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Class 11 Maths Chapter 13 Statistics Revision Notes

From analyzing students' exam scores and comparing business performance to interpreting scientific experiments and sports statistics, Statistics helps transform raw data into meaningful insights. In this chapter, you'll learn how to measure the spread and consistency of data using concepts such as mean deviation, variance, and standard deviation. These statistical tools are widely used in data analysis, research, finance, engineering, and competitive examinations, making this chapter an important foundation for higher mathematics and data science.

At ALLEN, our expert faculty have designed these Class 11 Maths Chapter 13: Statistics Revision Notes to make revision quick, structured, and exam-oriented. The notes include concise theory, important formulas, shortcut calculation techniques, solved illustrations, common mistakes, and PYQ-based insights. Whether you're preparing for CBSE Board exams, JEE Main, or JEE Advanced, these notes will help you strengthen conceptual understanding and improve problem-solving speed.

1.0Chapter Snapshot

Topic	Overview
Measures of Dispersion	Understanding the spread of data
Range	Simplest measure of variability
Mean Deviation	Average deviation from central values
Variance	Average of squared deviations
Standard Deviation	Most widely used measure of dispersion
Coefficient of Variation	Comparing consistency between datasets
Analysis of Data	Interpretation of statistical measures
Applications	Data analysis and decision-making

2.0Chapter 13 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution

Statistics: Definition, Formulas, Central Tendency & Solved JEE Problems

Statistics is a mathematical branch that deals with collecting, analyzing, interpreting, presenting, and organizing data. It involves techniques for making inferences and decisions based on data, helping us understand patterns, trends, and relationships in various fields such as science, economics, and social sciences.

1.0Introduction to Statistics

Introduction to Statistics is a foundational course or text that provides an initial understanding of the principles, methods, and applications of statistics. This field of study covers techniques for collecting, organizing, analyzing, interpreting, and presenting data, as well as making informed decisions based on statistical findings. An introduction to statistics aims to provide learners with the fundamental resources, skills and concepts necessary for navigating and applying statistical methods in various disciplines and real-world situations.

Statistics, in essence, entails the examination and manipulation of data. As previously discussed, it involves the analysis and computation of numerical data. Let's now explore additional interpretations of statistics provided by various authors.

As per the Merriam-Webster dictionary, statistics are defined as the arrangement of classified facts representing the conditions of a population, particularly those that can be expressed numerically or organized in tables.

Statistician Sir Arthur Lyon Bowley defines statistics as numerical statements of facts from any area of investigation placed in relation to one another.

2.0Central Tendency in Statistics

Central Tendency in Statistics is a concept used to identify the central or average value within a data set. It provides a single value that represents the typical or central value of the data distribution. The main three measures of central tendency are the mean, median, and mode.

1. Mean: Also known as the average, the mean is calculated by summing up all the values in the data set and dividing them by the total number of values. It is sensitive to extreme values, making it useful for symmetric distributions.

2. Median: The median is determined as the central value within a dataset once the values have been arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle values. It is less affected by extreme values and is useful for skewed distributions.

3. Mode: The mode is the value that occurs most frequently in the data set. It is useful for identifying the most common value or values in a distribution, regardless of whether the data is numerical or categorical.

These measures provide valuable insights into the central tendencies of a data set, helping researchers and analysts understand its characteristics and make informed decisions.

Related Video:

Arithmetic Mean

For Ungrouped Distribution

The arithmetic mean, often simply called the mean, is a measure of central tendency of statistics that represents the average of a set of numerical values. It is calculated by adding up all the values in the data set and then dividing the sum by the total number of values.

The formula for calculating the arithmetic mean $\overline{x}$ of a data set x₁, x₂, ..., x_n with n observations is:

$\overset{x}{ˉ} = \frac{x _{1} + x _{2} + \dots . + x _{n}}{n}$

In simpler terms, you add up all the values and then divide by how many values there are. The mean provides a single value that summarizes the central tendency of the data set. It is widely used in various fields for summarizing and analyzing data. However, it can be influenced by extreme values (outliers) in the data set, so it's important to consider the context of the data when interpreting the mean.

For Ungrouped Frequency Distribution

For ungrouped frequency distributions, where each data point has a corresponding frequency (how often it occurs), the arithmetic mean is calculated slightly differently.

Let's say we have a set of n data points x₁, x₂, ..., x_n with corresponding frequencies f₁, f₂, ..., f_n. To find the mean $\overline{x}$ , you follow these steps:

1. Multiply each data point by its frequency.

2. Sum up all these products.

3. Divide the sum by the total frequency (sum of all frequencies).

The formula for calculating the mean for ungrouped frequency distributions is:

$\overline{x} = \frac{f _{1} x _{1} + f _{2} x _{2} + \dots . + f _{n} x _{n}}{( f _{1} + f _{2} + \dots .. + f _{n} )}$

$\overset{x}{ˉ} = \frac{\sum _{i = 1}^{n} f _{i} x _{i}}{\sum _{i = 1}^{n} f _{i}} or \overset{x}{ˉ} = \frac{\sum _{i = 1}^{n} f _{i} x _{i}}{N}$

Where N= $\sum_{i = 1}^{n} f_{i}$

In other words, you are finding the weighted average of the data points, where each data point is weighted by its frequency. This accounts for the fact that some values may occur more frequently than others in the data set.

Median

The median serves as an additional measure of central tendency, similar to the mean, but it represents the middle value of a data set when arranged in ascending or descending order. To find the median:

1. Arrange the data set in either ascending or descending order.

2. When the number of observations (n) is odd, the median corresponds to the value positioned at the center of the ordered list.

