1. Medians

2. Medians of Ungrouped Data
When a set of data is arranged in ascending / descending order, the datum in the middle is the median of the data.
For example,
The heights of these 5 girls are in ascending order.
140 cm
138 cm
The middle datum is 135 cm. Therefore, it is the median.
135 cm
130 cm
133 cm

3. In general, for a group of n data arranged in ascending / descending order:
If n is an odd number,
If n is an even number,

4. Can you find the median for each of the following sets of data?
(b) 0.3, 0.7, 1, 0.2, 0.5, 1.1
(b) 0.3, 0.7, 1, 0.2, 0.5, 1.1
(a) Arrange the data in ascending order:
1, 2, 4, 7, 9
Number of data = 5
Median
datum th 2 1 5
the ç è æ + =
datum
3rd
the = 4 =

5. Can you find the median for each of the following sets of data?
(b) 0.3, 0.7, 1, 0.2, 0.5, 1.1
(b) Arrange the data in ascending order:
0.2, 0.3, 0.5, 0.7, 1, 1.1
Number of data = 6
Median
datum th 2 6
the 1 ú û ù ê ë é ç è æ + =
datum)
4th
the
datum
3rd 2 1 + = ( 2
0.7
0.5 + =
0.6 =

6. Follow-up question Find the median of 8, 20, 4, 14, 8, 11, 12, 0.
Solution
Arrange the data in ascending order:
0, 4, 8, 8, 11, 12, 14, 20
Number of data = 8
datum th 1 2 8
the
Median ú û ù ê ë é ç è æ + =
datum
5th
the
4th 2 1 + = ) ( 2 11 8 + =
9.5 =

7. Medians of Grouped Data
We can find the median of a set of grouped data from a cumulative frequency polygon / curve by the following steps.
Step 1. Find the total frequency (n) of the grouped data.
Step 2. Find the value on the horizontal axis corresponding to the cumulative frequency of (i.e. n  50%). This value is the required median.

8. Medians of Grouped Data
We can find the median of a set of grouped data from a cumulative frequency polygon / curve by the following steps.
Step 1. Find the total frequency (n) of the grouped data.
Step 2. Find the value on the horizontal axis corresponding to the cumulative frequency of (i.e. n  50%). This value is the required median.
Cumulative frequency
Heights of a group of students 10 20
Height (cm) 30
144.5
149.5
154.5
159.5
164.5
139.5
169.5 40
The number of
students (i.e. the
total frequency)
= 40.

9. Medians of Grouped Data
We can find the median of a set of grouped data from a cumulative frequency polygon / curve by the following steps.
Step 1. Find the total frequency (n) of the grouped data.
Cumulative frequency
Heights of a group of students 10 20
Height (cm) 30
144.5
149.5
154.5
159.5
164.5
139.5
169.5 40
The number of
students (i.e. the
total frequency)
= 40.

10. Medians of Grouped Data
We can find the median of a set of grouped data from a cumulative frequency polygon / curve by the following steps.
Step 1. Find the total frequency (n) of the grouped data.
Step 2. Find the value on the horizontal axis corresponding to the cumulative frequency of (i.e. n  50%). This value is the required median.

11. Medians of Grouped Data
Step 2. Find the value on the horizontal axis corresponding to the cumulative frequency of (i.e. n  50%). This value is the required median.
Cumulative frequency
Heights of a group of students 10 20
Height (cm) 30
144.5
149.5
154.5
159.5
164.5
139.5
169.5 40 =
20) 2 40 (
median

12. From the figure, the median height of the students = 151.5 cm
Cumulative frequency
Heights of a group of students 10 20
Height (cm) 30
144.5
149.5
154.5
159.5
164.5
139.5
169.5 40 =
20) 2 40 (
median
From the figure, the median height of the students
= cm

13. Follow-up question
The following cumulative frequency polygon shows the test marks of 30 students.
Cumulative frequency
Test marks of 30 students 30 20 =
15) 2 30 ( 10 30 40 50 60 70 80 90
Marks
Find the median mark of the students.
Solution