Handout 40

Quantitative Statistics For Research

            Before any statistical measure can be computed, you must summarize the scores or results.  This is called a frequency distribution table.  This is done by listing the scores, in order from highest to the lowest in one column and the number of occurrences of that score in a second column.  Then the total number of occurrences in the frequency column are totaled, which is referred to as “n”.  For example, if 3 students got a 1.0, 4 students got a 1.25, 5 students got a 1.5, 3 students got a 1.75, 1 student got a 2.0, 6 students got a 2.25, 4 students got a 2.50, 1 student got a 2.75, 6 students got a 3.0, 7 students got a 3.5, and 2 students got a 5.0, our frequency distribution table would look something like this:

  1.                     Score        Frequency

  2.                     1.00              3

  3.                     1.25              4

  4.                     1.50              5

  5.                     1.75              3

  6.                     2.0                5

  7.                     2.25              6

  8.                     2.50              4

  9.                     2.75              4

  10.                     3.0                                6

  11.                     3.50              7

  12.                     5.0                2 .

  13.                                   n = 49

            This kind of information can be very useful, but there are several ways to display this data in a more useful manner.  We may choose to use a graph using an X-axis and a Y-axis.  This is called a frequency polygon.  To do this, (1) produce a frequency distribution table like the one above.  Group scores, if necessary, into intervals.  Then (2) place all the possible scores or groups on the horizontal axis at equal intervals, starting with the lowest on the left.  Now, (3) place the frequencies associated with each score on the vertical axis, at equal intervals, starting with zero at the bottom.  You will end up with something like this:

Y-axis

7                 *          

6                             *                                  *     

5                                                                             *                     * 

4                                         *          *                                                         * 

3                                                                                        *                                   * 

2      *

1                                                                         

      5.0     3.50     3.00     2.75     2.50     2.25     2.00     1.75     1.50     1.25     1.00

                                                     X-axis

Now you have a visual picture of the distribution table!

Averages are also called a measure of central tendency.  Averaging allow the researcher to summarize the information with a single number.  You average using one of three methods: the mode, the median, and the mean.

The MODE is the most frequent score in a distribution.  In our example, it is the score obtained by more students than any other score.  In our example, the MODE AVERAGE  SCORE would be 3.50

The MEDIAN is the middle point where 50% of the scores lie both above and bellow.  The median average is the physical midpoint between the highest and the lowest score, regardless of how many students received any particular score.  In our example, the MEDIAN AVERAGE SCORE is 2.25 because it is in the geographic middle with 5 scores higher and 5 scores lower.

The MEAN is the average of all the scores in a distribution.  To discover the mean average, add all the scores in the distribution and divide it by the total number of scores in the distribution.  This is much easier than it sounds.  Remember, n = 49 in the first example of the distribution table.

  1.                 Score        Frequency

  2.                 1.00       X       3   =   3

  3.                 1.25       X       4   =   5

  4.                 1.50       X       5   =   7.5

  5.                 1.75       X       3   =   5.25

  6.                 2.0         X       5   =   10

  7.                 2.25       X       6   =   13.5

  8.                 2.50       X       4   =   10

  9.                 2.75       X       4   =   11

  10.                 3.0         X       6   =   18

  11.                 3.50       X       7   =   24.5

  12.                 5.0         X       2   =   10                                   

  13.                               n = 49          117.75      117.75 divided by 49  =  2.40

In this group of students, the MEAN AVERAGE SCORE is 2.40 (approx ½ way between a grade of 2.25 and 2.50

A measure of central tendency is useful for summarizing information.  However, two distributions may have the same means and medians, but still be very different in terms of the VARIABILITY.  Consider the following distributions:

    A  =  19, 20, 25, 32, 39

    B  =  2, 3, 25, 30, 75

The mean of both A and B is 27, and the median is both 25.  But the spread differs considerably.  The scores in A are close together while the scores in B are more spread out.  Therefore, we must be able to measure the SPREAD, or VARIABILITY.

The RANGE represents the distance between the highest and lowest scores in the distribution.  This is very simple to compute:

the highest score minus the lowest score  =  the range.

Therefore, the range of A is 20  ( 39 – 19 = 20 )

           And the range of B is 73  (75 – 2 = 73)

REMEMBER, you should always include a frequency table in any statistical presentation as this is the basic display of statistical data.   

 Crossbreak Tables show a relationship between two categorical variables of interest.  Consider the following example: 

                   .       Administrators                         Teachers      .

                    Christian      Muslim              Christian      Muslim        Total

Male                50                20                       150              80              30

Female             20                10                       150            120            300 

Total                 70                 30                       300            200            600

            In this example, we are able to compare the relationship between three variables within a school district - the number of administrators vs. teachers, the number of Christians vs. Muslims, and the number of women vs. men all within the same statistical framework.  We can expand this more levels if we wish, giving us a detailed view of the statistical situation we are describing. 

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