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To calculate the median of grouped data, you must first arrange the data in ascending order, then locate the middle value. If there are two middle values, you must take the mean (average) of these two values to obtain the median. If there are an even number of values, the median will be the mean of the two middle values.
Often we may want to calculate the median of data that is grouped in some way.
Recall that the represents the value that lies directly in the middle of a dataset, when all of the values are arranged from smallest to largest.
For example, suppose we have the following grouped data:
While it’s not possible to calculate the exact median since we don’t know the , it is possible to estimate the median using the following formula:
Median of Grouped Data = L + W[(N/2 – C) / F]
where:
- L: Lower limit of median class
- W: Width of median class
- N: Total Frequency
- C: Cumulative frequency up to median class
- F: Frequency of median class
Note: The median class is the class that contains the value located at N/2. In the example above, there are N = 23 total values. Thus, the median value is the one in position 23/2 = 11.5, which would be located in the class 21-30.
The following examples show how to calculate the median of grouped data in different scenarios.
Example 1: Calculate the Median of Grouped Data
Suppose we have the following frequency distribution that shows the exam scored receive by 40 students in a certain class:
In this example, there are N = 40 total values. Thus, the median value lies in the class where 40/2 = 20 is located. The 20th largest value would be located in the 71-80 class.
Knowing this, we can calculate the following values:
- L: Lower limit of median class: 71
- W: Width of median class: 9
- N: Total Frequency: 40
- C: Cumulative frequency up to median class: 12
- F: Frequency of median class: 15
We can plug these values into the formula to calculate the median of the distribution:
- Median = L + W[(N/2 – C) / F]
- Median = 71 + 9[(40/2 – 12) / 15]
- Median = 75.8
We estimate that the median exam score is 75.8.
Example 2: Calculate the Median of Grouped Data
Suppose we have the following frequency distribution that shows the number of points scored per game by 60 basketball players:
In this example, there are N = 60 total values. Thus, the median value lies in the class where 60/2 = 30 is located. The 30th largest value would be located in the 11-20 class.
Knowing this, we can calculate the following values:
- L: Lower limit of median class: 11
- W: Width of median class: 9
- N: Total Frequency: 60
- C: Cumulative frequency up to median class: 8
- F: Frequency of median class: 25
We can plug these values into the formula to calculate the median of the distribution:
- Median = L + W[(N/2 – C) / F]
- Median = 11 + 9[(60/2 – 8) / 25]
- Median = 18.92
We estimate that the median exam score is 18.92.
The following tutorials explain how to perform other common operations with grouped data: