Median

The median is the value of the middle item of a distribution once all the items have been arranged in order of magnitude. The middle item of an order of items is calculated as

^item.

Example 3.5:

(a) The median of the following nine values

8
6
9
12
15
6
3
20
11

is found by taking the middle (the fifth one) in the array.

3
6
6
8
9
11
12
15
20

The median is 9.

(b) Consider the following array

8
6
7
2
1
11
3
2
5

21
2
2
2
3
5
6
7
8

The median is 4 because, with an even number of items, we have to take the arithmetic mean of the two middle ones . . .

^(i.e. ^).

Finding the Median of an Ungrouped Frequency Distribution

The median of an ungrouped frequency distribution is found in a similar way. Consider the following distribution.

Value
x
Frequency
f
Cumulative Frequency
F
8
3
3
12
7
10
16
12
22
17
8
30
19
5
35

The median would be the . . .

^item.

The 18^th item has a value of 16 as we can see from the cumulative frequencies in the right hand column of the above table.

Finding a Median of a Grouped Frequency Distribution

We can establish the median of a grouped frequency distribution from an ogive.

Example 3.6: Construct an ogive of the following frequency distribution and hence establish the median.

Class
(MWK)
Frequency
f
Cumulative Frequency
F
340 - < 370
17
17
370- < 400
9
26
400 - < 430
9
25
430 - < 460
3
38
460 - < 490
2
40

The median is at . . .

^item.

Reading from the horizontal axis on the ogive the value of the median is approximately 380.

We can also establish the median of a grouped frequency by formula.

^{median =}

Where l = lower boundary of the median class, i = Class width of the median class n = Sample size, F = Cumulative frequency up to the median class, and f = Frequency of the median class.

Example 3.7: Find the median by formula for the following distribution:

Class Boundaries
f
F
95 – 14.5
4
4
14.5 – 19.5
6
10
19.5 – 24.5
8
18
24.5 – 29.5
4
22
29.5 – 34.5
3
25

Median class is 19.5 – 24.5