3.If the total number of observations (n) is even, the median is calculated as the arithmetic mean of the two middle values.

If n is the number of observations:

For Ungrouped Distribution and Ungrouped Frequency Distribution

For odd n: Median = Value of $(\frac{n + 1}{2}) (\frac{n + 1}{2}) t h$ observation.
For even n: Median = $\frac{Value of ( \frac{n}{2} ) thobservation + Value of ( \frac{n}{2} + 1 ) th observation}{2}$ .

The median is often preferred over the mean when the data set contains outliers or is skewed, as it is less affected by extreme values. It provides a better representation of the typical value in such cases.

For Grouped Frequency Distribution

When dealing with a grouped frequency distribution, where data is organized into intervals or classes along with their corresponding frequencies, finding the median involves a slightly different approach.

Here's how to find the median for a grouped frequency distribution:

Find the Median Class: Determine the class interval containing the median. This is usually the class where the cumulative frequency exceeds half of the total frequency.
Calculate Median: Use the formula for finding the median within a class interval:

Median = $l + \frac{( \frac{N}{2} - F )}{f} \times h$

where:

$ℓ$ = Lower boundary of the median class

N = Total frequency

F = Cumulative frequency of the class before the median class

f = Frequency of the class interval that contains the median.

h = Width of the class interval

This formula calculates the median by interpolation within the median class, considering the width of the class interval and the cumulative frequencies.

If the median class is not a continuous interval (e.g., if it's a single value), you can still use interpolation within that interval using similar principles.

Once you find the median, it represents the middle value of the data set. It's a useful measure of central tendency, especially for data sets with continuous values.

Mode

Mode is the value of variate, which has maximum Frequency.

(i) for ungrouped distribution

The value of the variable which repeated the maximum number of times.

(ii) for ungrouped frequency distribution.

The value of the variate has the maximum frequency.

(iii) for grouped frequency distribution.

First, we find the class with the maximum frequency. This is a modal class, then

Mode = $ℓ + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} \times h$

Where

$ℓ$ = lower limit of modal class.

f₀ = frequency of modal class.

f₁ = frequency of the class preceding modal class.

f₂ = frequency of the class succeeding modal class.

h = class interval of modal class.

1.0Relation between Mean, Median and Mode

In a moderately asymmetrical distribution following relation between mean, median and mode of a distribution.

It is known as the Empirical Formula.

Mode = 3 Median – 2 Mean

2.0Range

The range is a simple measure of variability in a data set, representing the difference between the maximum and minimum values. It provides a quick understanding of the spread or dispersion of data points.

The formula for calculating the range is straightforward:

Range = Maximum Value – Minimum Value

For example, if you have a data set {5, 10, 15, 20, 25}, the range would be:

Range = 25 – 5 = 20

The range in this case is 20, indicating that the data values span a range of 20 units.

While the range gives a sense of the spread of data, it can be greatly influenced by outliers and may not fully represent the variability within the data set.

3.0Variance and Standard Deviation

Variance

The Mean of the square of deviation of each variate from their mean is called variance, it is denoted by σ².

Standard Deviation (S.D.)

The positive square root of the variance is referred to as the standard deviation. It is denoted by σ.

i.e., $= S.D. = + variance$

Formulae for variance

(i) For ungrouped distribution

$σ^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n}$

(ii) For frequency distribution

$σ^{2} = \frac{\sum f _{i} ( x _{i} - x ˉ ) ^{2}}{N}$

4.0Statistics Formulas

Here are some Statistics Formula

Statistics	Formulae
Mean(Arithmetic mean)	$\overset{x}{ˉ} = \frac{x _{1} + x _{2} + \dots + x _{n}}{n}$
Median	For odd n: Median = Value of $(\frac{n + 1}{2}) t h$ observation.
	For even n: Median = $\frac{Value of ( \frac{n}{2} ) thobservation + Value of ( \frac{n}{2} + 1 ) th observation}{2}$
Mode	Mode $= ℓ + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} \times h$
Range	Range = Maximum Value − Minimum Value
Variance	$σ^{2} = \frac{\sum f _{i} ( x _{i} - x ˉ ) ^{2}}{N}$
Standard Deviation	$S.D. = + variance$

5.0Importance of Statistics

The importance of statistics cannot be overstated, as it plays a crucial role in various facts of research, decision-making, and everyday life. Here are some key reasons why statistics are important:

Data-driven Decision Making: Statistics provide a systematic framework for collecting, organizing, analyzing, and interpreting data. By applying statistical methods, individuals and organizations can base their decisions on empirical evidence rather than relying on intuition or guesswork, thanks to statistics.
Prediction and Forecasting: Through the examining past data and recognizing trends and patterns, industries such as stock market analysis, weather forecasting, and economic forecasting are examples of areas where statistical methods are used to make predictions about future events.
Risk Management: Statistics play a crucial role in risk assessment and management. By analyzing probabilities and distributions, individuals and organizations can assess and mitigate risks in various contexts, including insurance, finance, and healthcare.
Healthcare and Medicine: Statistics play a critical role in healthcare and medicine by analyzing clinical data, evaluating treatment effectiveness, and conducting epidemiological studies. Medical research relies on statistical analyses to identify risk factors, assess outcomes, and develop new treatments.
Democracy and Citizenship: Access to reliable and accurate statistical information is essential for informed citizenship and democratic participation. Statistics provide transparency, accountability, and evidence-based information for policymakers, stakeholders, and the general public.

