Median Formula: A Complete Guide to Calculating Central Tendency

Discover how to calculate the median using the median formula in this informative article. The median is a measure of central tendency that is commonly used in statistics to describe the middle value of a set of numbers. Learn how to find the median using step-by-step instructions, as well as its advantages and limitations as a statistical measure.

The median formula in statistics measures the central tendency of the data. It is defined as the middle value or the average of the two middle values (if the dataset has an even number of values). It is an essential tool for analyzing data types, such as income, height, test scores, etc. Median is preferred over the mean since it is less vulnerable to outliers and skewness. 

In this article, we will briefly discuss the median, how to calculate it, its properties, advantages and disadvantages, and finally, some practice problems to understand the concepts better.

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Now, let’s begin.

Table of Content

• Median
  • Formula
    • The number of Data Points is Even
    • The number of Data Points is Odd
• How to Find the Median of the Grouped Data
• Properties of Median

What is the Median?

If the list has an even number of data values, the median is the middle value or the average of two middle values. In simple terms, the median divides the group into two halves. Similar to the mean, it represents a large number of data points with a single value. 

To find the median of any dataset, the data points must be arranged in an order (i.e., ascending or descending).

Let’s take an example to find the median value.

Example: Consider the dataset: 42, 21, 34, 65, 90, 45, 109.

Find the median.

Solution

In the above dataset, there are 7 data points, so firstly, arrange the data points in ascending order (lowest to greatest), i.e., 

21, 34, 42, 45, 65, 90, 109.

Here, the 4the value will be the middle value.

Hence, the median is 45.

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Median Formula

To calculate the median, you have to follow simple steps:

• Arrange the data point in order (either ascending or descending)
• Count the total number of observation.

Case-1: If the number of data points is even

Median = [(n/2)^th term + ((n/2) + 1)^th term] / 2

Example: Consider the dataset that contains the test score of 8 students: {73, 80, 85, 88, 91, 92, 94, 97}.

Solution: 

There are 8 data points (i.e., n = 8) that are already arranged in ascending order.

Now, Applying the median formula, we get the following:

Median = [(8/2)th term + ((8/2)+1)th term] / 2

=> Median = [4th term + 5th term] / 2 = (88 + 91) / 2 = 179 / 2 = 89.5

=> Median = 89.5

Case-2: If the number of data points is odd

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

Example: Consider the dataset that contains the running times of 9 athletes in a race: {10.2, 10.1, 9.8, 10.5, 10.6, 9.9, 10.3, 10.4, 10.7}.

Solution

There are 9 data points (i.e., n = 9).

Firstly arrange these data points in ascending order, i.e., {9.8, 9.9, 10.1, 10.2, 10.3, 10.4, 10.5, 10.6, 10.7}

Now, applying the median formula, we get 

Median = [(9+1)/2]th term = 5th term

=> Median = 10.3

How to find the Median of the Grouped Data?

To find the median of the grouped data, you have to follow some simple steps:

• Find the cumulative frequency of the data.
  • Determine the total frequency of all groups.
• Divide the total frequency by 2 to find the data’s midpoint.
  • Find the group that contains the midpoint. 
• This will be the median group.

Calculate the Median using the following:

Median = L + ((n/2 – F) / f) x W

where,

L: Lower Limit of the median group

n: Total Frequency

F: Cumulative Frequency of the group before the median group

f: frequency of the median group

W: width of the median group

Now, let’s take an example to find the median for the grouped data.

Example: Consider the frequency table for the heights of students in a class:

Solution: 

Step-1: Find the cumulative frequency of the data

Step-2: Total Frequency = 35

Step-3: Midpoint = 35/2 = 17.5

Now, from the above table, we get that the mid-point lies in groups 150-155.

Step-4: Calculating the mean using the formula:

Median = L + ((n/2 – F) / f ) x w = 150 + ((35/2 – 8) / 12) x 5

=> Median = 150 + ((17.5 – 8) / 12 ) x 5 = 150 + 3.958 = 153.958

=> Median = 153.958

Properties of Median

• The median is a measure of central tendency, i.e., it indicates where the centre of the data is located.
• If the dataset has an odd number of values, the median is the middle value, whereas if the number of data points is even, the median is the average of two middle values. 
• The median is not affected by extreme values or outliers.
• Unlike the mean, the median doesn’t require the data to be normally distributed in order.
• Medians can be used with both categorical as well as numerical data.

Conclusion

In conclusion, the median is one of the important statistical tools that calculate the middle value of the dataset. Unlike the mean. it is not influenced by outliers and skewed data. However, it has its limitation as it may not always accurately reflect the overall characteristics of the dataset. In this article, we have briefly covered all these.
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