Median- Measures of Central Tendency

Median: In statistics, central tendency serves as a concise dataset summary. It represents the midpoint of the data distribution using a single value, although it doesn't detail individual data points. Instead, it provides an overview of the entire dataset. To calculate the median, one must organize the values in ascending order, allowing for a clear understanding of the dataset's middle point.

Median Definition

In the field of mathematics, the median signifies the central value within a sorted numerical list. This middle number is determined by arranging the numbers in ascending order. Once the numbers are properly organized, the middle value in the list is identified as the median of the dataset.

Median Formula

The formula varies based on whether the given dataset has an odd or even number of observations. To determine it use the following formulas:

For Odd Number of Observations:

If the dataset contains an odd number of values, the median is the middle value when the data is arranged in order.

Median= (n+1/2) th term

For Even Number of Observations:

Median= ((n/2) th term + (n/2+1) th term)/2

When there is an even number of values, it is calculated by averaging the two middle numbers after sorting the data in numerical order.

In both cases, 'n' represents the total number of observations in the dataset.

Calculation of Median

To calculate it for different types of data sets, such as individual data, discrete data, or frequency distributions, specific methods are applied:

For Individual Data:

• First, sort the raw data in ascending or descending order.
• Determine the total number of observations, denoted as N.
• If N is an odd number, the median is simply the value at position (N+1)/2.
• If N is even, use the formula: Median = [(N/2)th term + ((N/2)+1)th term] / 2.

For Discrete Data:

• Start by sorting the data.
• Convert simple frequencies into cumulative frequencies, with the last value labeled as N (the sum of frequencies).
• Calculate (N+1)/2.
• The median is the value corresponding to the cumulative frequency just greater than (N+1)/2.

For Frequency Distributions:

• Arrange the classes in ascending or descending order.
• Calculate cumulative frequencies, with the last value labeled as N.
• Find N/2.
• Identify the "class," which is the class with a cumulative frequency just greater than N/2.
• Finally, use the formula: Median = L + (h/f) * [(N/2) - C], where:

L is the lower limit of the median class.

h is the class size.

f is the frequency of the median class.

N is the total sum of frequencies.

C is the cumulative frequency just before the median class.

Advantages of Using the Median

Simplicity in Calculation and Comprehension: It is easy to calculate and easy to grasp. In many cases, you can identify it with a quick visual inspection.

Immunity to Outliers: It is not influenced by extreme values, such as exceptionally large or small data points. It depends on the position of values in the dataset, rather than their actual values

Precise Definition: It provides a specific and well-defined value, making it a reliable measure of central tendency.

Suitability for Qualitative Data: It is especially useful when dealing with qualitative data, where ranking is more relevant than precise measurements.

Applicability to Open-Ended Distributions: Even if the extreme values are unknown, it can still be calculated as long as the number of items is known. This is in contrast to the mean.

Graphical Representation: It can be graphically represented using ogive curves, a capability not shared by the arithmetic mean.

Limitations of Using the Median

Data Arrangement Requirement: Calculating it necessitates arranging data in ascending or descending order, which can be time-consuming for a large dataset.

Limited Consideration of Observations: It is a positional average and does not take into account the actual values of the data. It also neglects extreme values.

Not Universally Representative: Since it doesn't rely on all observations, it may not be a good representative when there is significant variation in the data.