6.0Limitations of Statistics

Here are some limitations of statistics:

Limited Scope: Statistics can only provide insights into quantitative aspects of a phenomenon. It may not capture qualitative or contextual information, leading to a partial understanding of the subject matter.
Sampling Bias: Statistical analyses are often based on samples rather than entire populations. If the sample is not representative of the population, the results may not be generalizable.
Assumptions: Many statistical methods rely on certain assumptions about the data, such as normality or independence. Failure to adhere to these assumptions can result in inaccurate outcomes.

7.0Solved Examples on Statistics

Example 1: In a group of 30 observations the mean of the first 10 is 12 and the last 20 is 9, then find the mean of the whole distribution.

Solution:

Sum of first 10 observation = 10 × 12 = 120

Sum of last 20 observation = 20 × 9 = 180

Sum of 30 observation = 120 + 180 = 300

Mean = $\frac{\sum x _{i}}{n} = \frac{300}{30} = 10$

Example 2: Find the A.M. of the following frequency distribution.

$x_{i} f_{i} 548511614101720$

Solution:

Here $N = \sum f_{i} = 4 + 5 + 6 + 10 + 20 = 45$

$\sum f_{i} x_{i} = (5 \times 4) + (8 \times 5) + (11 \times 6) + (14 \times 10) + (17 \times 20) = 606$

$∴ \overset{x}{ˉ} = \frac{Σ f _{i} x _{i}}{N} = \frac{606}{45} = 13.47$

Example 3: Find the median of following frequency distribution.

$x_{i} f_{i} 4235266211729478578675$

Solution:

$x_{i} 42475257626772 f_{i} 38681159 C \cdot f 3111725364150$

Here N=50 (even)

Median = $\frac{( \frac{N}{2} ) ^{th} term + ( \frac{N}{2} + 1 ) ^{th} term}{2}$

$= \frac{2 5 ^{th} term + 2 6 ^{th} term}{2}$

$= \frac{57 + 62}{2} = \frac{119}{2} = 59.5$

Example 4: Find the median of the following frequency distribution.

$Class f_{i} 0 - 5 6 5 - 10 10 10 - 15 8 15 - 20 9 20 - 25 7$

Solution:

$Class 0 - 5 5 - 10 10 - 15 15 - 20 20 - 25 f_{i} 610897 C . f . 616243340$

Here

$\frac{N}{2} = \frac{40}{2} = 20$

Median Class is 10 – 15

∴ Median = $10 + \frac{( 20 - 16 )}{8} \times 5$

= 10+2.5=12.5

Example 5: Find the mode of the following frequency distribution.

$Class f_{i} 0 - 10 8 10 - 20 30 20 - 30 40 30 - 40 10 40 - 50 12$

Solution:

Here model class is 20–30 (which has maximum frequency)

∴ Mode = $l + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} h$

=20+ $\frac{( 40 - 30 )}{( 2 \times 40 - 30 - 10 )} \times 10$

= 20+2.5

= 22.5

Example 6: Find variance of first 'n' even natural number is:

Solution:

x_i → 2,4,6,.....,2n

$= \frac{\sum x _{i}}{n} = \frac{n ( n + 1 )}{n} = n + 1$

Variance $(x_{i}) = \frac{\sum x _{i}^{2}}{n} - (\overset{x}{ˉ})^{2}$

$= \frac{2 ^{2} + 4 ^{2} + \dots . ( 2 n ) ^{2}}{n} - (n + 1)^{2}$

$= 4 \times \frac{n ( n + 1 ) ( 2 n + 1 )}{6 n} - (n + 1)^{2}$

$= \frac{4}{6} (n + 1) (2 n + 1) - (n + 1)^{2} = \frac{n ^{2} - 1}{3}$

Example 7: If the mean and variance of data a, b, 8, 5, and 10 are 6 and 6.8 respectively. Then the value of a and b is:

Solution:

Mean = 6

$\frac{a + b + 8 + 5 + 10}{5} = 6$

a + b = 7 ...(i)

and $v a r (x_{i}) = 6.8$

$\frac{a ^{2} + b ^{2} + 8 ^{2} + 5 ^{2} + 1 0 ^{2}}{5} - (6)^{2} = 6.8$

a²+ b²= 25 ...(ii)

from (i) and (ii)

${a = 3, b = 4 a = 4, b = 3$

8.0Statistics Practice Questions

1. The mean of discrete observations y₁, y₂,......,y_n is given by

(A) $\frac{\sum _{i = 1}^{n} y _{i}}{n}$

(B) $\frac{\sum _{i = 1}^{n} y _{i}}{\sum _{i = 1}^{n} i}$

(C) $\frac{\sum _{i = 1}^{n} y _{i} f _{i}}{n}$

(D) $\frac{\sum _{i = 1}^{n} y _{i} f _{i}}{\sum _{i = 1}^{n} f _{i}}$

2. The range for the given set of observations 2, 3, 5, 9, 8, 7, 6, 5, 7, 4, 3 is

(A) 11 (B) 7 (C) 5.5 (D) 6

3. What is the standard deviation of the set of observations 32, 28, 29, 30, 31?

(A) 1.6 (B) $2$

(C) 2 (D) None of these

4. The variance of the data 2, 4, 6, 8, 10 is

(A) 8 (B) 9 (C) 6 (D) 7

Also Read:

Central Tendency	Correlation and Regression	Coefficient of Correlation
Standard Deviation	Mean Deviation	Variance and Standard Deviation
Correlation Coefficient	Mean Median Mode Questions	Median of Grouped Data

1.0Learning Outcomes

After completing these revision notes, you will be able to:

Understand the importance of measures of dispersion.
Calculate mean deviation for grouped and ungrouped data.
Compute variance and standard deviation accurately.
Compare datasets using the coefficient of variation.
Interpret statistical measures for real-life situations.
Solve numerical problems involving statistical calculations.
Apply statistical concepts in JEE Main, JEE Advanced, and CBSE Board examinations.
Build a strong foundation for probability and higher statistical concepts.

