The Median
l When a data
set is ordered, it is called a data array.
l The median is
defined to be the midpoint of the data array.
l The symbol
used to denote the median is MD.
The
Median - Example
The weights (in pounds) of seven army
recruits are 180, 201, 220, 191, 219, 209, and 186. Find the median.
l Arrange the
data in order and select the middle point.
l Data array:
180, 186, 191, 201, 209, 219, 220.
l The median, MD
= 201.
l In the
previous example, there were an odd number of values in the data set. In this case it is easy to select the middle
number in the data array.
l When there is
an even number of values in the data set, the median is obtained by taking the
average of the two middle numbers.
Example
l Six customers
purchased the following number of magazines: 1, 7, 3, 2, 3, 4. Find the median.
l Arrange the
data in order and compute the middle point.
l Data array: 1,
2, 3, 3, 4, 7.
l The median, MD
= (3 + 3)/2 = 3.
Example
l The ages of 10
college students are: 18, 24, 20, 35, 19, 23, 26, 23, 19, 20. Find the median.
l Arrange the
data in order and compute the middle point.
Example
l Data array:
18, 19, 19, 20, 20, 23, 23, 24, 26, 35.
l The median, MD
= (20 + 23)/2 = 21.5.
The Median-Ungrouped
Frequency Distribution
l For an
ungrouped frequency distribution, find the median by examining the cumulative
frequencies to locate the middle value.
l If n is
the sample size, compute n/2.
Locate the data point where n/2 values fall below and n/2
values fall above.
l Sony Appliance
recorded the number of VCRs sold per week over a one-year period. The data is given below.
|
Net Set Sold |
Frequency |
|
1 |
4 |
|
2 |
9 |
|
3 |
6 |
|
4 |
2 |
|
5 |
3 |
l To locate the
middle point, divide n by 2; 24/2 = 12.
l Locate the
point where 12 values would fall below and 12 values will fall above
l Consider the
cumulative distribution.
l The 12th and
13th values fall in class 2. Hence MD =
2.
Net Set Sold
|
Frequency
|
Cumulative Frequency
|
1
|
4
|
|
2
|
9
|
13
|
3
|
6
|
19
|
4
|
2
|
21
|
5
|
3
|
24
|
This class contains the 5th through
the 13th values.
|
This class contains the 5th through
the 13th values. |
The Median for a
Grouped Frequency Distribution
The Median Can be computed as
Median = l + h/f ( N/2
– C )
l = lower boundary
h = width of median group
f = frequency
N = ∑f
C = Cumulative frequency preceding median group.
Example
Given
the table below find the median
|
Class |
Frequency |
|
15.5-20.5 |
3 |
|
20.5-25.5 |
5 |
|
25.5-30.5 |
4 |
|
30.5-35.5 |
3 |
|
35.5-40.5 |
2 |
Table with Cumulative Frequency
|
Class |
Frequency |
Cumulative
Frequency |
|
15.5-20.5 |
3 |
3 |
|
20.5-25.5 |
5 |
8 |
|
25.5-30.5 |
4 |
12 |
|
30.5-35.5 |
3 |
15 |
|
35.5-40.5 |
2 |
17 |
l To locate the
halfway point, divide n by 2; 17/2 = 8.5 approximately
equals to 9.
l Find the class
that contains the 9th value. This will
be the median class.
l Consider the
cumulative distribution.
l The median
class will then be 25.5 – 30.5.
N = 17 , c = 8
h = 5 , l = 25.5
Median = l + h/f ( N/2
– C )
Median =
Median = 26.125