Can the mean be influenced by outliers?

Yes, the mean is sensitive to outliers since extreme values can skew the result.

Can the mode be missing in a data set?

Yes, a data set can have no mode if no value repeats.

What is the difference between the mean, median, & mode?

The mean is the average, the median is the middle value, and the mode is the most frequent value.

How to find Mean and Median?

Mean: Add up all the values and divide by the number of values. Median: Order the values from lowest to highest and pick the middle value or average the two middle values if there is an even number of values.

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Mean

Master Central Tendency in Minutes: Learn how to find the average or balance point of data arranged in intervals. Master class marks, understand the three direct calculation methods (Direct, Assumed Mean, and Step-Deviation), and solve high-yield board exam.

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

1.0Learning Outcomes

After completing this lesson, you will be able to:

Define raw data vs. grouped frequency distributions.
Calculate the Class Mark (midpoint) for any given class interval.
Compute the mean using the Direct Method.
Reduce large arithmetic numbers using the Assumed Mean Method.
Simplify complex calculation steps using the Step-Deviation Method.
Solve board exam problems containing missing frequency values.

Mean Introduction

In statistics, the Mean is the average value of a given data, making it easier to compare and understand the data. It is an important measure of central tendency apart from Median and mode. In simple words, Mean is a value that is closer to all the values present in given data that we use to summarise that data.

Mean Example: Imagine a set of scores for a class of students, ranging from high to low. To determine the overall performance of the class, you can calculate the mean, a single value that represents the average score that offers a clear understanding of how the class performed as a whole.

2.0Introduction to Mean

Mean Definition

The mean of a given set of data can be defined as the sum of all the values of data divided by the total number of values. It is also referred to as the Average of the data. The general mean formula is:

$M e an = \frac{S u m o f a ll v a l u es}{N u mb er o f Va l u es}$

Calculation of Mean

Calculating Mean for Ungrouped Data:

The mean for ungrouped data is calculated as the sum of all individual data values divided by the total number of data points, giving a very simple average for the whole set. The mean formula for an ungrouped data say (x₁, x₂, x₃, x₄ …..,x_n) can be written as:

$M e an = \frac{x _{1} + x _{2} + x _{3} + x _{4} \dots .. + x _{n}}{n}$

Calculating Mean for Grouped Data:

Grouped data is the data that is arranged in intervals or classes. To calculate the mean of grouped data, you use the midpoint of each class, the class mark(x_i), and the frequency(f_i) of each class, multiply them together, and then divide by the total number of observations. Mathematically, the same can written as:

$M e an (\overset{x}{ˉ}) = \frac{\sum f _{i} x _{i}}{\sum f _{i}}$

3.0Other Values of Central Tendency

As mentioned earlier, the mean is not the only measure of central tendency. Median and Mode are the two other measures of central tendency equally important as the mean.

Median

The median helps identify the middle point of the data, making it useful in situations where the data is not symmetrically distributed. Unlike the mean, which is affected by large values, the median gives a more accurate measure of central tendency in these cases.

How to find median for:

For Ungrouped data: When there is an odd number of values, the median is the middle value, and when there is an even number, the median is the average of the two middle values.
For Grouped Data: For Grouped data, the Median can be found by the following formula:

$M e d ian = L + (\frac{\frac{N}{2} - CF}{f}) \times h$

Here:

L = Lower boundary of the median class
N = Total number of observations
CF = Cumulative frequency of the class just before the median class
f = Frequency of the median class
h = Class width (difference between the upper and lower boundaries of the median class)

Mode:

The mode is the specific value which occurs most frequently in a dataset. Mode is useful for identifying the most common value in a given dataset. To calculate the Mode:

For Ungrouped Data: It is the most number of times occurring value in a given data.
For grouped Data: The mode of grouped data can be calculated by the following formula:

$M o d e = L + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h$

Here,

L = Lower boundary of the modal class
f₁ = Frequency of the modal class
f₀ = Frequency of the class just before the modal class
f₂ = Frequency of the class after the modal class
h = Class width

4.0Relation Between Mean, Median and Mode

The relation between mean, median, and mode can be expressed using the following empirical formula:

$M o d e = 3 M e d ian -2 M e an$

5.0Practice Problems

Problem 1: Find the mean for the following data: 10, 12, 14, 15, 15, 18, 20, 20, 20, 25