2.05-Minute Quick Revision

Must-Revise Concepts

Measures of Dispersion
Range
Mean Deviation
Variance
Standard Deviation
Coefficient of Variation
Absolute Deviation
Squared Deviation
Frequency Distribution
Data Interpretation

Important Formula Recall

$Range = Maximum Value - Minimum Value$
Mean Deviation about Mean
$MD = \frac{\sum ∣ x - x ˉ ∣}{n}$
Variance: $σ^{2} = \frac{\sum ( x - x ˉ ) ^{2}}{n}$
Standard Deviation: $σ = \frac{\sum ( x - x ˉ ) ^{2}}{n}$
Coefficient of Variation: $C V = \frac{σ}{x ˉ} \times 100$

Important Properties

Standard deviation is always non-negative.
A smaller standard deviation indicates greater consistency in the data.
Variance is the square of the standard deviation.
Mean deviation uses absolute deviations, whereas variance uses squared deviations.
Coefficient of variation is useful for comparing datasets with different means.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Mean Deviation	High	Moderate
Variance	Very High	Moderate
Standard Deviation	Very High	Moderate
Coefficient of Variation	High	Moderate
Grouped Data Problems	High	Moderate
Statistical Interpretation	Moderate	Easy–Moderate

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Statistics in JEE Main, JEE Advanced, and CBSE examinations.

5.0Common Mistakes & JEE Tips

Although Statistics is largely formula-based, students often lose marks due to calculation errors and incorrect formula selection. Careful computation and systematic presentation are the keys to scoring well.

Common Mistake	Correct Approach
Confusing variance with standard deviation	Remember that standard deviation is the square root of variance.
Ignoring class frequencies in grouped data	Multiply each observation by its corresponding frequency before calculating sums.
Using the wrong mean while finding deviations	Calculate the mean accurately before finding mean deviation or variance.
Forgetting to square deviations	Variance requires squared deviations, not absolute deviations.
Arithmetic mistakes in lengthy calculations	Solve step-by-step and verify intermediate values.
Incorrect use of class marks	Convert class intervals into class marks before computation.
Omitting units while interpreting results	Mention units wherever applicable during interpretation.
Using the wrong coefficient of variation formula	Divide standard deviation by the mean and multiply by 100.

6.0ALLEN Faculty Tips

Memorize all statistical formulas before attempting numerical problems.
Organize calculations neatly to avoid arithmetic errors.
Learn shortcut methods for grouped data calculations.
Practice both grouped and ungrouped data questions regularly.
Focus on interpreting the meaning of statistical measures, not just calculations.
Compare datasets using the coefficient of variation whenever required.
Solve Previous Years' Questions (PYQs) to improve speed and accuracy.

7.0PYQ Trend Analysis

Statistics is a scoring chapter in JEE Main and is occasionally tested in JEE Advanced. Questions are generally numerical and involve direct application of formulas related to variance, standard deviation, and coefficient of variation.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Mean Deviation	1–2	Easy–Moderate
Variance	2–3	Moderate
Standard Deviation	2–3	Moderate
Coefficient of Variation	1–2	Moderate
Grouped Data Problems	1–2	Moderate
Mixed Statistics Questions	1–2	Moderate

8.0Most Common Question Types

Finding the range of a given dataset.
Calculating mean deviation about the mean or median.
Computing variance and standard deviation.
Solving grouped frequency distribution problems.
Comparing two datasets using the coefficient of variation.
Identifying the most consistent dataset based on statistical measures.
Interpreting measures of dispersion in practical situations.
Solving mixed numerical questions involving multiple statistical concepts.

9.0Smart Revision Strategy

Follow this revision sequence to revise the chapter efficiently:

Revise the definitions of all measures of dispersion.
Memorize the formulas for mean deviation, variance, standard deviation, and coefficient of variation.
Practice ungrouped data questions first.
Solve grouped frequency distribution problems.
Revise shortcut methods for lengthy calculations.
Compare datasets using the coefficient of variation.
Solve Previous Years' Questions (PYQs).
End with a timed practice set to improve calculation speed and accuracy.

Class 11 Maths Chapters List:- JEE & NEET Revision Notes

Chapter 1 Sets

Chapter 2 Relations and Functions

Chapter 3 Trigonometric Functions

Chapter 4 Complex Numbers and Quadratic Equations

Chapter 5 Linear Inequalities

Chapter 6 Permutations and Combinations

Chapter 7 Binomial Theorem

Chapter 8 Sequence and Series

Chapter 9 Straight Lines

Chapter 10 Conic Sections

Chapter 11 Introduction to Three-dimensional Geometry

Chapter 12 Limits and Derivatives

Chapter 13 Statistics

Chapter 14 Probability

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Class 11 Maths Chapter 13 Statistics Revision Notes

From analyzing students' exam scores and comparing business performance to interpreting scientific experiments and sports statistics, Statistics helps transform raw data into meaningful insights. In this chapter, you'll learn how to measure the spread and consistency of data using concepts such as mean deviation, variance, and standard deviation. These statistical tools are widely used in data analysis, research, finance, engineering, and competitive examinations, making this chapter an important foundation for higher mathematics and data science.

