Median of Numbers

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Statistics: Median

Master Continuous Frequency Distributions in Minutes: Learn how to calculate the exact median point when data is organized inside continuous class intervals. Master cumulative frequency tables, learn how to pinpoint the critical median class, and apply the continuous median formula accurately without confusing notation—all formatted in clean, plain text for easy copy-pasting.

Class: 10 Mathematics (CBSE)
Chapter: Statistics
Estimated Learning Time: 15–20 Minutes

1.0Learning Outcomes

After completing this lesson, you will be able to:

Understand how the median of grouped data differs from raw numerical lists.
Construct a clear Cumulative Frequency (cf) column from a standard frequency distribution.
Locate the critical Median Class interval using the N/2 boundary condition.
Apply the continuous Median formula correctly with all its corresponding parameters.
Solve board exam problems containing missing frequencies when the median is pre-determined.

Median of Numbers

In mathematics and statistics, data is a collection of numerical values or observations that help us understand patterns within. To summarise data effectively, we use specialised measured values, also known as the measures of central tendency—mean, median, and mode. Here, we will focus on the median of numbers — how to find it, its formula (for grouped and ungrouped data), and its importance in statistics and data analysis.

2.0Introduction to Median

The median in statistics is basically the value of a dataset that lies exactly in the middle of the dataset, when arranged in an ascending or descending order. Simply put, the median tells us the central position of the data — the point that divides the dataset into two halves.

It is unaffected by the outliers or the extreme values. That is, other values may vary widely, but they don’t affect the median unless they shift the order of the middle values.

3.0Why is the Median Important?

The median in statistics is particularly useful when working with real data. The median is not affected, unlike the mean, by very high or very low values (outliers). For instance, where the majority of students in a class got between 60 and 80 marks, but one student got 10 and another 100, the average will be inaccurate. The median is a better indication of what a "typical" score is like.

4.0How to Find the Median?

Just like other central tendencies, the Median is also calculated differently for grouped and ungrouped data. Here’s how to find the median for both types of data:

Median of Ungrouped Data

Ungrouped data refers to a collection of individual numbers that are not sorted or organised in any specific class intervals. To find the median of ungrouped data, follow these easy steps:

Step 1: Arrange the data in order from smallest to largest (in increasing order) or largest to smallest (in decreasing order).

Step 2: Count the number of values in a data set, which is generally denoted by n.

Step 3: Check the nature of n, whether it is odd or even:

For the odd number of values: If n is odd, the median is the middle value. Below is the formula to find the median if the number of values in a data set is odd, like 3, 5, 7, 9, 11,….
$M e d ian = (\frac{n + 1}{2})$ th value of the dataset
For an even number of values: If n is even, the median is the average of the two middle values. Here’s how to find the median of even numbers of values:
$Median = \frac{( \frac{n}{2} ) ^{th} term of the dataset + ( \frac{n}{2} + 1 ) ^{th} term of the dataset}{2}$

Median of Grouped Data

Grouped data is data that has been arranged into class intervals along with their corresponding frequencies. Since the exact values within each class are not given, we use a formula to estimate the median. Here’s the median formula for grouped data:

$Median = L + (\frac{\frac{n}{2} - CF}{f}) \times h$

Here:

L is the Lower Limit of the median class
n is the Total frequency
CF is the Cumulative frequency before the median class
f is the Frequency of the median class
h is the Class width (width can be found using the formula: Upper Limit – Lower Limit)

Cumulative Frequency Table

Cumulative frequency means the total number of values added up one after another. It shows how many values are less than or equal to a certain number or class in a data set. Here’s how the frequency table column for the data set is formed:

Begin with the first frequency as the initial CF.
Add each frequency to the prior CF to extend the column. For Example, if frequencies are 5, 8, 12 → CF becomes:
- 5 → 5
- 5 + 8 → 13
- 13 + 12 → 25

Fun Fact: The sum of all the frequencies of a dataset is always equal to the last cumulative frequency of the last class interval.