Solution: For mean:

The Sum of all the values = 10 + 12 + 14 + 15 + 15 + 18 + 20 + 20 + 20 + 25 = 169

Number of Values = 10

$M e an = \frac{S u m o f a ll v a l u es}{n u mb er o f v a l u es}$

$M e an = \frac{169}{10}$

Mean=16.9

Problem 2: Find the mean, median, and mode for the following data: 5, 7, 8, 12, 15, 19, 23

Solution:

For mean:

$M e an = \frac{5 + 7 + 8 + 12 + 15 + 19 + 23}{7}$

$M e an = \frac{89}{7}$

Mean = 12.71

For Median:

The number of values is odd n = 7

So the median = $\frac{n + 1}{2} = \frac{7 + 1}{2} = 4^{t h}$

Hence, the median of ungrouped data is 4_th term = 12

For Mode:

The highest value of the given ungrouped data is 23, so it is the mode of the data.

Problem 3: The following data represents the number of hours spent by students on homework per week (grouped data):

Hours (class Interval)	Frequency(f)
0 - 10	5
10 - 20	8
20 - 30	12
30 - 40	10
40 - 50	4

Find the mean, median, and mode.

Solution:

1. Mean

Hours (class Interval)	Classmark	Frequency(f)	f_ix_i
0 - 10	5	5	25
10 - 20	15	8	120
20 - 30	25	12	300
30 - 40	35	10	350
40 - 50	45	4	180
Total		39	975

$M e an (\overset{x}{ˉ}) = \frac{\sum f _{i} x _{i}}{\sum f _{i}}$

$M e an (\overset{x}{ˉ}) = \frac{975}{39} = 25$

2. Mode:

For mode f₁ = 12 because it occurs the most times. So, f₀ = 8, f₂ = 10, h = 10, L = 20

$M o d e = L + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h$

$M o d e = 20 + (\frac{12 - 8}{2 \times 12 - 8 - 10} \times 10$

$M o d e = 20 + \frac{4}{6} \times 10$

$M o d e = 20 + 6.67 = 26.67$

3. Median:

Hours (class Interval)	Frequency(f)	CF
0 - 10	5	5
10 - 20	8	13
20 - 30	12	25
30 - 40	10	35
40 - 50	4	39

N = 39/2 = 19.5

CF = 13 (since 19.5 is between 13 and 25)

L = 20, f = 12, h = 10

$M e d ian = L + (\frac{\frac{N}{2} - CF}{f}) \times h$

$M e d ian = 20 + (\frac{19.5 - 13}{12}) \times 10$

$M e d ian = 20 + (\frac{6.5}{12}) \times 10$

$M e d ian = 20 + 5.417 = 25.417$

6.0EUREKA by ALLEN – Learn Better, Score Higher

EUREKA by ALLEN is designed to simplify, enrich, and enhance your experience in Class 10. Through the use of fun and engaging video lessons, regular practice tests, and immediate help for any doubts you may have regarding the material; students have a firm understanding of the concepts they are studying and feel confident in their preparation for their board exams. No matter if you are attempting to receive a higher mark or develop a better understanding of your studies, EUREKA will support you as you continue to grow as a learner.

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 mean 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 arithmetic mean of ungrouped and grouped data, the Direct Method, Assumed Mean Method, and Step-Deviation Method, and prepare effectively for examinations.

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

8.0Mean – 30 Second Quick Revision

Mean is the average of observations.
Mean = Sum of Observations ÷ Number of Observations
Considers every value in the dataset.
Affected by very large or very small values.
Used to represent central tendency.
Most commonly used average.
Remember: Add all values, then divide by total values

9.0Recommended Next Topics

Finding the Median of Grouped Data (Cumulative Frequency charts)
Finding the Mode of Grouped Data (Modal class identification)
Empirical Relationship between Mean, Median, and Mode
Cumulative Frequency Curves (Ogive graphs)

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.

Mean

Master Central Tendency in Minutes: Learn how to find the average or balance point of data arranged in intervals. Master class marks, understand the three direct calculation methods (Direct, Assumed Mean, and Step-Deviation), and solve high-yield board exam.