At ALLEN, our expert faculty have designed these Class 11 Maths Chapter 13: Statistics Revision Notes to make revision quick, structured, and exam-oriented. The notes include concise theory, important formulas, shortcut calculation techniques, solved illustrations, common mistakes, and PYQ-based insights. Whether you're preparing for CBSE Board exams, JEE Main, or JEE Advanced, these notes will help you strengthen conceptual understanding and improve problem-solving speed.

1.0Chapter Snapshot

Topic	Overview
Measures of Dispersion	Understanding the spread of data
Range	Simplest measure of variability
Mean Deviation	Average deviation from central values
Variance	Average of squared deviations
Standard Deviation	Most widely used measure of dispersion
Coefficient of Variation	Comparing consistency between datasets
Analysis of Data	Interpretation of statistical measures
Applications	Data analysis and decision-making

2.0Chapter 13 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution

Statistics: Definition, Formulas, Central Tendency & Solved JEE Problems

Statistics is a mathematical branch that deals with collecting, analyzing, interpreting, presenting, and organizing data. It involves techniques for making inferences and decisions based on data, helping us understand patterns, trends, and relationships in various fields such as science, economics, and social sciences.

1.0Introduction to Statistics

Introduction to Statistics is a foundational course or text that provides an initial understanding of the principles, methods, and applications of statistics. This field of study covers techniques for collecting, organizing, analyzing, interpreting, and presenting data, as well as making informed decisions based on statistical findings. An introduction to statistics aims to provide learners with the fundamental resources, skills and concepts necessary for navigating and applying statistical methods in various disciplines and real-world situations.

Statistics, in essence, entails the examination and manipulation of data. As previously discussed, it involves the analysis and computation of numerical data. Let's now explore additional interpretations of statistics provided by various authors.

As per the Merriam-Webster dictionary, statistics are defined as the arrangement of classified facts representing the conditions of a population, particularly those that can be expressed numerically or organized in tables.

Statistician Sir Arthur Lyon Bowley defines statistics as numerical statements of facts from any area of investigation placed in relation to one another.

2.0Central Tendency in Statistics

Central Tendency in Statistics is a concept used to identify the central or average value within a data set. It provides a single value that represents the typical or central value of the data distribution. The main three measures of central tendency are the mean, median, and mode.

1. Mean: Also known as the average, the mean is calculated by summing up all the values in the data set and dividing them by the total number of values. It is sensitive to extreme values, making it useful for symmetric distributions.

2. Median: The median is determined as the central value within a dataset once the values have been arranged in ascending or descending order. If there is an even number of values, the median is the average of the two middle values. It is less affected by extreme values and is useful for skewed distributions.

3. Mode: The mode is the value that occurs most frequently in the data set. It is useful for identifying the most common value or values in a distribution, regardless of whether the data is numerical or categorical.

These measures provide valuable insights into the central tendencies of a data set, helping researchers and analysts understand its characteristics and make informed decisions.

Related Video:

Arithmetic Mean

For Ungrouped Distribution

The arithmetic mean, often simply called the mean, is a measure of central tendency of statistics that represents the average of a set of numerical values. It is calculated by adding up all the values in the data set and then dividing the sum by the total number of values.

The formula for calculating the arithmetic mean $\overline{x}$ of a data set x₁, x₂, ..., x_n with n observations is:

$\overset{x}{ˉ} = \frac{x _{1} + x _{2} + \dots . + x _{n}}{n}$

In simpler terms, you add up all the values and then divide by how many values there are. The mean provides a single value that summarizes the central tendency of the data set. It is widely used in various fields for summarizing and analyzing data. However, it can be influenced by extreme values (outliers) in the data set, so it's important to consider the context of the data when interpreting the mean.

For Ungrouped Frequency Distribution

For ungrouped frequency distributions, where each data point has a corresponding frequency (how often it occurs), the arithmetic mean is calculated slightly differently.

Let's say we have a set of n data points x₁, x₂, ..., x_n with corresponding frequencies f₁, f₂, ..., f_n. To find the mean $\overline{x}$ , you follow these steps:

1. Multiply each data point by its frequency.

2. Sum up all these products.

3. Divide the sum by the total frequency (sum of all frequencies).

The formula for calculating the mean for ungrouped frequency distributions is:

$\overline{x} = \frac{f _{1} x _{1} + f _{2} x _{2} + \dots . + f _{n} x _{n}}{( f _{1} + f _{2} + \dots .. + f _{n} )}$

$\overset{x}{ˉ} = \frac{\sum _{i = 1}^{n} f _{i} x _{i}}{\sum _{i = 1}^{n} f _{i}} or \overset{x}{ˉ} = \frac{\sum _{i = 1}^{n} f _{i} x _{i}}{N}$

Where N= $\sum_{i = 1}^{n} f_{i}$

In other words, you are finding the weighted average of the data points, where each data point is weighted by its frequency. This accounts for the fact that some values may occur more frequently than others in the data set.

Median

The median serves as an additional measure of central tendency, similar to the mean, but it represents the middle value of a data set when arranged in ascending or descending order. To find the median:

1. Arrange the data set in either ascending or descending order.

2. When the number of observations (n) is odd, the median corresponds to the value positioned at the center of the ordered list.

3.If the total number of observations (n) is even, the median is calculated as the arithmetic mean of the two middle values.

If n is the number of observations:

For Ungrouped Distribution and Ungrouped Frequency Distribution

For odd n: Median = Value of $(\frac{n + 1}{2}) (\frac{n + 1}{2}) t h$ observation.
For even n: Median = $\frac{Value of ( \frac{n}{2} ) thobservation + Value of ( \frac{n}{2} + 1 ) th observation}{2}$ .