5.0Solved Examples for Median

Problem 1: The following table gives the frequencies of monthly incomes of workers. Find the median income.

Monthly Income	Cumulative Frequency
Less than 5000	4
Less than 10,000	12
Less than 15,000	27
Less than 20,000	43
Less than 25,000	50

Solution: Let’s first convert a frequency table:

Monthly Income	Cumulative Frequency	Frequency (f)
0 – 5000	4	4
5000 – 10,000	12	12 – 4 = 8
10,000 – 15,000	27	27 – 12 = 15
15,000 – 20,000	43	43 – 27 = 16
20,000 – 25,000	50	50 – 43 = 7
Total(n)		50

According to the newly calculated frequency table $\frac{n}{2} = 25$ , therefore,

l = 10,000
f = 15
CF = 12
h = 5000

Now using the formula:

$Median = L + (\frac{\frac{n}{2} - CF}{f}) \times h$

$M e d ian = 10, 000 + (\frac{25 - 12}{15}) \times 5000$

$M e d ian = 10, 000 + (\frac{13000}{3})$

Median= 10,000+ 4333.33=Rs. 14,333.33

Problem 2: The table below shows the marks obtained by students in a test. Two frequencies are missing, represented by x and y. The total number of students is 60, and the median is given as 32.5. Find the values of x and y.

Marks	Number of Students
0 – 10	5
10 – 20	x
20 – 30	15
30 – 40	y
40 – 50	10
50 – 60	5

Solution: First, construct the cumulative frequency table:

Marks	Number of Students(f)	Cumulative Frequency
0 – 10	5	5
10 – 20	x	5 + x
20 – 30	15	20 + x
30 – 40	y	20 + x + y
40 – 50	10	30 + x + y
50 – 60	5	35 + x + y
Total	60

According to the new frequency table $\frac{n}{2} = 30$ , therefore:

l = 30
h = 10
f = y
CF = 13+x

The total number of students (f) = 60

35 + x + y = 60

x + y = 25 ………………………….(1)

Now, using the formula for median:

$Median = L + (\frac{\frac{n}{2} - CF}{f}) \times h$

$32.5 = 30 + (\frac{30 - ( 13 + x )}{y}) \times 10$

$2.5 = \frac{17 - x}{y} \times 10$

y= (17-x)4

y=68-4x

4x+y=68 …(2)

Eliminating equation 2 from 1

x + y = 25

4x+y = 68

________

–3x = – 33

x = 11

Now, put this value of x in equation 1:

11 + y = 25

y = 25 – 11 = 14

Hence, the value of x and y is 11 and 14, respectively.

Problem 3: A group of 11 students appeared for a rapid-fire quiz. Their scores out of 20 are as follows, but one score is missing: 12, 15, 10, 13, 14, 16, 18, 11, 17, 19, x. If the median score is known to be 15, find the missing score x.

Solution: According to the question:

n = 11

Median or the order: 15

Median of odd number of values = $(\frac{n + 1}{2})$ th value of the dataset

Median = $(\frac{11 + 1}{2})$ th=6th term

Now, arrange the given numbers in increasing order: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, x

Let's put x in the end. But according to our observation, the 6th term should be 15, but according to our assumed order, it is 16.

Hence, our assumption is incorrect, and x is the 6th term, which is 15, in the observation.

6.0EUREKA by ALLEN – Designed for Better Board Results

To accomplish your Class 10 objectives using EUREKA by ALLEN, a strong online learning resource dedicated to assisting students in achieving their maximum potential in board exams, EUREKA combines experienced teachers with specific practice materials, AI-driven support systems, and ongoing evaluations to maintain student readiness for examinations while also encouraging a thorough comprehension of the subject matter.

