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Statistics Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Q: What is Statistics in Class 11 Maths?

Statistics in Class 11 Maths is the branch of mathematics that deals with the collection, organisation, analysis, interpretation, and presentation of data. It covers measures of central tendency (mean, median, mode), measures of dispersion (range, mean deviation, variance, standard deviation), and coefficient of variation. It is an important chapter for JEE and board exams.

Q: What is the difference between mean, median and mode?

Mean is the arithmetic average of all data values. Median is the middle value when data is arranged in order. Mode is the most frequently occurring value. For a moderately skewed distribution, the empirical relation is: Mode = 3 Median − 2 Mean.

Q: Can I download the Statistics formula sheet PDF for free?

Yes. You can download the complete Statistics formula sheet PDF for free using the download button on this page. It covers frequency distribution, measures of central tendency, measures of dispersion, coefficient of variation, and all key formulas in one comprehensive place for quick revision before JEE and board exams.

By Rohit Gupta Aug 20, 2026 17 min read

Statistics Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Statistics — Competishun

Statistics Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Mean · Median · Mode · Variance · Standard Deviation · Coefficient of Variation

Download Formula Sheet PDF

Statistics is one of the most scoring and practically useful chapters in Class 11 Maths. It carries moderate weightage in JEE and is a guaranteed source of marks in board exams. The chapter deals with the collection, organisation, analysis, and interpretation of data, providing tools to make sense of raw numbers.

This chapter covers measures of central tendency (mean, median, mode), measures of dispersion (range, mean deviation, variance, standard deviation), and coefficient of variation. It also includes frequency distributions, cumulative frequencies, and graphical methods like histograms. Understanding these concepts is essential not just for exams, but for making sense of data in everyday life.

This page gives you the complete guide to Statistics with all concepts explained in depth. You will find clear definitions, formulas, visual diagrams, and practice questions. Download the free PDF below and keep it handy for quick revision before your JEE Main, board exams, or any other test.

Central TendencyMean · Median · Mode

DispersionVariance · SD · Range

Coefficient of VariationCompare Consistency

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Download the Statistics Complete Guide PDF

Get all Statistics concepts, formulas, measures of central tendency and dispersion, and coefficient of variation in one clean PDF, free. Perfect for JEE and board revision.

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What is Statistics?

Definition: Statistics is the branch of mathematics that deals with the collection, organisation, analysis, interpretation, and presentation of data. It provides tools to summarise large amounts of information and draw meaningful conclusions.

Statistics is everywhere. From the marks of students in a class to the heights of players in a team, from stock market trends to weather forecasts, statistics helps us understand the world around us. In the context of JEE and board exams, the chapter focuses on two main areas:

Measures of Central Tendency

These tell us where the "centre" of the data lies. The three main measures are the mean (average), the median (middle value), and the mode (most frequent value).

Measures of Dispersion

These tell us how spread out the data is. The main measures are range, mean deviation, variance, and standard deviation.

Key Insight: Statistics is not just about memorising formulas. It is about understanding what the numbers tell us. A good statistician can look at a dataset and quickly understand its centre, spread, and shape.

Glossary of Statistics Terms — Complete A to Z

Before diving deep into each topic, let's understand the key terminology used in this chapter:

Term	Definition
Data	A collection of facts, numbers, or measurements.
Raw Data	Ungrouped data in its original form.
Frequency Distribution	Organised data showing how often each value occurs.
Class Interval	A range of values in grouped data.
Class Mark (xᵢ)	The midpoint of a class interval: (lower limit + upper limit) / 2.
Frequency (fᵢ)	The number of observations in a class.
Cumulative Frequency (cf)	The running total of frequencies. "Less than" and "more than" types.
Mean (x̄)	The arithmetic average of all data values.
Median (M)	The middle value when data is arranged in order.
Mode (Z)	The most frequently occurring value.
Range	The difference between the largest and smallest values.
Mean Deviation (MD)	The average of the absolute deviations from the mean or median.
Variance (σ²)	The average of the squared deviations from the mean.
Standard Deviation (σ)	The square root of variance.
Coefficient of Variation (CV)	A unitless measure of relative dispersion: (σ / x̄) × 100%.
Mastering these terms is essential for understanding Statistics. They will be used throughout this guide.