The median is often preferred over the mean when the data set contains outliers or is skewed, as it is less affected by extreme values. It provides a better representation of the typical value in such cases.

For Grouped Frequency Distribution

When dealing with a grouped frequency distribution, where data is organized into intervals or classes along with their corresponding frequencies, finding the median involves a slightly different approach.

Here's how to find the median for a grouped frequency distribution:

Find the Median Class: Determine the class interval containing the median. This is usually the class where the cumulative frequency exceeds half of the total frequency.
Calculate Median: Use the formula for finding the median within a class interval:

Median = $l + \frac{( \frac{N}{2} - F )}{f} \times h$

where:

$ℓ$ = Lower boundary of the median class

N = Total frequency

F = Cumulative frequency of the class before the median class

f = Frequency of the class interval that contains the median.

h = Width of the class interval

This formula calculates the median by interpolation within the median class, considering the width of the class interval and the cumulative frequencies.

If the median class is not a continuous interval (e.g., if it's a single value), you can still use interpolation within that interval using similar principles.

Once you find the median, it represents the middle value of the data set. It's a useful measure of central tendency, especially for data sets with continuous values.

Mode

Mode is the value of variate, which has maximum Frequency.

(i) for ungrouped distribution

The value of the variable which repeated the maximum number of times.

(ii) for ungrouped frequency distribution.

The value of the variate has the maximum frequency.

(iii) for grouped frequency distribution.

First, we find the class with the maximum frequency. This is a modal class, then

Mode = $ℓ + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} \times h$

Where

$ℓ$ = lower limit of modal class.

f₀ = frequency of modal class.

f₁ = frequency of the class preceding modal class.

f₂ = frequency of the class succeeding modal class.

h = class interval of modal class.

1.0Relation between Mean, Median and Mode

In a moderately asymmetrical distribution following relation between mean, median and mode of a distribution.

It is known as the Empirical Formula.

Mode = 3 Median – 2 Mean

2.0Range

The range is a simple measure of variability in a data set, representing the difference between the maximum and minimum values. It provides a quick understanding of the spread or dispersion of data points.

The formula for calculating the range is straightforward:

Range = Maximum Value – Minimum Value

For example, if you have a data set {5, 10, 15, 20, 25}, the range would be:

Range = 25 – 5 = 20

The range in this case is 20, indicating that the data values span a range of 20 units.

While the range gives a sense of the spread of data, it can be greatly influenced by outliers and may not fully represent the variability within the data set.

3.0Variance and Standard Deviation

Variance

The Mean of the square of deviation of each variate from their mean is called variance, it is denoted by σ².

Standard Deviation (S.D.)

The positive square root of the variance is referred to as the standard deviation. It is denoted by σ.

i.e., $= S.D. = + variance$

Formulae for variance

(i) For ungrouped distribution

$σ^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n}$

(ii) For frequency distribution

$σ^{2} = \frac{\sum f _{i} ( x _{i} - x ˉ ) ^{2}}{N}$

4.0Statistics Formulas

Here are some Statistics Formula

Statistics	Formulae
Mean(Arithmetic mean)	$\overset{x}{ˉ} = \frac{x _{1} + x _{2} + \dots + x _{n}}{n}$
Median	For odd n: Median = Value of $(\frac{n + 1}{2}) t h$ observation.
	For even n: Median = $\frac{Value of ( \frac{n}{2} ) thobservation + Value of ( \frac{n}{2} + 1 ) th observation}{2}$
Mode	Mode $= ℓ + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} \times h$
Range	Range = Maximum Value − Minimum Value
Variance	$σ^{2} = \frac{\sum f _{i} ( x _{i} - x ˉ ) ^{2}}{N}$
Standard Deviation	$S.D. = + variance$

5.0Importance of Statistics

The importance of statistics cannot be overstated, as it plays a crucial role in various facts of research, decision-making, and everyday life. Here are some key reasons why statistics are important:

Data-driven Decision Making: Statistics provide a systematic framework for collecting, organizing, analyzing, and interpreting data. By applying statistical methods, individuals and organizations can base their decisions on empirical evidence rather than relying on intuition or guesswork, thanks to statistics.
Prediction and Forecasting: Through the examining past data and recognizing trends and patterns, industries such as stock market analysis, weather forecasting, and economic forecasting are examples of areas where statistical methods are used to make predictions about future events.
Risk Management: Statistics play a crucial role in risk assessment and management. By analyzing probabilities and distributions, individuals and organizations can assess and mitigate risks in various contexts, including insurance, finance, and healthcare.
Healthcare and Medicine: Statistics play a critical role in healthcare and medicine by analyzing clinical data, evaluating treatment effectiveness, and conducting epidemiological studies. Medical research relies on statistical analyses to identify risk factors, assess outcomes, and develop new treatments.
Democracy and Citizenship: Access to reliable and accurate statistical information is essential for informed citizenship and democratic participation. Statistics provide transparency, accountability, and evidence-based information for policymakers, stakeholders, and the general public.

6.0Limitations of Statistics

Here are some limitations of statistics:

Limited Scope: Statistics can only provide insights into quantitative aspects of a phenomenon. It may not capture qualitative or contextual information, leading to a partial understanding of the subject matter.
Sampling Bias: Statistical analyses are often based on samples rather than entire populations. If the sample is not representative of the population, the results may not be generalizable.
Assumptions: Many statistical methods rely on certain assumptions about the data, such as normality or independence. Failure to adhere to these assumptions can result in inaccurate outcomes.

