MEASURES OF FREQUENCY

  I.  Frequency. 
        A.  Measures of frequency indicate how often a particular attribute of a 
            variable occurs in a set of observations.
	      1.  Common frequency measures include counts, proportions, percentages,
                  ratios, and rates.
              2.  Frequency measures are often presented in frequency tables in which
                  all the attributes of a variable are listed along with a count of 
                  the number of observations that have each attribute.
        B.  Frequency tables for categorical variables
              1.  Listing attributes.
                    a.  List of attributes must be exhaustive.
                    b.  List of attributes must be mutually exclusive.
              2.  Counting occurrences.
        C.  Frequency tables for numerical variables.
              1.  Frequency tables for numerical variables with a limited number of
                  attributes (<10).
                    a.  List each possible numerical value.
                    b.  List how often each value occurs.
              2.  Frequency tables for numerical variables with several attributes.
                    a.  When a numerical variable has several attibutes it may not be
                        practical to list them all.  Instead, class intervals should 
                        be created.  A class interval is a grouping of several attributes
                        into a single category or class.
                    b.  Constructing class intervals
                          i.  Intervals must be exhaustive and mutually exclusive.
                         ii.  Intervals should be constructed so that the data are 
                              summarized effectively.
                        iii.  Ideally, intervals should be equal in width, but 
                              there are exceptions.
                         iv.  One way of determining interval width is to take the 
                              desired number of intervals (10-12 max but typically 
                              3-5) and dividing it into the range plus 1. 
                          v.  Class intervals may also be created by dividing the 
                              distribution into roughly equal parts, say quarters.  
                              This requires producing frequencies for the full 
                              numerical data and using this information to create 
                              class intervals. 
                         vi.  Class intervals may also be created by using logical 
                              breaking points inherent in the data.
                    c.  Class intervals for continuous variables.
                          i.  Stated limits and real limits.
                         ii.  Mid-points.
  
 II.  Relative Frequency.
        A.  In addition to counts, frequency tables often present relative frequencies
            as well.
        B.  A relative frequency is a statistic that represents the proportion of the 
            total set of observations that fall within a given category.
        C.  There are several ways of expressing relative frequency.
              1.  Proportions.
              2.  Percentages.
              3.  Ratios.
              4.  Rates.
                                                    
III.  Cumulative frequencies and cumulative relative frequencies.
        A.  Frequency tables may also display cumulative frequencies and cumulative
            relative frequencies.
        B.  The cumulative frequency of a category is a count of all the observations
            that fall in that category plus all observations in the categories that 
            precede it.      
        B.  The cumulative relative frequency is the proportion of responses that fall 
            in a given category plus the proportion in all categories that precede it.
            Typically it is expressed as a percent.                                

 IV.  Graphic presentations of frequency distributions.
        A.  Pie charts of categorical data and discrete numerical data.
        B.  Bargraphs of categorical data and discrete numerical data.
        C.  Histograms of continuous numerical data.
        D.  Frequency polygons (line graphs) and curves of continuous numerical data.
        E.  The use of frequency polygons and curves with discrete numerical data.