Computing Median for Grouped Data

This section explains the concept of median for both ungrouped and grouped data, including its significance and the methods of calculation.

In this section, we delve into how to find the median from both ungrouped and grouped data. The median is a positional average that divides a data set into two equal parts. This section highlights methods for calculating the median effectively, with relevant examples and formulas.

Computing Median for Grouped Data

The median serves as a positional average that denotes the middle value in a data series, effectively separating the data into two equal halves. It is represented using the symbol M. In this section, we explore how to compute the median for both ungrouped and grouped data:

Median for Ungrouped Data

When dealing with ungrouped scores, the first step involves arranging the data in either ascending or descending order. The median can then be found by locating the central observation. The formula to compute the median for ungrouped data is:

$M = \frac{(N + 1)}{2} \text{th item}$\

Where N is the total number of observations.

Example:

For mountain peak heights:

• Heights: 8,126 m, 8,611 m, 7,817 m, 8,172 m, 8,076 m, 8,848 m, 8,598 m.
• Sorted: 7,817; 8,076; 8,126; 8,172; 8,598; 8,611; 8,848.
• Median: 8,172 m (4th item).

Median for Grouped Data

For grouped data, the calculation involves determining the median class, defined by its cumulative frequency. The formula used in this context is:

$M = l + \frac{f \cdot \left( \frac{N}{2} - c \right)}{f}$\

• l = Lower limit of the median class.
• f = Frequency of the median class.
• N = Total frequency.
• c = Cumulative frequency of the class preceding the median class.

Example:

Given a frequency distribution:

• Class: 50-60, Frequency: 3
• 60-70, Frequency: 7
• 70-80, Frequency: 11
• 80-90, Frequency: 16
• 90-100, Frequency: 8
• 100-110, Frequency: 5

1. Compute cumulative frequencies.
2. Identify the median class.
3. Apply the formula.

Understanding the median is crucial in statistics as it offers valuable insight into data distribution, especially when compared to mean and mode.