report the measure that
uses as much of the available information as possible.  This means report
the mean for numerical variables, report the median for ordinal variables,
and report the mode for nominal variables.
B.  However, there are exceptions.
1.  If a numerical distribution is highly skewed, the median should be
reported in addition to or instead of the mean.
2.  If a numerical distribution is bimodal, the modes should be reported
in addtion to or instead of the mean.
3.  For variables with three or fewer attributes, report the mode
regardless of level of measurement.
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