To find the median of arrayed (grouped) data we need to follow the following steps:
Step I: Arrange the grouped data in ascending or descending order, and form a frequency table.
Step II: Prepare a cumulative-frequency table of the data.
Step III: Select the cumulative frequency that is just greater than \(\frac{N}{2}\), where N is the total number of observations (variates). Then find the median as follows.
The variate whose cumulative frequency is the selected cumulative frequency, is the median of the data.
If \(\frac{N}{2}\) is equal to the cumulative frequency of a variate then
median = mean of this variate and the variate just greater than it.
Solved Examples on Find the Median of Grouped Data /Arrayed Data:
1. Find the median of the following distribution.
Variate
2
5
6
8
10
Number of Students
3
2
5
4
2
Solution:
Here, the frequency distribution is given.
The cumulative-frequency table of the distribution is
Variates 2 5 6 8 10 |
Frequency 3 2 5 4 2 N = 16 |
Cumulative Frequency 3 5 10 14 16 |
Here, \(\frac{N}{2}\) = \(\frac{16}{2}\) = 8.
The cumulative frequency just greater than 8 is 10.
The variate whose cumulative frequency is 10 is 6.
Therefore, the median = 6.
2. Find the median of the arrayed data given below.
10, 11, 11, 12, 12, 12, 13, 14, 14, 15, 15, 15, 15, 16, 16, 17, 18, 19, 19, 20.
Solution:
Putting the data in a frequency table, we have the cumulative frequencies as below.
Here, the total frequency N = 20.
So, \(\frac{N}{2}\) = \(\frac{20}{2}\) = 10.
The cumulative frequency just greater than 10 is 13 and the corresponding variates is 15. So, the median = 15.
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