7.0Solved Examples on Statistics

Example 1: In a group of 30 observations the mean of the first 10 is 12 and the last 20 is 9, then find the mean of the whole distribution.

Solution:

Sum of first 10 observation = 10 × 12 = 120

Sum of last 20 observation = 20 × 9 = 180

Sum of 30 observation = 120 + 180 = 300

Mean = $\frac{\sum x _{i}}{n} = \frac{300}{30} = 10$

Example 2: Find the A.M. of the following frequency distribution.

$x_{i} f_{i} 548511614101720$

Solution:

Here $N = \sum f_{i} = 4 + 5 + 6 + 10 + 20 = 45$

$\sum f_{i} x_{i} = (5 \times 4) + (8 \times 5) + (11 \times 6) + (14 \times 10) + (17 \times 20) = 606$

$∴ \overset{x}{ˉ} = \frac{Σ f _{i} x _{i}}{N} = \frac{606}{45} = 13.47$

Example 3: Find the median of following frequency distribution.

$x_{i} f_{i} 4235266211729478578675$

Solution:

$x_{i} 42475257626772 f_{i} 38681159 C \cdot f 3111725364150$

Here N=50 (even)

Median = $\frac{( \frac{N}{2} ) ^{th} term + ( \frac{N}{2} + 1 ) ^{th} term}{2}$

$= \frac{2 5 ^{th} term + 2 6 ^{th} term}{2}$

$= \frac{57 + 62}{2} = \frac{119}{2} = 59.5$

Example 4: Find the median of the following frequency distribution.

$Class f_{i} 0 - 5 6 5 - 10 10 10 - 15 8 15 - 20 9 20 - 25 7$

Solution:

$Class 0 - 5 5 - 10 10 - 15 15 - 20 20 - 25 f_{i} 610897 C . f . 616243340$

Here

$\frac{N}{2} = \frac{40}{2} = 20$

Median Class is 10 – 15

∴ Median = $10 + \frac{( 20 - 16 )}{8} \times 5$

= 10+2.5=12.5

Example 5: Find the mode of the following frequency distribution.

$Class f_{i} 0 - 10 8 10 - 20 30 20 - 30 40 30 - 40 10 40 - 50 12$

Solution:

Here model class is 20–30 (which has maximum frequency)

∴ Mode = $l + \frac{( f _{0} - f _{1} )}{( 2 f _{0} - f _{1} - f _{2} )} h$

=20+ $\frac{( 40 - 30 )}{( 2 \times 40 - 30 - 10 )} \times 10$

= 20+2.5

= 22.5

Example 6: Find variance of first 'n' even natural number is:

Solution:

x_i → 2,4,6,.....,2n

$= \frac{\sum x _{i}}{n} = \frac{n ( n + 1 )}{n} = n + 1$

Variance $(x_{i}) = \frac{\sum x _{i}^{2}}{n} - (\overset{x}{ˉ})^{2}$

$= \frac{2 ^{2} + 4 ^{2} + \dots . ( 2 n ) ^{2}}{n} - (n + 1)^{2}$

$= 4 \times \frac{n ( n + 1 ) ( 2 n + 1 )}{6 n} - (n + 1)^{2}$

$= \frac{4}{6} (n + 1) (2 n + 1) - (n + 1)^{2} = \frac{n ^{2} - 1}{3}$

Example 7: If the mean and variance of data a, b, 8, 5, and 10 are 6 and 6.8 respectively. Then the value of a and b is:

Solution:

Mean = 6

$\frac{a + b + 8 + 5 + 10}{5} = 6$

a + b = 7 ...(i)

and $v a r (x_{i}) = 6.8$

$\frac{a ^{2} + b ^{2} + 8 ^{2} + 5 ^{2} + 1 0 ^{2}}{5} - (6)^{2} = 6.8$

a²+ b²= 25 ...(ii)

from (i) and (ii)

${a = 3, b = 4 a = 4, b = 3$

8.0Statistics Practice Questions

1. The mean of discrete observations y₁, y₂,......,y_n is given by

(A) $\frac{\sum _{i = 1}^{n} y _{i}}{n}$

(B) $\frac{\sum _{i = 1}^{n} y _{i}}{\sum _{i = 1}^{n} i}$

(C) $\frac{\sum _{i = 1}^{n} y _{i} f _{i}}{n}$

(D) $\frac{\sum _{i = 1}^{n} y _{i} f _{i}}{\sum _{i = 1}^{n} f _{i}}$

2. The range for the given set of observations 2, 3, 5, 9, 8, 7, 6, 5, 7, 4, 3 is

(A) 11 (B) 7 (C) 5.5 (D) 6

3. What is the standard deviation of the set of observations 32, 28, 29, 30, 31?

(A) 1.6 (B) $2$

(C) 2 (D) None of these

4. The variance of the data 2, 4, 6, 8, 10 is

(A) 8 (B) 9 (C) 6 (D) 7

Also Read:

Central Tendency	Correlation and Regression	Coefficient of Correlation
Standard Deviation	Mean Deviation	Variance and Standard Deviation
Correlation Coefficient	Mean Median Mode Questions	Median of Grouped Data

1.0Learning Outcomes

After completing these revision notes, you will be able to:

Understand the importance of measures of dispersion.
Calculate mean deviation for grouped and ungrouped data.
Compute variance and standard deviation accurately.
Compare datasets using the coefficient of variation.
Interpret statistical measures for real-life situations.
Solve numerical problems involving statistical calculations.
Apply statistical concepts in JEE Main, JEE Advanced, and CBSE Board examinations.
Build a strong foundation for probability and higher statistical concepts.