Key Features of EUREKA Class 10 Courses:
AI-enabled doubt solving 24/7 Interactive and personalized learning experience Story-led concept explanations Board exam-focused question practice Instant assessments and feedback Smart progress reports NCERT and CBSE syllabus coverage Flexible self-paced learning Expert faculty mentorship

Explore Now

7.0Supporting Study Materials

This study material CBSE Notes and NCERT Solutions for the Chapter "Statistics" on Median Topics is designed according to the latest CBSE Class 10 Mathematics syllabus and NCERT guidelines. It provides clear explanations of key concepts, definitions, formulas, and important questions to help students understand the median of ungrouped and grouped data, cumulative frequency (less than and more than type), the empirical relationship between mean, median, and mode, and prepare effectively for examinations.

CBSE Class 10 Maths Notes Chapter 13 Statistics	NCERT Solutions for Class 10 Maths Chapter 13: Statistics

8.0Median of Numbers – 30 Second Quick Revision

Median is the middle value of an ordered dataset.
Arrange data in ascending or descending order first.
For odd observations: Middle term is the median.
For even observations: Average of two middle terms.
Not affected by extreme values.
Represents the central position of data.
Remember: Median = Middle Value After Arranging Data

9.0Recommended Next Topics

Finding the Mode of Grouped Data (Modal class tracking)
Mean of Grouped Data (Direct, Assumed Mean, and Step-Deviation methods)
The Empirical Relationship formula between Mean, Median, and Mode
Cumulative Frequency Curves (Less-than and More-than Ogive graphing)

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Statistics: Median

Master Continuous Frequency Distributions in Minutes: Learn how to calculate the exact median point when data is organized inside continuous class intervals. Master cumulative frequency tables, learn how to pinpoint the critical median class, and apply the continuous median formula accurately without confusing notation—all formatted in clean, plain text for easy copy-pasting.

Class: 10 Mathematics (CBSE)
Chapter: Statistics
Estimated Learning Time: 15–20 Minutes

1.0Learning Outcomes

After completing this lesson, you will be able to:

Understand how the median of grouped data differs from raw numerical lists.
Construct a clear Cumulative Frequency (cf) column from a standard frequency distribution.
Locate the critical Median Class interval using the N/2 boundary condition.
Apply the continuous Median formula correctly with all its corresponding parameters.
Solve board exam problems containing missing frequencies when the median is pre-determined.

Median of Numbers

In mathematics and statistics, data is a collection of numerical values or observations that help us understand patterns within. To summarise data effectively, we use specialised measured values, also known as the measures of central tendency—mean, median, and mode. Here, we will focus on the median of numbers — how to find it, its formula (for grouped and ungrouped data), and its importance in statistics and data analysis.

2.0Introduction to Median

The median in statistics is basically the value of a dataset that lies exactly in the middle of the dataset, when arranged in an ascending or descending order. Simply put, the median tells us the central position of the data — the point that divides the dataset into two halves.

It is unaffected by the outliers or the extreme values. That is, other values may vary widely, but they don’t affect the median unless they shift the order of the middle values.

3.0Why is the Median Important?

The median in statistics is particularly useful when working with real data. The median is not affected, unlike the mean, by very high or very low values (outliers). For instance, where the majority of students in a class got between 60 and 80 marks, but one student got 10 and another 100, the average will be inaccurate. The median is a better indication of what a "typical" score is like.

4.0How to Find the Median?

Just like other central tendencies, the Median is also calculated differently for grouped and ungrouped data. Here’s how to find the median for both types of data:

Median of Ungrouped Data

Ungrouped data refers to a collection of individual numbers that are not sorted or organised in any specific class intervals. To find the median of ungrouped data, follow these easy steps:

Step 1: Arrange the data in order from smallest to largest (in increasing order) or largest to smallest (in decreasing order).

Step 2: Count the number of values in a data set, which is generally denoted by n.

