1. Modes and Modal Classes

2. Modes The mode of a set of data is the datum with the highest
frequency.
(a) Find the mode of 12, 18, 13, 10, 12, 10, 10.
For example:
Since the datum ‘10’, which occurs 3 times, has the highest frequency among the set of data, the mode is 10.
(b) Find the mode of 1, 1, 2, 3, 4, 4, 5.
Since the data ‘1’ and ‘4’, both of which occur 2 times, have the highest frequency among the set of data, the modes are 1 and 4.

3. Follow-up question
Find the modes for the following sets of data. (a) 11, 21, 36, 98, 21, 21 (b) –1, –1, 3, 3, 8, 13, 17
(c) 16, 23, 25, 29, 31, 33
Solution
(a) Since the datum ‘21’, which occurs 3 times, has the highest frequency among the data, the mode is 21.
(b) Since the data ‘–1’ and ‘3’, both of which occur 2 times, have the highest frequency among the data, the modes are –1 and 3.

4. Follow-up question (cont’d)
Find the modes for the following sets of data. (a) 11, 21, 36, 98, 21, 21 (b) –1, –1, 3, 3, 8, 13, 17
(c) 16, 23, 25, 29, 31, 33
Solution
(c) Since all data have the same frequency, there are no modes in the set of data.
From (b), we see that a data set may have more than 1 mode.
From (c), we see that a data set may have no modes.

5. Modal Classes of Grouped Data
The following table shows the studying time of 40 students for a Mathematics test.
Time (hours)
1 – 2
3 – 4
5 – 6
7 – 8
Frequency 11 8 14 7
I do not know the frequency of each individual datum. How can I find the mode?
You can find the modal class, i.e. the class interval with the highest frequency.
According to the above table, 5 hours – 6 hours is the modal class.

6. Follow-up question
The following data show the ages of 20 readers in a library:
According to the above data, construct a frequency distribution table with class intervals 20 – 29, 30 – 39, etc. Then find the modal class of the ages of the 20 readers.

7. Follow-up question (cont’d)
Solution
Frequency distribution table of the ages of the 20 readers:
Age
Tally
Frequency
20 – 29 2
30 – 39
40 – 49 5
50 – 59
60 – 69 6
From the above table, the modal class of the ages is 60 – 69.