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.