Data and Frequency Distribution

Definition: Data can be either raw (ungrouped) or organised into a frequency distribution, where data is grouped into classes (class intervals) with their corresponding frequencies.

When data is raw, it is simply a list of numbers. However, for large datasets, it is more convenient to group the data into class intervals and create a frequency distribution table. This table shows:

Class Intervals: The ranges into which data is grouped.
Class Mark (xᵢ): The midpoint of each class interval: xᵢ = (lower limit + upper limit) / 2.
Frequency (fᵢ): The number of observations falling into each class.
Cumulative Frequency (cf): The running total of frequencies. There are two types: "less than" and "more than" cumulative frequencies.

Example: Frequency Distribution Table

Class Interval	Class Mark (xᵢ)	Frequency (fᵢ)	Cumulative Frequency (cf)
0-10	5	5	5
10-20	15	8	13
20-30	25	12	25
30-40	35	7	32
40-50	45	3	35
The class width (h) is the difference between the upper and lower limits of a class. For the above example, h = 10.

Important: The choice of class intervals can affect the appearance and interpretation of the data. There is no single "correct" way to group data, but the intervals should be of equal width and cover all the data.

Measures of Central Tendency — Mean, Median, Mode

Definition: Measures of central tendency are single values that describe the centre of a dataset. They tell us where the "middle" of the data lies. The three main measures are the mean, median, and mode.

1. Arithmetic Mean (x̄)

The arithmetic mean is the sum of all data values divided by the number of values. It is the most common measure of central tendency.

Formulas:
Direct Method: x̄ = Σxᵢ / n (for ungrouped data)
For Grouped Data: x̄ = Σfᵢxᵢ / Σfᵢ
Assumed Mean Method: x̄ = A + (Σfᵢdᵢ / Σfᵢ), where dᵢ = xᵢ − A
Step-Deviation Method: x̄ = A + h × (Σfᵢuᵢ / Σfᵢ), where uᵢ = (xᵢ − A) / h

2. Median (M)

The median is the middle value when the data is arranged in ascending or descending order. It divides the data into two equal halves.

Formulas:
Ungrouped Data (n odd): M = value of the ((n+1)/2)th term
Ungrouped Data (n even): M = average of the (n/2)th and (n/2 + 1)th terms
Grouped Data: M = l + [(N/2 − cf) / f] × h
Where l = lower limit of median class, N = total frequency, cf = cumulative frequency before median class, f = frequency of median class, h = class width.

3. Mode (Z)

The mode is the value that appears most frequently in the dataset. A dataset can have more than one mode (bimodal, multimodal) or no mode at all.

Formula for Grouped Data:
Z = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h
Where l = lower limit of modal class, f₁ = frequency of modal class, f₀ = frequency of class preceding modal class, f₂ = frequency of class succeeding modal class, h = class width.

4. Empirical Relation

For a moderately skewed distribution, there is an empirical relation between the mean, median, and mode:

Empirical Formula: Mode = 3 × Median − 2 × Mean

Key Insight: The mean is sensitive to extreme values (outliers), while the median is robust to outliers. The mode is the only measure of central tendency that can be used with categorical data.

Graphical Mode — Finding Mode from a Histogram

Definition: The mode can also be found graphically from a histogram. The modal class is the tallest rectangle. The mode is found by drawing lines from the top corners of the modal bar to the adjacent bars; the point where they intersect, when dropped to the x-axis, gives the mode.

Finding Mode from a Histogram

4 0-10

8 10-20

7 30-40

3 40-50

The tallest bar (20-30) is the modal class. The mode lies within this class.

Important: The graphical method of finding the mode is a visual way to understand where the data is concentrated. The modal class is the class interval with the highest frequency.

Measures of Dispersion — Range, Mean Deviation, Variance, Standard Deviation

Definition: Measures of dispersion tell us how spread out the data is. They quantify the variability or scatter in the dataset. The main measures are range, mean deviation, variance, and standard deviation.

