Median in Statistics

The median is the middle value of the dataset when arranged in ascending or descending order.

• If the dataset has an odd number of values, the median is the middle value.
• If the dataset has an even number of values, the median is the average of the two middle values.

Example: Suppose we have the heights of 5 friends as 167 cm, 169 cm, 171 cm, 174 cm, 179 cm. When arranged in order, the middle value is 171 cm, so the median height is 171 cm.

Median divides the data into two equal halves. It shows the middle point of a data set and gives a better idea of the “typical” value, especially when the data has very high or very low values.

Why median is useful?

• It is not affected by outliers (unlike mean).
• It shows the central position of data.
• It is useful for skewed distributions (e.g., incomes, house prices).
• It helps in ranking and comparing data sets.

Median Example

Various examples of the median are:

Example 1: Median salary of five friends, where the individual salary of each friend is,

• 74,000,
• 82,000,
• 75,000,
• 96,000,
• 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.

Example 2: Median Age of a Group- Consider a group of people's 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 Value 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.

Median of Ungrouped Data

The median formula is calculated by two methods,

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

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

Example: Find the Median of the following data,

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

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

Comparing with the given data we get,

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

Median = 20 + [(15 - 13)/6] × 10

= 20 + (2/6) x 10

= 60/3 + 10/3

= 20 + 3.3333 = 23.33 (approx)

Thus, the median mark of the class test is 23.33

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.

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

The 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

Median = 50/2

Median = 25

Thus, median of 23 and 27 is 25.

Also Check

• Statistics
• Measure of central tendency
• 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,

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

Comparing with the given data we get,

• l = 40
• n = 30
• f = 9
• h = 10
• cf = 12

Median = 20 + [(15 - 12)/6]×10
= 40 - (3/9) x 20
= 40 +6.6667
= 46.6667

Thus, the median mark of the class test is 46.67.

Example 4: Find the median number of hours studied per week

The following table shows the distribution of the number of hours spent studying per week by a group of students:

Solution:

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

Hours Studied (Per week)	0 - 5	5 - 10	10 - 15	15 - 20	20 - 25
Frequency	8	15	25	12	10
Cumulative Frequency	0 + 8 = 8	8 + 15 = 23	23 + 25 = 48	48 + 12 = 60	60 + 10 = 70

n = ∑f_i = 8 + 15 + 25 + 12 + 10 = 70(even)

n/2 = 70/2 = 35

Median Class = 10 - 15

Now using the formula,

Median = l + [(n/2 – cf) / f]×h

Comparing with the given data we get,

• l = 10
• n = 70
• f = 25
• h = 5
• cf = 23

Median = 10 + [(35 - 23)/25]×5

= 10 - (12/15) x 5
= 10 - (0.48) x 5
= 10 + 2.4
= 12.4

Thus, the median number of hours per week is 12.4 hours.