Median of Grouped Data

Median of grouped data is the median of the data that is continuous and in the form of frequency distribution. Median is the middlemost value of the given data that separates the lower half of the data from the higher half. While calculating the median of grouped data, the following things are involved:

• Median class
• Cumulative frequency
• Median of grouped data formula

Let us learn more on how to calculate the median of grouped data and its formula along with a few examples to understand this concept better.

Median of grouped data is the middle value of the data that is arranged in ascending order and written in a continuous manner. The data is in the form of a frequency distribution table that divides the higher level of data from the lower level of data. One of the simplest methods of finding the median of grouped data is by using the formula. As finding the middle value or median of a grouped data might be tough. Therefore, to find the median for grouped data we can use the following steps and formula:

• Step 1: Find the total number of observations.
• Step 2: Define the class size, and divide the data into different classes.
• Step 3: Calculate the cumulative frequency of each class.
• Step 4: Identify the class in which the median falls. (Median Class is the class where (n + 1)/2^th term lies.)
• Step 5: Find the lower limit of the median class(l), and the cumulative frequency of the median class (c).
• Step 6: Then apply the median formula median = l + (n/2 - c)/f × h.

Where,

• l = lower limit of median class
• n = total number of observations
• c = cumulative frequency of the preceding class
• f = frequency of median class
• h = class size (upper limit - lower limit)

Definition of Median

Median is the middlemost value in a given data set after it is arranged in ascending order. If the total number of items in the list is odd, then after arranging the values in ascending order the middlemost value is taken as the median i.e. the median is given by the [(n+1)/2]^th term, where 'n' is the total number of observations. If the number of items in the data set is even, then the average of the two middle values is taken, that is, [(n/2)^th term + ((n/2) + 1)^th term] / 2, where 'n' is the total number of observations.

For example: Let's consider the data: 48, 20, 50, 69, 73. What is the median?

Solution:

Arranging in ascending order, we get 20, 48, 50, 69, 73. Here, n (no.of observations) = 5

So, to find the median of odd data we use the formula:

[(n+1)/2] = (5 + 1)/2 = 6/2 = 3

Therefore, Median = 3^rd observation

Median = 50.

Finding the median of any given data is simple since the median is the middlemost value of the data. Since the data is grouped, it is divided into class intervals. Let us learn the steps to finding the median of grouped data.

• Step 1: Construct the frequency distribution table with class intervals and frequencies.
• Step 2: Calculate the cumulative frequency of the data by adding the preceding cumulative value of the frequency with the current value.
• Step 3: Find the value of n by adding the values in frequency (which is nothing but the last value of the cumulative frequency column)
• Step 4: Find the median class. If n is odd, the median is the (n+1)/2^th value. If n is even, then the median will be the average of the n/2^th and the (n/2 +1)^th observation.
• Step 5: Find the lower limit of the class interval and the cumulative frequency.
• Step 6: Apply the formula for median in statistics: Median = l + [(n/2−c)/f] × h

Let us look at an example to understand this better.

For Example: Calculate the median for the following data:

Solution:

We need to calculate the cumulative frequencies to find the median.

n = Last value of cf = 80, n/2 = 80/2 = 40

Since n is even, we will find the average of the n/2^th and the (n/2 +1)^th observation i.e. the cumulative frequency greater than 40 is 63 and the class is 40 - 60. Hence, the median class is 40 - 60.

l = 40, f = 37, c = 26, h = 20

Using Median formula:

Median = l + [(n/2−c)/f] × h

= 40 + [(80/2−26)/37] × 20

= 40 + [(40−26)/37] × 20

= 40 + [14/37] × 20

= 40 + 7.57

= 47.57

Therefore, the median is 47.57.

Before starting to use the median of grouped data formula, find the median class by using the above procedure. Then the formula to calculate the median of grouped data is l + [(n/2−c)/f] × h, where:

• l = lower limit of median class
• n = total number of observations
• c = cumulative frequency of the preceding (of median class) class
• f = frequency of median class
• h = size of each class

☛Related Topics:

• Mode
• Average
• Mean
• Geometric Mean

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How to Find the Median of Grouped Data?

To find the median of grouped data, the data should have been formed in a frequency distribution table in ascending order with the values being continuous. For grouped data, a frequency distribution is required to find the median and is done by using the formula. Median = l + [(n/2−c)/f] × h.

☛Also Check:

• Mean of Grouped Data
• Mode of Grouped Data

What is the Median Formula Class 10 of Grouped Data?

The formula for the median of grouped data depends on the observations, the class size, the frequency, and the cumulative frequency. The formula to calculate the median (in class 10) is l + [(n/2−c)/f] × h.

Where,

• l = lower limit of median class
• n = total number of observations
• c = cumulative frequency of the preceding class
• f = frequency of median class
• h = class size

How to Find the Median Class?

To find the median class, first find the total number of observations (n).

• If n is odd, then the class containing (n + 1)/2^th value is the median class.
• If n is even, then the class containing the average of (n/2)^th value and ((n/2)+1)^th values is the median class.

In general, the formula of median class 10 is the class containing (n + 1)/2^th value.

How to Find the Median From Frequency Table?

The steps to find the median from frequency table (of grouped data) is as follows:

• Find the median class by finding the class with (n + 1)/2^th term.
• Its lower limit is denoted by l and its frequency is denoted by f.
• Calculate the total frequency and denote it by n.
• Calculate the cumulative frequency of the previous class of median class and denote it by c.
• Calculate 'h' which is the size of any class interval.
• Substitute all these values in the median of grouped data formula l + [(n/2−c)/f] × h.

What is Meant By Median in Statistics?

The value of the middle-most observation obtained after arranging the data in ascending or descending order is called the median of the data. When describing a set of data, the central position of the data set is identified and used further in the median formula. This is known as the central measure of tendency. The median is an important measure of central tendency.

Why Median is Called Positional Average?

The median falls in the middle when the data is arranged in an increasing or decreasing order. Hence the median is referred to as positional average. The median is the exact middle value, in case of odd number of data points whereas for even number of data points, the median is the average of the two middle values.