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

1.0Learning Outcomes

After completing this lesson, you will be able to:

Define raw data vs. grouped frequency distributions.
Calculate the Class Mark (midpoint) for any given class interval.
Compute the mean using the Direct Method.
Reduce large arithmetic numbers using the Assumed Mean Method.
Simplify complex calculation steps using the Step-Deviation Method.
Solve board exam problems containing missing frequency values.

Mean Introduction

In statistics, the Mean is the average value of a given data, making it easier to compare and understand the data. It is an important measure of central tendency apart from Median and mode. In simple words, Mean is a value that is closer to all the values present in given data that we use to summarise that data.

Mean Example: Imagine a set of scores for a class of students, ranging from high to low. To determine the overall performance of the class, you can calculate the mean, a single value that represents the average score that offers a clear understanding of how the class performed as a whole.

2.0Introduction to Mean

Mean Definition

The mean of a given set of data can be defined as the sum of all the values of data divided by the total number of values. It is also referred to as the Average of the data. The general mean formula is:

$M e an = \frac{S u m o f a ll v a l u es}{N u mb er o f Va l u es}$

Calculation of Mean

Calculating Mean for Ungrouped Data:

The mean for ungrouped data is calculated as the sum of all individual data values divided by the total number of data points, giving a very simple average for the whole set. The mean formula for an ungrouped data say (x₁, x₂, x₃, x₄ …..,x_n) can be written as:

$M e an = \frac{x _{1} + x _{2} + x _{3} + x _{4} \dots .. + x _{n}}{n}$

Calculating Mean for Grouped Data:

Grouped data is the data that is arranged in intervals or classes. To calculate the mean of grouped data, you use the midpoint of each class, the class mark(x_i), and the frequency(f_i) of each class, multiply them together, and then divide by the total number of observations. Mathematically, the same can written as:

$M e an (\overset{x}{ˉ}) = \frac{\sum f _{i} x _{i}}{\sum f _{i}}$

3.0Other Values of Central Tendency

As mentioned earlier, the mean is not the only measure of central tendency. Median and Mode are the two other measures of central tendency equally important as the mean.

Median

The median helps identify the middle point of the data, making it useful in situations where the data is not symmetrically distributed. Unlike the mean, which is affected by large values, the median gives a more accurate measure of central tendency in these cases.

How to find median for:

For Ungrouped data: When there is an odd number of values, the median is the middle value, and when there is an even number, the median is the average of the two middle values.
For Grouped Data: For Grouped data, the Median can be found by the following formula:

$M e d ian = L + (\frac{\frac{N}{2} - CF}{f}) \times h$

Here:

L = Lower boundary of the median class
N = Total number of observations
CF = Cumulative frequency of the class just before the median class
f = Frequency of the median class
h = Class width (difference between the upper and lower boundaries of the median class)

Mode:

The mode is the specific value which occurs most frequently in a dataset. Mode is useful for identifying the most common value in a given dataset. To calculate the Mode:

For Ungrouped Data: It is the most number of times occurring value in a given data.
For grouped Data: The mode of grouped data can be calculated by the following formula:

$M o d e = L + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h$

Here,

L = Lower boundary of the modal class
f₁ = Frequency of the modal class
f₀ = Frequency of the class just before the modal class
f₂ = Frequency of the class after the modal class
h = Class width

4.0Relation Between Mean, Median and Mode

The relation between mean, median, and mode can be expressed using the following empirical formula:

$M o d e = 3 M e d ian -2 M e an$

5.0Practice Problems

Problem 1: Find the mean for the following data: 10, 12, 14, 15, 15, 18, 20, 20, 20, 25

Solution: For mean:

The Sum of all the values = 10 + 12 + 14 + 15 + 15 + 18 + 20 + 20 + 20 + 25 = 169

Number of Values = 10

$M e an = \frac{S u m o f a ll v a l u es}{n u mb er o f v a l u es}$

$M e an = \frac{169}{10}$

Mean=16.9

Problem 2: Find the mean, median, and mode for the following data: 5, 7, 8, 12, 15, 19, 23

Solution:

For mean:

$M e an = \frac{5 + 7 + 8 + 12 + 15 + 19 + 23}{7}$

$M e an = \frac{89}{7}$

Mean = 12.71

For Median:

The number of values is odd n = 7

So the median = $\frac{n + 1}{2} = \frac{7 + 1}{2} = 4^{t h}$

Hence, the median of ungrouped data is 4_th term = 12

For Mode:

The highest value of the given ungrouped data is 23, so it is the mode of the data.