2.05-Minute Quick Revision

Must-Revise Concepts

Measures of Dispersion
Range
Mean Deviation
Variance
Standard Deviation
Coefficient of Variation
Absolute Deviation
Squared Deviation
Frequency Distribution
Data Interpretation

Important Formula Recall

$Range = Maximum Value - Minimum Value$
Mean Deviation about Mean
$MD = \frac{\sum ∣ x - x ˉ ∣}{n}$
Variance: $σ^{2} = \frac{\sum ( x - x ˉ ) ^{2}}{n}$
Standard Deviation: $σ = \frac{\sum ( x - x ˉ ) ^{2}}{n}$
Coefficient of Variation: $C V = \frac{σ}{x ˉ} \times 100$

Important Properties

Standard deviation is always non-negative.
A smaller standard deviation indicates greater consistency in the data.
Variance is the square of the standard deviation.
Mean deviation uses absolute deviations, whereas variance uses squared deviations.
Coefficient of variation is useful for comparing datasets with different means.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Mean Deviation	High	Moderate
Variance	Very High	Moderate
Standard Deviation	Very High	Moderate
Coefficient of Variation	High	Moderate
Grouped Data Problems	High	Moderate
Statistical Interpretation	Moderate	Easy–Moderate

4.0Formula Handbook

The following formulas and standard results are essential for solving questions from Statistics in JEE Main, JEE Advanced, and CBSE examinations.

5.0Common Mistakes & JEE Tips

Although Statistics is largely formula-based, students often lose marks due to calculation errors and incorrect formula selection. Careful computation and systematic presentation are the keys to scoring well.

Common Mistake	Correct Approach
Confusing variance with standard deviation	Remember that standard deviation is the square root of variance.
Ignoring class frequencies in grouped data	Multiply each observation by its corresponding frequency before calculating sums.
Using the wrong mean while finding deviations	Calculate the mean accurately before finding mean deviation or variance.
Forgetting to square deviations	Variance requires squared deviations, not absolute deviations.
Arithmetic mistakes in lengthy calculations	Solve step-by-step and verify intermediate values.
Incorrect use of class marks	Convert class intervals into class marks before computation.
Omitting units while interpreting results	Mention units wherever applicable during interpretation.
Using the wrong coefficient of variation formula	Divide standard deviation by the mean and multiply by 100.

6.0ALLEN Faculty Tips

Memorize all statistical formulas before attempting numerical problems.
Organize calculations neatly to avoid arithmetic errors.
Learn shortcut methods for grouped data calculations.
Practice both grouped and ungrouped data questions regularly.
Focus on interpreting the meaning of statistical measures, not just calculations.
Compare datasets using the coefficient of variation whenever required.
Solve Previous Years' Questions (PYQs) to improve speed and accuracy.

7.0PYQ Trend Analysis

Statistics is a scoring chapter in JEE Main and is occasionally tested in JEE Advanced. Questions are generally numerical and involve direct application of formulas related to variance, standard deviation, and coefficient of variation.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Mean Deviation	1–2	Easy–Moderate
Variance	2–3	Moderate
Standard Deviation	2–3	Moderate
Coefficient of Variation	1–2	Moderate
Grouped Data Problems	1–2	Moderate
Mixed Statistics Questions	1–2	Moderate

8.0Most Common Question Types

Finding the range of a given dataset.
Calculating mean deviation about the mean or median.
Computing variance and standard deviation.
Solving grouped frequency distribution problems.
Comparing two datasets using the coefficient of variation.
Identifying the most consistent dataset based on statistical measures.
Interpreting measures of dispersion in practical situations.
Solving mixed numerical questions involving multiple statistical concepts.

9.0Smart Revision Strategy

Follow this revision sequence to revise the chapter efficiently:

Revise the definitions of all measures of dispersion.
Memorize the formulas for mean deviation, variance, standard deviation, and coefficient of variation.
Practice ungrouped data questions first.
Solve grouped frequency distribution problems.
Revise shortcut methods for lengthy calculations.
Compare datasets using the coefficient of variation.
Solve Previous Years' Questions (PYQs).
End with a timed practice set to improve calculation speed and accuracy.

Frequently Asked Questions

Is Statistics an easy chapter for JEE Main?

Which formulas should I memorize before the exam?

Can I complete the Statistics chapter in one day?

Which chapters should I study before learning Statistics?

What should I practice after completing ALLEN's Statistics Revision Notes?

Join ALLEN!

Class 11 Maths Chapter 13 Statistics Revision Notes

1.0Chapter Snapshot

2.0Chapter 13 Smart Study Material by ALLEN Experts: Download PDFs

On this page

Statistics: Definition, Formulas, Central Tendency & Solved JEE Problems

1.0Introduction to Statistics

2.0Central Tendency in Statistics

Arithmetic Mean

Median

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page

Frequently Asked Questions

Is Statistics an easy chapter for JEE Main?

Which formulas should I memorize before the exam?

Can I complete the Statistics chapter in one day?

Which chapters should I study before learning Statistics?

What should I practice after completing ALLEN's Statistics Revision Notes?

Join ALLEN!

Class 11 Maths Chapter 13 Statistics Revision Notes

1.0Chapter Snapshot

2.0Chapter 13 Smart Study Material by ALLEN Experts: Download PDFs

On this page

Statistics: Definition, Formulas, Central Tendency & Solved JEE Problems

1.0Introduction to Statistics

2.0Central Tendency in Statistics

Arithmetic Mean

Median

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page