Frequency Distributions

1. Ungrouped
   1. Example - [Minitab]
   2. Rules
   3. Review of the Mathematics Involved
2. Grouped
   1. Example - [Minitab]
   2. Rules

Practice Problems (includes Graphing from the next section) (Answers)
Homework

A frequency distribution is a procedure for describing a set of data. There are two types: ungrouped and grouped.

I. Ungrouped (also called a "Tally")

1. Example - [Minitab]

   The following data represent the number of hours of TV viewing per week (X) for 20 people.
   
   

   7	5	4	7	4	6	6	6	5	4
   6	7	2	7	5	5	6	2	2	7

   What follows is an ungrouped frequency distribution for the above data.
   
   

   X	f	fr=p	%	Cf	Cp	C%
   7	5	(5/20=).25	25	N=20	p=1.00	%=100
   6	5	.25	25	15	.75	75
   5	4	.20	20	10	.50	50
   4	3	.15	15	6	.30	30
   3	0	.00	0	3	.15	15
   2	3	.15	15	3	.15	15
   åf=20=N		åp=1.00	å%=100

   How frequently the scores are distributed within the distribution is clearly shown. For example, you can quickly see that half of the people watched 6 or more hours of TV. It is shown in different ways (i.e., f, p, & %). Notice also the Cumulative frequency columns. The concept refers to the frequency of scores falling at or below the upper exact limit. Lastly, note that decimal usage within each column is consistent.

2. Rules
   1. Locate the extreme values (the lowest score or X_L and the highest score or X_H) and make a column that lists all the values from X_H to X_L (Note that X_H is at the top of the column and X_Lis at the bottom. This is a convention which will make some things easier later).
   2. Make an adjacent column labeled "frequency" (f). Count the frequency of each X and put it in the f column. This is called a tally of the scores. It is a good idea when doing this to put a slash through the number so you know it has been counted. (If you make a mistake, you can go back and use a slash in the opposite direction.) Check that the åf = N and put this in a row at the bottom of the column.
   3. Create another column called "relative frequency" or "proportion" (f_r = p = f/N). As a general rule, proportions should be expressed in hundredths. Check that the åf_r » 1 and show it at the bottom of the column.
   4. Create another column called "percent" (% = f_r * 100). As a general rule, percents should be expressed without decimal places (or sometimes with one). Show that å% » 100 at the bottom of the column.
   5. Create another column called "cumulative frequency."
      To do this, start at X_L and work your way up by adding all frequencies for the scores at or below the score you are interested in. Thus:
      
      For X_L, the Cf = f
      For X_L+1, the Cf = f of X_L + f of X_L+1, etc.
      The Cf of X_H = N.
   6. Create another column called "cumulative proportion" (Cp = Cf/N).
   7. Could create another column called "cumulative percent" (C% = Cp * 100).
   
   
3. Review of the Mathematics Involved
   1. f_r = p = f/N.
   2. % = f_r * 100.
   3. åp » 1.00 and å% » 100. "»" rather "=" is used because of rounding error.
   4. Cp = Cf/N (& the Cp for X_H » 1.00)
   5. C% = Cp * 100 (& the C% for X_H » 100).

Useful for large data sets and because they make the form or shape of the distribution more obvious. A disadvantage, though, is that the scores lose their individual identity.

1. Example - [Minitab]
   The following data are the expected scores on Exam 1 (N=25).
   
   

   95	88	81	79	73
   92	88	81	79	72
   92	86	81	77	67
   91	85	80	77	62
   89	84	80	74	61

   A simple tally or ungrouped frequency distribution would not be very helpful in this case. What follows is a grouped frequency distribution for this data.

   Interval (Stated or Apparent limits)	Real or Exact limits	Mid- point	f	p	%	Cf	Cp	C%
   95-99	94.5-99.5	97	1	.04	4	N=25	p=1	%=100
   90-94	89.5-94.5	92	3	.12	12	24	.96	96
   85-89	84.5-89.5	87	5	.20	20	21	.84	84
   80-84	79.5-84.5	82	6	.24	24	16	.64	64
   75-79	74.5-79.5	77	4	.16	16	10	.40	40
   70-74	69.5-74.5	72	3	.12	12	6	.24	24
   65-69	64.5-69.5	67	1	.04	4	3	.12	12
   60-64	59.5-64.5	62	2	.08	8	2	.08	8
   åf=25				åp=1	å%=100

   Note how easy it is to see the form or shape of the distribution. For example, it is easy to derive:

   Number Grade	Letter Grade	% of Class
   90s	A	16
   80s	B	44
   70s	C	28
   60s	D	12
   100

   However, using just the grouped frequency distribution we wouldn't be able to tell exactly what the highest score is (i.e., the individual scores have lost their identity).

2. Rules
   1. Compute the Range
      
      (Consider the scores 4, 5, & 6. R = 6 - 4 + 1 = 3 scores.)
   2. Use about 10 to 20 groups or intervals.
   3. The width of a group or interval (i) must be an odd number.
   4. Make the lowest apparent limit divisible by i. This interval should contain X_L.
   5. Use the formula:
      
      to help you determine an appropriate number of groups or intervals to group the data into. Note that the formula is a guide and is not engraved in stone. To work with the formula, start by using the lowest possible value of i (i.e., 3) and divide it into R.

   So, in our example R = X_H - X_L + 1 = 95 - 61 + 1 = 35.

   If i=3, then the # of groups = 11.67 or 12.
   If i=5, then the # of groups = 7.
   If i=7, then the # of groups = 5.
   Notes:
   
   1. if i=1, then we are dealing with an ungrouped frequency distribution.
      
   2. the # of groups are ALWAYS rounded up. Thus, this is one place (I believe the only place) where we do not follow the odd/even rounding rules.

   Since we have limited space on the screen, an i of 5 was chosen (& thus 7 groups). Note however, that since 61 is not divisible by i, a lowest stated limit of 60 was used which explains why the number of groups we actually used was 8 rather than 7.