1. Range

The range is the simplest measure of dispersion. It is the difference between the largest and smallest values in the dataset.

Formula: Range = Maximum Value − Minimum Value
Coefficient of Range: (Max − Min) / (Max + Min)

Limitation: The range only uses the two extreme values and is highly sensitive to outliers. It does not tell us anything about the distribution of the values in between.

2. Mean Deviation (MD)

The mean deviation is the average of the absolute deviations from a central value (usually the mean or the median). It is a more robust measure of dispersion than the range.

Formulas:
About Mean: MD = Σ|xᵢ − x̄| / n
About Median: MD = Σ|xᵢ − M| / n
Grouped Data: MD = Σfᵢ|xᵢ − x̄| / Σfᵢ
Coefficient of MD: MD / x̄ (or MD / M)

Key Insight: The mean deviation is least when taken about the median. This is a useful property when choosing which central value to use.

3. Variance (σ²)

Variance is the average of the squared deviations from the mean. It is the most commonly used measure of dispersion in statistics.

Formulas:
Ungrouped Data: σ² = Σ(xᵢ − x̄)² / n
Grouped Data: σ² = Σfᵢ(xᵢ − x̄)² / Σfᵢ
Alternative Formula: σ² = (Σxᵢ² / n) − (x̄)²
Step-Deviation Method: σ² = h² × [ (Σfᵢuᵢ² / Σfᵢ) − (Σfᵢuᵢ / Σfᵢ)² ]

4. Standard Deviation (σ)

The standard deviation is the square root of the variance. It is expressed in the same units as the original data, making it easier to interpret.

Formula: σ = √(σ²)

5. Comparison of Dispersion Measures

Measure	Definition	Pros	Cons
Range	Max − Min	Simple to calculate	Uses only extremes, sensitive to outliers
Mean Deviation	Average absolute deviation	Easy to understand, robust	Not widely used in advanced statistics
Variance	Average squared deviation	Mathematically tractable	Units are squared
Standard Deviation	Square root of variance	Same units as data, widely used	Sensitive to outliers
Standard deviation is the most widely used measure of dispersion in JEE and board exams.

RMS and Minimisation Properties

Definition: The Root Mean Square (RMS) of a set of numbers is the square root of the mean of their squares. The RMS is minimum at the mean, and the mean deviation is minimum at the median.

Formulas:
RMS: RMS = √(Σxᵢ² / n)
Minimisation:
Σ(xᵢ − a)² is minimum at a = x̄ (the mean)
Σ|xᵢ − a| is minimum at a = M (the median)

Important: These minimisation properties are fundamental in statistics. They explain why the mean is used in variance calculations (since variance is the average of squared deviations) and why the median is used in mean deviation calculations.

Change of Origin and Scale — The Effect on Measures

Definition: If we change the origin (add or subtract a constant) or scale (multiply or divide by a constant) the data, the measures of central tendency and dispersion change in predictable ways.

Transformation	Effect on Mean	Effect on Variance	Effect on Standard Deviation
Change of Origin: yᵢ = xᵢ + a	ȳ = x̄ + a	σ² (unchanged)	σ (unchanged)
Change of Scale: yᵢ = bxᵢ	ȳ = b x̄	σ² → b² σ²	σ → \|b\| σ
General: yᵢ = a + bxᵢ	ȳ = a + b x̄	σ² → b² σ²	σ → \|b\| σ
Dispersion is unchanged by a shift of origin (adding or subtracting a constant). Scaling changes the dispersion by the scale factor.

Key Insight: This property is very useful in JEE problems. If you add a constant to all data values, the variance and standard deviation remain the same. If you multiply all data values by a constant, the variance is multiplied by the square of the constant.

Coefficient of Variation — Comparing Consistency

Definition: The coefficient of variation (CV) is a unitless measure of relative dispersion. It is defined as the ratio of the standard deviation to the mean, expressed as a percentage.

Formula: CV = (σ / x̄) × 100%

The coefficient of variation is used to compare the consistency or variability of two or more datasets with different means or units. A smaller CV indicates more consistent data (less variability relative to the mean).

Important: Compare consistency using CV, not raw standard deviation. Two datasets can have the same standard deviation but very different means, making the standard deviation alone misleading.

