Median

Median is the middle value of any data when arranged in ascending or descending order. Suppose we have the height of 5 friends as, 171 cm, 174 cm, 167 cm, 169 cm, and 179 cm, then the median height of the friends is calculated as, first arranging the data in ascending order, 167 cm, 169 cm, 171 cm, 174 cm, 179 cm. Now clearly observing the data we see that 171 cm is the middle term in the given data thus, we can say that the median height of the friends is, 171 cm.

A Median is a middle value for sorted data. The sorting of the data can be done either in ascending order or descending order. A median divides the data into two equal halves. Median is among one of the three measures of the central tendency and finding the median gives us very useful insight into the given set of data. In this article, we will learn about the median, its formula for grouped and ungrouped data, examples, and others in detail.

Median Definition

Median is defined as the middle term of the given set of data if the data is arranged either in ascending or descending order. Suppose we are given the weight of three girls of a class as 49 kg, 62 kg, and 56 kg then the median weight of them is calculated by first arranging the data in any order, let’s arrange data in ascending order as 49 kg, 56 kg, and 62 kg then by observing we can say that, 56 kg is the middle term in the given data set. So the median of the data set is 56 kg. Median is one of the three measures of the central tendency. The three measures of the central tendency are,

• Mean
• Median
• Mode

In this article, we will only study about Median. Read more on Mean and Mode.

Median Example

Various examples of the median are:

• Median salary of five friends, where the individual salary of each friend is, 74,000, 82,000, 75,000, 96,000, and 88,000. First arranged in ascending order 74,000, 75,000, 82,000, 88,000, and 96,000 then by observing the data we get the median salary as 82,000.
• Median Age of a Group: Consider a group of people ages 25, 30, 27, 22, 35, and 40. First, arrange the ages in ascending order: 22, 25, 27, 30, 35, 40. The median age is the middle value, which is 30 in this case.
• Median Test Scores: In a class, the test scores of 10 students are 78, 85, 90, 72, 91, 68, 80, 95, 87, and 81. Arrange them in ascending order: 68, 72, 78, 80, 81, 85, 87, 90, 91, and 95. Since there are an even number of scores, the median is the average of the two middle values, which are 81 and 85. The median test score is (81 + 85) / 2 = 83.

Median Formula

As we know median is the middle term of any data, and finding the middle term when the data is linearly arranged is very easy, the method of calculating the median varies when the given number of data is even or odd, for example, if we have 3(odd-numbered) data 1, 2, and 3 then 2 is the middle term as it has one number to its left and one number to its right. So finding the middle term is quite simple, but when we are given with even number of data(say 4 data sets), 1, 2, 3, and 4, then finding the median is quite tricky as by observing we can see that there is no single middle term then for finding the median we use a different concept.

Here, we will learn about the median of grouped and ungrouped data in detail.

Median of Ungrouped Data

Median formula is calculated by two methods,

• Median Formula (when n is Odd)
• Median Formula (when n is Even)

Now let’s learn about these formulas in detail.

Median Formula (When n is Odd)

If the number of values (n value) in the data set is odd then the formula to calculate the median is,

Median Formula (When n is Even)

If the number of values (n value) in the data set is even then the formula to calculate the median is:

Median of Grouped Data

Grouped data is the data where the class interval frequency and cumulative frequency of the data are given. The median of the grouped data median is calculated using the formula,

Median = l + [(n/2 – cf) / f]×h

Where,

• l is Lower Limit of Median Class
• n is Number of Observations
• f is Frequency of Median Class
• h is Class Size
• cf is Cumulative Frequency of Class Preceding Median Class

We can understand the use of the formula by studying the example discussed below,

Example: Find the Median of the following data,

If the marks scored by the students in a class test out of 50 are,

Marks	0-10	10-20	20-30	30-40	40-50
Number of Students	5	8	6	6	5

Solution:

For finding the Median we have to build a table with cumulative frequency as,

Marks	0-10	10-20	20-30	30-40	40-50
Number of Students	5	8	6	6	5
Cumulative Frequency	0+5 = 5	5+8 = 13	13+6 = 19	19+6 = 25	25+5 = 30

n = ∑f_i = 5+8+6+6+5 = 30(even)

n/2 = 30/2 = 15

Median Class = 20-30

Now using the formula,

Median = l + [(n/2 – cf) / f]×h

where

• l is Lower Limit of Median Class
• n is Number of Observations
• f is Frequency of Median Class
• h is Class Size
• cf is Cumulative Frequency of Class Preceding Median Class

Comparing with the given data we get,

• l = 20
• n = 30
• f = 6
• h = 10
• cf = 13

Median = 20 + [(15 – 10)/6]×10

= 20 + 5/3

= 60/3 + 5/3

= 65/3 = 21.67 (approx)

Thus, the median mark of the class test is 21.67

How to Find Median?

To find the median of the data we can use the steps discussed below,

Step 1: Arrange the given data in ascending or descending order.

Step 2: Count the number of data values(n)

Step 3: Use the formula to find the median if n is even, or the median formula when n is odd, accordingly based on the value of n from step 2.

Step 4: Simplify to get the required median.