Problem 3: The following data represents the number of hours spent by students on homework per week (grouped data):

Hours (class Interval)	Frequency(f)
0 - 10	5
10 - 20	8
20 - 30	12
30 - 40	10
40 - 50	4

Find the mean, median, and mode.

Solution:

1. Mean

Hours (class Interval)	Classmark	Frequency(f)	f_ix_i
0 - 10	5	5	25
10 - 20	15	8	120
20 - 30	25	12	300
30 - 40	35	10	350
40 - 50	45	4	180
Total		39	975

$M e an (\overset{x}{ˉ}) = \frac{\sum f _{i} x _{i}}{\sum f _{i}}$

$M e an (\overset{x}{ˉ}) = \frac{975}{39} = 25$

2. Mode:

For mode f₁ = 12 because it occurs the most times. So, f₀ = 8, f₂ = 10, h = 10, L = 20

$M o d e = L + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h$

$M o d e = 20 + (\frac{12 - 8}{2 \times 12 - 8 - 10} \times 10$

$M o d e = 20 + \frac{4}{6} \times 10$

$M o d e = 20 + 6.67 = 26.67$

3. Median:

Hours (class Interval)	Frequency(f)	CF
0 - 10	5	5
10 - 20	8	13
20 - 30	12	25
30 - 40	10	35
40 - 50	4	39

N = 39/2 = 19.5

CF = 13 (since 19.5 is between 13 and 25)

L = 20, f = 12, h = 10

$M e d ian = L + (\frac{\frac{N}{2} - CF}{f}) \times h$

$M e d ian = 20 + (\frac{19.5 - 13}{12}) \times 10$

$M e d ian = 20 + (\frac{6.5}{12}) \times 10$

$M e d ian = 20 + 5.417 = 25.417$

6.0EUREKA by ALLEN – Learn Better, Score Higher

EUREKA by ALLEN is designed to simplify, enrich, and enhance your experience in Class 10. Through the use of fun and engaging video lessons, regular practice tests, and immediate help for any doubts you may have regarding the material; students have a firm understanding of the concepts they are studying and feel confident in their preparation for their board exams. No matter if you are attempting to receive a higher mark or develop a better understanding of your studies, EUREKA will support you as you continue to grow as a learner.

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 mean 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 arithmetic mean of ungrouped and grouped data, the Direct Method, Assumed Mean Method, and Step-Deviation Method, and prepare effectively for examinations.

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

8.0Mean – 30 Second Quick Revision

Mean is the average of observations.
Mean = Sum of Observations ÷ Number of Observations
Considers every value in the dataset.
Affected by very large or very small values.
Used to represent central tendency.
Most commonly used average.
Remember: Add all values, then divide by total values

9.0Recommended Next Topics

Finding the Median of Grouped Data (Cumulative Frequency charts)
Finding the Mode of Grouped Data (Modal class identification)
Empirical Relationship between Mean, Median, and Mode
Cumulative Frequency Curves (Ogive graphs)

Frequently Asked Questions

Can the mean be influenced by outliers?

Can the mode be missing in a data set?

What is the difference between the mean, median, & mode?

How to find Mean and Median?

Frequently Asked Questions

Can the mean be influenced by outliers?

Can the mode be missing in a data set?

What is the difference between the mean, median, & mode?

How to find Mean and Median?

Join ALLEN!

Mean

1.0Learning Outcomes

Mean Introduction

2.0Introduction to Mean

Mean Definition

Calculation of Mean

3.0Other Values of Central Tendency

4.0Relation Between Mean, Median and Mode

5.0Practice Problems

6.0EUREKA by ALLEN – Learn Better, Score Higher

7.0Supporting Study Materials

8.0Mean – 30 Second Quick Revision

9.0Recommended Next Topics

Table of Contents

Join ALLEN!

Mean

1.0Learning Outcomes

Mean Introduction

2.0Introduction to Mean

Mean Definition

Calculation of Mean

3.0Other Values of Central Tendency

4.0Relation Between Mean, Median and Mode

5.0Practice Problems

6.0EUREKA by ALLEN – Learn Better, Score Higher

7.0Supporting Study Materials

8.0Mean – 30 Second Quick Revision

9.0Recommended Next Topics

Table of Contents