Step 3: Check the nature of n, whether it is odd or even:

For the odd number of values: If n is odd, the median is the middle value. Below is the formula to find the median if the number of values in a data set is odd, like 3, 5, 7, 9, 11,….
$M e d ian = (\frac{n + 1}{2})$ th value of the dataset
For an even number of values: If n is even, the median is the average of the two middle values. Here’s how to find the median of even numbers of values:
$Median = \frac{( \frac{n}{2} ) ^{th} term of the dataset + ( \frac{n}{2} + 1 ) ^{th} term of the dataset}{2}$

Median of Grouped Data

Grouped data is data that has been arranged into class intervals along with their corresponding frequencies. Since the exact values within each class are not given, we use a formula to estimate the median. Here’s the median formula for grouped data:

$Median = L + (\frac{\frac{n}{2} - CF}{f}) \times h$

Here:

L is the Lower Limit of the median class
n is the Total frequency
CF is the Cumulative frequency before the median class
f is the Frequency of the median class
h is the Class width (width can be found using the formula: Upper Limit – Lower Limit)

Cumulative Frequency Table

Cumulative frequency means the total number of values added up one after another. It shows how many values are less than or equal to a certain number or class in a data set. Here’s how the frequency table column for the data set is formed:

Begin with the first frequency as the initial CF.
Add each frequency to the prior CF to extend the column. For Example, if frequencies are 5, 8, 12 → CF becomes:
- 5 → 5
- 5 + 8 → 13
- 13 + 12 → 25

Fun Fact: The sum of all the frequencies of a dataset is always equal to the last cumulative frequency of the last class interval.

5.0Solved Examples for Median

Problem 1: The following table gives the frequencies of monthly incomes of workers. Find the median income.

Monthly Income	Cumulative Frequency
Less than 5000	4
Less than 10,000	12
Less than 15,000	27
Less than 20,000	43
Less than 25,000	50

Solution: Let’s first convert a frequency table:

Monthly Income	Cumulative Frequency	Frequency (f)
0 – 5000	4	4
5000 – 10,000	12	12 – 4 = 8
10,000 – 15,000	27	27 – 12 = 15
15,000 – 20,000	43	43 – 27 = 16
20,000 – 25,000	50	50 – 43 = 7
Total(n)		50

According to the newly calculated frequency table $\frac{n}{2} = 25$ , therefore,

l = 10,000
f = 15
CF = 12
h = 5000

Now using the formula:

$Median = L + (\frac{\frac{n}{2} - CF}{f}) \times h$

$M e d ian = 10, 000 + (\frac{25 - 12}{15}) \times 5000$

$M e d ian = 10, 000 + (\frac{13000}{3})$

Median= 10,000+ 4333.33=Rs. 14,333.33

Problem 2: The table below shows the marks obtained by students in a test. Two frequencies are missing, represented by x and y. The total number of students is 60, and the median is given as 32.5. Find the values of x and y.

Marks	Number of Students
0 – 10	5
10 – 20	x
20 – 30	15
30 – 40	y
40 – 50	10
50 – 60	5

Solution: First, construct the cumulative frequency table:

Marks	Number of Students(f)	Cumulative Frequency
0 – 10	5	5
10 – 20	x	5 + x
20 – 30	15	20 + x
30 – 40	y	20 + x + y
40 – 50	10	30 + x + y
50 – 60	5	35 + x + y
Total	60

According to the new frequency table $\frac{n}{2} = 30$ , therefore:

l = 30
h = 10
f = y
CF = 13+x

The total number of students (f) = 60

35 + x + y = 60

x + y = 25 ………………………….(1)

Now, using the formula for median:

$Median = L + (\frac{\frac{n}{2} - CF}{f}) \times h$

$32.5 = 30 + (\frac{30 - ( 13 + x )}{y}) \times 10$

$2.5 = \frac{17 - x}{y} \times 10$

y= (17-x)4

y=68-4x

4x+y=68 …(2)

Eliminating equation 2 from 1

x + y = 25

4x+y = 68

________

–3x = – 33

x = 11

Now, put this value of x in equation 1:

11 + y = 25

y = 25 – 11 = 14

Hence, the value of x and y is 11 and 14, respectively.