Study the following example to get an idea about the steps used.

Example: Find the median of given data set 30, 40, 10, 20, and 50

Solution:

Median of the data 30, 40, 10, 20, and 50 is,

Step 1: Order the given data in ascending order as:

10, 20, 30, 40, 50

Step 2: Check if n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value.

Step 3: Here, n = 5 (odd)

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

Median = [(5 + 1)/2]^th term = 3^3r term

            = 30

Thus, the median is 30.

Application of Median Formula

Median formula has various applications, this can be understood with the following example, in a cricket match the scores of the five batsmen A, B C, D, and E are 29, 78, 11, 98, and 65 then the median run of the five batsmen is,

First arrange the run in ascending order as, 11, 29, 65, 78, and 98. Now by observing we can clearly see that the middle term is 65. thus the median run score is 65.

Median of Two Numbers

For two numbers finding the middle term is a bit tricky as for two numbers there is no middle term, so we find the median as we find the mean by adding them and then dividing it by two. Thus, we can say that the median of the two numbers is the same as the mean of the two numbers. Thus, the median of the two numbers a and b is,

Median = (a + b)/2

Now let’s understand this using an example, find the median of the following 23 and 27

Solution:

Median = (23 + 27)/2 = 50/2 = 25

Thus, the median of 23 and 27 is 25.

Read More,

• Mean, Median, and Mode

Solved Examples on Median

Example 1: Find the median of the given data set 60, 70, 10, 30, and 50

Solution:

Median of the data 60, 70, 10, 30, and 50 is,

Step 1: Order the given data in ascending order as:

10, 30, 50, 60, 70

Step 2: Check if n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value.

Step 3: Here, n = 5 (odd)

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

Median = [(5 + 1)/2]^th term = 3^rd term

            = 50

Example 2: Find the median of the given data set 13, 47, 19, 25, 75, 66, and 50

Solution:

Median of the data 13, 47, 19, 25, 75, 66, and 50 is,

Step 1: Order the given data in ascending order as:

13, 19, 25, 47, 50, 66, 75

Step 2: Check if n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value.

Step 3: Here, n = 7 (odd)

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

Median = [(7 + 1)/2]^th term = 4^th term

            = 47

Example 3: Find the Median of the following data,

If the marks scored by the students in a class test out of 100 are,

Marks	0-20	20-40	40-60	60-80	80-100
Number of Students	5	7	9	4	5

Solution:

For finding the Median we have to build a table with cumulative frequency as,

Marks	0-20	20-40	40-60	60-80	80-100
Number of Students	5	7	9	4	5
Cumulative Frequency	0+5 = 5	5+7 = 12	12+9 = 21	21+4 = 25	25+5 = 30

n = ∑f_i = 5+7+9+4+5 = 30(even)

n/2 = 30/2 = 15

Median Class = 40-60

Now using the formula,

Median = l + [(n/2 – cf) / f]×h

Where

• l is Lower Limit of Median Class
• n is Number of Observations
• f is Frequency of Median Class
• h is Class Size
• cf is Cumulative Frequency of Class Preceding Median Class

Comparing with the given data we get,

• l = 40
• n = 30
• f = 9
• h = 10
• cf = 21

Median = 20 + [(15 – 21)/6]×10

= 40 – 1/10

= 40 – 0.1

= 39.9

Thus, the median mark of the class test is 39.9

FAQs on Median

Q1: What is Median?

Answer:

The median is defined as the middle term of the given data when the data is arranged in, ascending or descending order.

Q2: What is the Relation between Mean, Median, and Mode?

Answer:

The relationship between mean median and mode is:

Mode = 3 Median – 2 Mean

Q3: How to Find the Median of Even Number of Data?

Answer:

The formula for calculating the median when the given ‘n’ is an even number,

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

Q4: How to Find the Median of Odd Number of Data?

Answer:

The formula for calculating the median when the given ‘n’ is an odd number,

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

Q5: How to Find the Median of Grouped Data?

Answer:

The formula for calculating the median of grouped data is,

Median = l + [(n/2 – cf) / f]×h

where

• l is Lower Limit of Median Class
• n is Number of Observations
• f is Frequency of Median Class
• h is Class Size
• cf is Cumulative Frequency of Class Preceding Median Class

Q6: How to find Median in Statistics?

Answer:

To find median in statistics, we can use the following steps:

• Step 1: Arrange the data in ascending order (from smallest to largest).
• Step 2: If the dataset has an odd number of values, the median is the middle value.
• Step 3: If the dataset has an even number of values, the median is the average of the two middle values.

Basic Programs Related to Median

• Maximize the median of an array
• Minimum Increment/Decrement to make array elements equal
• Minimum sum of differences with an element in an array
• Median of two sorted arrays of different sizes | Set 1 (Linear)
• Median of two sorted arrays with different sizes in O(log(min(n, m)))

More problems Related to Median

• Median and Mode using Counting Sort
• Minimum number of elements to add to make median equals x
• Decode a median string to the original string
• Median after K additional integers
• Find median in row-wise sorted matrix
• Find median of BST in O(n) time and O(1) space
• Median in a stream of integers (running integers

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