How to Find the Median of Ungrouped Data?

To find the median of ungrouped data, arrange the data in ascending or descending order and then determine the middle value.

Steps to Calculate the Median

Here's a detailed breakdown of how to find the median of ungrouped data:

1. Arrange the Data: Organize the data points either in ascending order (smallest to largest) or descending order (largest to smallest). The order doesn't affect the final median value.

2. Determine the Number of Observations (n): Count the total number of data points in your dataset. This value, denoted as 'n', is crucial for determining the median's position.

3. Calculate the Median: The method for calculating the median depends on whether 'n' is odd or even.

   • If 'n' is odd: The median is the value located at the (n + 1) / 2 position in the ordered data. For example, if you have 7 data points, the median is the (7 + 1) / 2 = 4th value.

   • If 'n' is even: The median is the average of the two middle values. These values are located at positions n / 2 and (n / 2) + 1. For example, if you have 8 data points, the median is the average of the 8 / 2 = 4th and (8 / 2) + 1 = 5th values.

Examples

Example 1: Odd Number of Observations

Data: 12, 3, 5, 8, 2, 7, 9

1. Arrange in ascending order: 2, 3, 5, 7, 8, 9, 12
2. Number of observations (n): 7 (odd)
3. Median position: (7 + 1) / 2 = 4
4. Median: 7 (the 4th value)

Example 2: Even Number of Observations

Data: 4, 1, 7, 3, 9, 6

1. Arrange in ascending order: 1, 3, 4, 6, 7, 9
2. Number of observations (n): 6 (even)
3. Median positions: 6 / 2 = 3 and (6 / 2) + 1 = 4
4. Median: (4 + 6) / 2 = 5 (average of the 3rd and 4th values)

In summary, finding the median involves ordering the data and then locating the middle value (or the average of the two middle values) based on whether the number of data points is odd or even.