Problem 3: A group of 11 students appeared for a rapid-fire quiz. Their scores out of 20 are as follows, but one score is missing: 12, 15, 10, 13, 14, 16, 18, 11, 17, 19, x. If the median score is known to be 15, find the missing score x.

Solution: According to the question:

n = 11

Median or the order: 15

Median of odd number of values = $(\frac{n + 1}{2})$ th value of the dataset

Median = $(\frac{11 + 1}{2})$ th=6th term

Now, arrange the given numbers in increasing order: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, x

Let's put x in the end. But according to our observation, the 6th term should be 15, but according to our assumed order, it is 16.

Hence, our assumption is incorrect, and x is the 6th term, which is 15, in the observation.

6.0EUREKA by ALLEN – Designed for Better Board Results

To accomplish your Class 10 objectives using EUREKA by ALLEN, a strong online learning resource dedicated to assisting students in achieving their maximum potential in board exams, EUREKA combines experienced teachers with specific practice materials, AI-driven support systems, and ongoing evaluations to maintain student readiness for examinations while also encouraging a thorough comprehension of the subject matter.

Key Features of EUREKA Class 10 Courses:
AI-enabled doubt solving 24/7 Interactive and personalized learning experience Story-led concept explanations Board exam-focused question practice Instant assessments and feedback Smart progress reports NCERT and CBSE syllabus coverage Flexible self-paced learning Expert faculty mentorship

Explore Now

7.0Supporting Study Materials

This study material CBSE Notes and NCERT Solutions for the Chapter "Statistics" on Median Topics is designed according to the latest CBSE Class 10 Mathematics syllabus and NCERT guidelines. It provides clear explanations of key concepts, definitions, formulas, and important questions to help students understand the median of ungrouped and grouped data, cumulative frequency (less than and more than type), the empirical relationship between mean, median, and mode, and prepare effectively for examinations.

CBSE Class 10 Maths Notes Chapter 13 Statistics	NCERT Solutions for Class 10 Maths Chapter 13: Statistics

8.0Median of Numbers – 30 Second Quick Revision

Median is the middle value of an ordered dataset.
Arrange data in ascending or descending order first.
For odd observations: Middle term is the median.
For even observations: Average of two middle terms.
Not affected by extreme values.
Represents the central position of data.
Remember: Median = Middle Value After Arranging Data

9.0Recommended Next Topics

Finding the Mode of Grouped Data (Modal class tracking)
Mean of Grouped Data (Direct, Assumed Mean, and Step-Deviation methods)
The Empirical Relationship formula between Mean, Median, and Mode
Cumulative Frequency Curves (Less-than and More-than Ogive graphing)

Frequently Asked Questions

1

Does the position of the median change with increasing or decreasing values?

2

Why do we arrange the data before finding the median?

3

How is the median class identified in grouped data?

4

Does a change in class width affect the median result?

Frequently Asked Questions

1

Does the position of the median change with increasing or decreasing values?

2

Why do we arrange the data before finding the median?

3

How is the median class identified in grouped data?

4

Does a change in class width affect the median result?

Join ALLEN!

Statistics: Median

1.0Learning Outcomes

Median of Numbers

2.0Introduction to Median

3.0Why is the Median Important?

4.0How to Find the Median?

Median of Ungrouped Data

Median of Grouped Data

5.0Solved Examples for Median

6.0EUREKA by ALLEN – Designed for Better Board Results

7.0Supporting Study Materials

8.0Median of Numbers – 30 Second Quick Revision

9.0Recommended Next Topics

Table of Contents

Join ALLEN!

Statistics: Median

1.0Learning Outcomes

Median of Numbers

2.0Introduction to Median

3.0Why is the Median Important?

4.0How to Find the Median?

Median of Ungrouped Data

Median of Grouped Data

5.0Solved Examples for Median

6.0EUREKA by ALLEN – Designed for Better Board Results

7.0Supporting Study Materials

8.0Median of Numbers – 30 Second Quick Revision

9.0Recommended Next Topics

Table of Contents