Comparing Distributions — Consistency and Spread

When comparing two distributions, the following rules apply:

Same mean: The distribution with the larger standard deviation is more spread out and less consistent.
Different means: Use the coefficient of variation (CV) to compare consistency. The distribution with the smaller CV is more consistent.
CV formula: CV = (σ / x̄) × 100%. A smaller CV means the data is more clustered around the mean.

Key Insight: The coefficient of variation is the only fair way to compare the variability of datasets with different units or scales. It puts the standard deviation in the context of the mean.

Skewness — The Shape of the Distribution

Definition: Skewness describes the asymmetry of a distribution. A distribution can be symmetric, right-skewed (positively skewed), or left-skewed (negatively skewed).

Symmetric

Perfectly balanced distribution.

Right-Skewed (Positively Skewed)

Mode < Median < Mean

Left-Skewed (Negatively Skewed)

Mean < Median < Mode

Key Insight: The relationship between mean, median, and mode tells us about the skewness of the distribution:
• Symmetric: Mean = Median = Mode
• Right-skewed: Mode < Median < Mean
• Left-skewed: Mean < Median < Mode

Practice Questions — From JEE and Boards

Question	Answer
Q1: Find the mean of the data: 2, 4, 6, 8, 10.	Mean = (2+4+6+8+10)/5 = 30/5 = 6
Q2: Find the median of: 3, 5, 1, 8, 7, 6.	Arranged: 1, 3, 5, 6, 7, 8. Median = (5+6)/2 = 5.5
Q3: Find the mode of: 2, 3, 4, 3, 2, 5, 3.	3 appears most frequently (3 times). Mode = 3
Q4: Calculate the variance of: 2, 4, 6, 8, 10.	x̄ = 6. σ² = [(2-6)²+(4-6)²+(6-6)²+(8-6)²+(10-6)²]/5 = (16+4+0+4+16)/5 = 40/5 = 8
Q5: Find the standard deviation of: 1, 3, 5, 7, 9.	x̄ = 5. σ² = [(−4)²+(−2)²+0+2²+4²]/5 = (16+4+0+4+16)/5 = 8. σ = √8 = 2√2
Q6: If the mean is 20 and the standard deviation is 5, what is the coefficient of variation?	CV = (5/20) × 100 = 25%
Q7: For a moderately skewed distribution, mode = 30 and mean = 24. Find the median.	Mode = 3×Median − 2×Mean → 30 = 3M − 48 → 3M = 78 → M = 26
Q8: If all values are increased by 5, what happens to the variance?	Variance remains unchanged (change of origin does not affect dispersion).
Practise these types of questions to become comfortable with applying Statistics concepts in exam scenarios.

All Statistics Formulas at a Glance

Formula	What It Means
x̄ = Σxᵢ / n	Mean (ungrouped)
x̄ = Σfᵢxᵢ / Σfᵢ	Mean (grouped)
x̄ = A + (Σfᵢdᵢ / Σfᵢ)	Assumed mean method
x̄ = A + h × (Σfᵢuᵢ / Σfᵢ)	Step-deviation method
M = l + [(N/2 − cf) / f] × h	Median (grouped)
Z = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h	Mode (grouped)
Mode = 3 Median − 2 Mean	Empirical relation
σ² = Σ(xᵢ − x̄)² / n	Variance (ungrouped)
σ² = Σfᵢ(xᵢ − x̄)² / Σfᵢ	Variance (grouped)
σ = √(σ²)	Standard deviation
CV = (σ / x̄) × 100%	Coefficient of variation
Range = Max − Min	Range
MD = Σ\|xᵢ − x̄\| / n	Mean deviation (about mean)
Memorise these formulas for Statistics. They are the key to scoring full marks in this chapter.

Common Mistakes in Statistics

Confusing mean, median, and mode: Mean is the average, median is the middle value, and mode is the most frequent value. Each has different properties and uses.
Forgetting the units of variance: Variance is in squared units. Standard deviation is in the same units as the data.
Misapplying the empirical relation: Mode = 3 Median − 2 Mean is only valid for moderately skewed distributions.
Using the wrong formula for grouped data: Always use class marks (xᵢ) when working with grouped data.
Forgetting that variance is unchanged by change of origin: Adding or subtracting a constant from all data values does not change the variance.
Comparing consistency using raw standard deviation: Use coefficient of variation (CV) when comparing datasets with different means.

