How to Calculate Median for Continuous Data

Median of Grouped Data

Median of grouped data is the data that is continuous and is in the form of frequency distribution. Median is the middlemost value of the given data that separates the higher half of the data from the lower half. While calculating the median of grouped data, the middle value is not known as the data is divided into class intervals. Let us learn more on how to calculate the median of grouped data, the median formula, and solve a few examples to understand this concept better.

1.	What is Median of Grouped Data?
2.	Steps to Find Median of Grouped Data
3.	Median of Grouped Data Formula
4.	FAQs on Median of Grouped Data

What is Median of Grouped Data?

Median of a grouped data is data that is arranged in ascending order and is 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/2 lies.)
• Step 5: Find the lower limit of the median class(l), and the cumulative frequency of the median class (c).

Where,

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

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 the 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.

Steps to Find Median of Grouped Data

Median of grouped data is in the form of a frequency distribution arranged in ascending order and is continuous. Find 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 value of the frequency with the current value.
• Step 3: Find the value of n by adding the values in frequency.
• Step 4: Find the median class. If n is odd, the median is the (n+1/2). And 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 for grouped data: 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:

Marks	0 - 10	10 - 30	30 - 60	60 - 80	80 - 90
Number of students	6	20	37	10	7

Solution:

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

Marks	Number of students	Cumulative frequency
0 - 20	6	0 + 6	6
20 - 40	20	6 + 20	26
40 - 60	37	26 + 37	63
60 - 80	10	63 + 10	73
80 - 100	7	73 + 7	80

N = sum 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 + [(37 - 26)/40] × 20

= 40 + (11/40) × 20

= 40 + (220/40)

= 40 + 5.5

= 45.5

Therefore, the median is 45.5.

Median of Grouped Data Formula

The formula to calculate the median of grouped data is the same as mentioned above. The data needs to be continuous and in the form of a frequency distribution and the median is calculated through the following sequence of steps.

• Find the total number of observations(n).
• Define the class size(h), and divide the data into different classes.
• Calculate the cumulative frequency of each class.
• Identify the class in which the median falls. (Median Class is the class where n/2 lies.)
• Find the lower limit of the median class(l), and the cumulative frequency of the median class (c).

Hence the formula is, Median = l + [(n/2−c)/f] × h

Related Topics:

Listed below are a few interesting topics related to the median of grouped data, take a look.

• Mode
• Average
• Mean
• Geometric Mean

Examples on Median of Grouped Data

1. Example 1: A survey on the heights (in cm) of 50 girls of class X was conducted at a school and the following data was obtained:

   Height (in cm)	120-130	130-140	140-150	150-160	160-170	Total
   Number of girls	2	8	12	20	8	50

   Find the median of the above grouped data.

   Solution:

   To find the median, we need cumulative frequencies.

   Consider the table:

   Class Intervals	No. of girls (fi)	Cumulative frequency (c)
   120-130	2	2
   130-140	8	2 + 8 = 10
   140-150	12	10 + 12 = 22
   150-160	20	22 + 20 = 42
   160-170	8	42 + 8 = 50

   n = 50, n/2 = 25

   Median class = 150 - 160

   l = 150, c = 22, f = 20, h = 10

   Median = l + [(n/2−c)/f] × h = 150 + [((50/2) - 22)/20] × 10 = 150 + 1.5 = 151.5

   Therefore, the Median = 151.5.

2. Example 2: The following table gives the weekly expenditure of 200 families. Find the median of the weekly expenditure.

   Weekly Expenditure ($)	0-1000	1000-2000	2000-3000	3000-4000	4000-5000	Total
   Number of Families	34	12	43	60	51	200

   Find the median of the above-grouped data.

   Solution:

   To find the median, we need cumulative frequencies.

   Consider the table:

   Weekly Expenditure	No. of families (fi)	Cumulative frequency (c)
   0 - 1000	34	34
   1000 - 2000	12	34 + 12 = 46
   2000 - 3000	43	46 + 43 = 89
   3000 - 4000	60	89 + 60 = 149
   4000 - 5000	51	159 + 51 = 200

   n = 200, n/2 = 200/2 = 100

   Median Class = 3000 - 4000

   l = 3000, c = 89, f = 60, h = 1000

   Median = l + [(n/2−c)/f] × h = 3000 + [(200/2 - 89)/60] × 1000 = 3000 + 183 = 3183.

   Therefore, the median is 3183.

3. Example 3: Josie noted the number of cakes she baked every day over the past week. The numbers were: 1, 3, 3, 4, 4, 6, 2. What is the median value of the cakes she baked?

   Solution:

   To find the median value of the number of baked cakes, we first arrange the numbers of cakes baked per day in a sequence and then pick the middlemost value.
   The original set of number of cakes baked per day: 1, 3, 3, 4, 4, 6, 2.
   The Ordered Set of the cakes baked per day: 1, 2, 3, 3, 4, 4, 6
   Count the number of observations(n) = 7.
   Since the number of observations is odd, median = middle value i.e. 4th value. Thus, median = 3.
   Therefore, the median value of cakes she baked is 3.

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Practice Questions on Median of Grouped Data

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FAQs on Median of Grouped Data

What is the Median of Grouped Data?

Median of grouped data is the data formed in a frequency distribution table in ascending order with the values being continuous. Median is considered as the middle value of any data with values in an even or odd manner. For a 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.

What is the Formula for Median of Grouped Data?

The formula for median of grouped data depends on the observations, the class size, the frequency, and the cumulative frequency. The formula to calculate the median 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 each class
• h = class size

What are the Steps to Find Median of Grouped Data?

The steps to find the median of grouped data is as follows:

• Construct the frequency distribution table with class intervals and frequencies.
• Calculate the cumulative frequency of the data by adding the preceding value of the frequency with the current value.
• Find the value of n by adding the values in frequency.
• Find the median class.
• Find the lower limit of the class interval and the cumulative frequency.
• Apply the formula for median for grouped data: Median = 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. 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.

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