Grouped Frequency Tables
Grouped Frequency Tables
Another way of arranging data is by grouping the observations into intervals and tabulating the frequencies for each interval. The result is called a grouped frequency table or grouped frequency distribution. In this distribution the intervals are called classes.
STEPS IN CONSTRUCTING A GROUPED FREQUENCY DISTRIBUTION
1. Determine the highest score H and the lowest score L. Get the range d which is the difference between the two scores
d = H – L
2. Estimate the class width w by dividing the range by a number of groupings or classes. If the quotient is not integer, round off the value to the nearest integer to obtain the class width. It is convenient if w is an odd number. (Note: The numbers 7 and 10 are chosen so that the number of intervals will neither be too few nor too many.)
3. Set up the class intervals. Start with either the lowest score or a convenient value slightly less than the lowest score. Then add the class width to the starting point to get the next interval. Do this until the highest score is contained in the last interval.
4. Tally the corresponding number of scores in each interval. Then summarize the results or sum up the tallies under the frequency column.
EXAMPLE: Construct a grouped frequency distribution for the data in these table
35 29 26
33 34 44
37 38 40
48 36 26
41 42 43
32 36 36
15 39 35
40 34 36
Solution:
Steps:
1. H = 48, L = 15; hence the range is d = 48 – 15 = 23
2. W=33/7 = 4.7, which can be rounded off to 5 (w = 5); an odd numbered w is convenient because the midpoint of the interval is an integer.
3. Starting with 15 and with w = 5, the classes are: 15-19, 20-24, 25-29, 30-34, 35-39, 40-44, 45-49.
4. Table 2.4 presents the grouped frequency distribution.
TABLE 2.4
GROUPED FREQUENCY DISTRIBUTION OF STUDENTS’ SCORES ON A MATH TEST
CLASS
CLASS INTERVALS
TALLY
FREQUENCY
1
2
3
4
5
6
7
15-19
20-24
25-29
30-34
35-39
40-44
45-49
I
_
III
IIII
IIIII-IIII
IIIII-I
I
1
0
3
4
9
6
1
The group frequency distribution gives more information about the gathered data. For example, the greatest number of frequencies is found in the fifth interval (35-39) and more than half of the class got scores between 30 and 39. Note, however, that some information can no longer be specified in a frequency distribution. For instance, it cannot be determine anymore how many students got a score of 35.
In a grouped frequency distribution, each interval is called a class or a class interval. From table
2.4, there are 7 classes. For each class, the lower class limit is the lowest number that can actually belong to the class. The upper class limit is the highest number indicated in each class. For example, in the second class, the lower class limit is 20 and the upper class limit is 24. The class boundaries
are the actual endpoints of the intervals. It is one-half the sum of the upper limit of one class and the lower limit of the class that immediately comes after it. Thus for the fourth interval, the lower class boundary is 29 + 30 = 59 = 29.5 while the upper class boundary is 34 + 35 = 39 = 34.5
2 2 2 2 the upper class boundary of a class is the lower class boundary of the next higher class. Class boundaries are useful in computing percentile ranks of a frequency distribution.
The class mark is the midpoint of each class. It is obtained by adding the class limits and dividing the sum by 2. Thus, for the third class, the class mark is 25 + 29 = 54 = 27
2 2 For the rest of the data, the class mark are 17, 22, 27, 32, 37, 42, and 47. Class marks are useful for computing the mean of a frequency distribution. The class boundaries and the class marks are given in table 2.5
TABLE 2.5
CLASS
CLASS INTERVALS
CLASS BOUNDARIES
CLASS MARKS
FREQUENCY
1
2
3
4
5
6
7
15-19
20-24
25-29
30-34
35-39
40-44
45-49
14.5-19.5
19.5-24.5
24.5-29.5
29.5-34.5
34.5-39.5
39.5-44.5
44.5-49.5
17
22
27
32
37
42
47
1
0
3
4
9
6
1
The above given values are useful when you want to make a visual presentation of the data.
Guidelines in constructing a grouped frequency distribution:
1. Each item in the data set should fall into only one class.
2. Include all the classes even if a class has a zero frequency.
3. Select convenient numbers for class width for all classes.
4. Use the same class width for all classes.
5. Try to use between 5 to 20 classes.
Critical Thinking Question: Imagine that you are skeptical about the idea an energy vortex and wanted to conduct a study to determine whether specific geographical locations caused these psychological reactions. You would need to have some participants stand on a reported vortex (experimental group) and others stand somewhere else (the control group)... what would you need do to with the control group in order to address the possibility of a placebo effect in the study?