Golden Rule: In Statistics, always check whether the data is grouped or ungrouped. Use the appropriate formulas and be careful with the units of variance and standard deviation.

Why Statistics Matters for JEE and Boards

Scoring chapter: Statistics is one of the most scoring chapters in Class 11 Maths. Questions are often direct and formula-based.
Moderate weightage: This chapter appears in 1-2 questions in JEE Main and is a guaranteed source of marks in board exams.
Foundation for higher studies: Understanding statistics is essential for data science, economics, psychology, and many other fields.
Practical relevance: Statistics is used everywhere, from business to medicine to politics. Learning it helps you understand the world around you.

Why this guide helps: A comprehensive Statistics guide with all concepts, definitions, formulas, and practice questions saves you time during revision and helps you quickly recall everything during the exam. You won't need to look anywhere else.

Get the Complete Statistics PDF for Free

Download the full Statistics guide with all concepts, definitions, formulas, and practice questions. Perfect for last-minute revision before JEE and board exams.

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Frequently Asked Questions — Statistics

What is Statistics in Class 11 Maths?

Statistics in Class 11 Maths is the branch of mathematics that deals with the collection, organisation, analysis, interpretation, and presentation of data. It covers measures of central tendency (mean, median, mode), measures of dispersion (range, mean deviation, variance, standard deviation), and coefficient of variation. It is an important chapter for JEE and board exams.

What is the difference between mean, median and mode?

Mean is the arithmetic average of all data values. Median is the middle value when data is arranged in order. Mode is the most frequently occurring value. For a moderately skewed distribution, the empirical relation is: Mode = 3 Median − 2 Mean.

What is the formula for variance and standard deviation?

Variance is the average of the squared deviations from the mean. For ungrouped data: σ² = Σ(xᵢ − x̄)² / n. Standard deviation is the square root of variance: σ = √(σ²). For grouped data, use σ² = Σfᵢ(xᵢ − x̄)² / Σfᵢ. Variance is also given by σ² = (Σxᵢ² / n) − (x̄)².

What is the coefficient of variation and why is it used?

The coefficient of variation (CV) is a unitless measure of relative dispersion: CV = (σ / x̄) × 100%. It is used to compare the consistency or variability of two or more datasets with different means or units. A smaller CV indicates more consistent data.

Can I download the Statistics formula sheet PDF for free?

Yes. You can download the complete Statistics formula sheet PDF for free using the download button on this page. It covers frequency distribution, measures of central tendency, measures of dispersion, coefficient of variation, and all key formulas in one comprehensive place for quick revision before JEE and board exams.

Statistics Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Download the Statistics Complete Guide PDF

What is Statistics?

Measures of Central Tendency

Measures of Dispersion

Glossary of Statistics Terms — Complete A to Z

Data and Frequency Distribution

Example: Frequency Distribution Table

Measures of Central Tendency — Mean, Median, Mode

1. Arithmetic Mean (x̄)

2. Median (M)

3. Mode (Z)

4. Empirical Relation

Graphical Mode — Finding Mode from a Histogram

Measures of Dispersion — Range, Mean Deviation, Variance, Standard Deviation

1. Range

2. Mean Deviation (MD)

3. Variance (σ²)

4. Standard Deviation (σ)

5. Comparison of Dispersion Measures

RMS and Minimisation Properties

Change of Origin and Scale — The Effect on Measures

Coefficient of Variation — Comparing Consistency

Comparing Distributions — Consistency and Spread

Skewness — The Shape of the Distribution

Symmetric

Right-Skewed (Positively Skewed)

Left-Skewed (Negatively Skewed)

Practice Questions — From JEE and Boards

All Statistics Formulas at a Glance

Common Mistakes in Statistics

Why Statistics Matters for JEE and Boards

Get the Complete Statistics PDF for Free