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CBSE Class 10 Mathematics: Statistics and Probability

Complete Study Material (2025 Edition)

CHAPTER 14: STATISTICS

1. INTRODUCTION TO STATISTICS

Statistics is the branch of mathematics that deals with collection, organization, analysis, interpretation, and presentation of data. It helps us make sense of large amounts of information and draw meaningful conclusions.

Key Terms

  • Data: Collection of facts, numbers, or information
  • Raw Data: Unorganized data as originally collected
  • Array: Arrangement of data in ascending or descending order
  • Range: Difference between highest and lowest values
  • Class Interval: Groups into which data is divided
  • Frequency: Number of times a particular value occurs

2. MEASURES OF CENTRAL TENDENCY

Central tendency gives us a single value that represents the entire dataset.

2.1 ARITHMETIC MEAN

Definition: The arithmetic mean is the sum of all observations divided by the number of observations.

For Ungrouped Data

Formula: Mean (x̄) = (x₁ + x₂ + x₃ + ... + xₙ)/n = Σx/n

For Grouped Data

Direct Method: Mean (x̄) = Σ(fᵢxᵢ)/Σfᵢ

Where:

  • fᵢ = frequency of ith class
  • xᵢ = mid-value of ith class
  • Σfᵢ = total frequency = n

Assumed Mean Method: Mean (x̄) = a + (Σfᵢdᵢ)/Σfᵢ

Where:

  • a = assumed mean
  • dᵢ = xᵢ - a (deviation from assumed mean)

Step Deviation Method: Mean (x̄) = a + h × (Σfᵢuᵢ)/Σfᵢ

Where:

  • h = class width
  • uᵢ = (xᵢ - a)/h

2.2 MEDIAN

Definition: The median is the middle value when data is arranged in ascending or descending order.

For Ungrouped Data

  • If n is odd: Median = ((n+1)/2)th observation
  • If n is even: Median = [n/2 th observation + (n/2 + 1)th observation]/2

For Grouped Data

Formula: Median = l + [(n/2 - CF)/f] × h

Where:

  • l = lower boundary of median class
  • n = total frequency
  • CF = cumulative frequency before median class
  • f = frequency of median class
  • h = class width

Finding Median Class: The class where cumulative frequency ≥ n/2

2.3 MODE

Definition: Mode is the value that appears most frequently in the dataset.

For Ungrouped Data

The value with highest frequency is the mode.

For Grouped Data

Formula: Mode = l + [(f₁ - f₀)/(2f₁ - f₀ - f₂)] × h

Where:

  • l = lower boundary of modal class
  • f₁ = frequency of modal class
  • f₀ = frequency of class before modal class
  • f₂ = frequency of class after modal class
  • h = class width

Finding Modal Class: The class with highest frequency

3. CUMULATIVE FREQUENCY

3.1 Cumulative Frequency Table

Cumulative frequency is the running total of frequencies.

Less than type: Shows number of observations less than upper boundary More than type: Shows number of observations more than lower boundary

3.2 Cumulative Frequency Curves (Ogives)

Less Than Ogive

  • Plot upper boundaries on x-axis
  • Plot cumulative frequencies on y-axis
  • Join points with smooth curve

More Than Ogive

  • Plot lower boundaries on x-axis
  • Plot cumulative frequencies on y-axis
  • Join points with smooth curve

Finding Median from Ogive: Draw horizontal line from n/2 on y-axis to meet the curve, then drop perpendicular to x-axis.

4. IMPORTANT RELATIONSHIPS

4.1 Empirical Relationship

For moderately skewed distribution: Mode = 3 Median - 2 Mean

4.2 Properties of Mean

  • Sum of deviations from mean = 0
  • Mean is affected by extreme values
  • Mean of combined groups = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂)

5. PRACTICE QUESTIONS - STATISTICS

MCQs (Multiple Choice Questions)

  1. The median of first 10 prime numbers is: a) 11 b) 12 c) 13 d) 14 Answer: b) 12
  2. If mean of x, x+2, x+4, x+6, x+8 is 11, then x is: a) 5 b) 7 c) 9 d) 11 Answer: b) 7
  3. The mode of data 3, 4, 5, 6, 6, 7, 8 is: a) 5 b) 6 c) 7 d) 8 Answer: b) 6
  4. In the formula for median of grouped data, CF represents: a) Class frequency b) Cumulative frequency before median class c) Total frequency d) Class width Answer: b) Cumulative frequency before median class
  5. The sum of deviations from arithmetic mean is: a) Maximum b) Minimum c) Zero d) Cannot be determined Answer: c) Zero

Fill in the Blanks

  1. The middle value of arranged data is called _______. Answer: Median
  2. In step deviation method, u = _______. Answer: (x - a)/h
  3. The empirical relation between mean, median and mode is _______. Answer: Mode = 3 Median - 2 Mean
  4. The class with highest frequency is called _______ class. Answer: Modal
  5. Cumulative frequency curve is also called _______. Answer: Ogive

Very Short Answer Questions (1 Mark)

  1. Q: Find the mean of first 5 natural numbers. A: Mean = (1+2+3+4+5)/5 = 15/5 = 3
  2. Q: What is the median of 2, 4, 6, 8, 10? A: Median = 6 (middle value)
  3. Q: If mode = 25 and median = 20, find mean using empirical formula. A: Mode = 3 Median - 2 Mean 25 = 3(20) - 2 Mean 25 = 60 - 2 Mean Mean = 17.5

Short Answer Questions (2-3 Marks)

  1. Q: Find the mean of the following data using assumed mean method: Class: 0-10, 10-20, 20-30, 30-40, 40-50 Frequency: 5, 8, 12, 10, 5 Solution: Let assumed mean a = 25
    ClassMid-value (x)Frequency (f)d = x - 25fd
    0-1055-20-100
    10-20158-10-80
    20-30251200
    30-40351010100
    40-5045520100
    Σf = 40, Σfd = 20 Mean = a + Σfd/Σf = 25 + 20/40 = 25 + 0.5 = 25.5
  2. Q: Find the median class for the following data: Class: 0-5, 5-10, 10-15, 15-20, 20-25 Frequency: 3, 7, 12, 8, 5 Solution:
    ClassFrequencyCumulative Frequency
    0-533
    5-10710
    10-151222
    15-20830
    20-25535
    n = 35, n/2 = 17.5 Cumulative frequency just greater than 17.5 is 22 Therefore, median class is 10-15

Long Answer Questions (4-5 Marks)

  1. Q: The following table shows the distribution of heights of 60 students:
    Height (cm)150-155155-160160-165165-170170-175175-180
    Frequency812181084
    Find the median height. Solution:
    ClassFrequencyCumulative Frequency
    150-15588
    155-1601220
    160-1651838
    165-1701048
    170-175856
    175-180460
    n = 60, n/2 = 30 Cumulative frequency just greater than 30 is 38 Median class = 160-165 l = 160, CF = 20, f = 18, h = 5 Median = l + [(n/2 - CF)/f] × h = 160 + [(30 - 20)/18] × 5 = 160 + (10/18) × 5 = 160 + 2.78 = 162.78 cm

Higher Order Thinking Skills (HOTS) Questions

  1. Q: The arithmetic mean of the following frequency distribution is 50. Find the value of k.
    Class0-2020-4040-6060-8080-100
    Frequency172832k19
    Solution: Let the frequencies be f₁ = 17, f₂ = 28, f₃ = 32, f₄ = k, f₅ = 19 Mid-values: x₁ = 10, x₂ = 30, x₃ = 50, x₄ = 70, x₅ = 90 Mean = Σfx/Σf = 50 (17×10 + 28×30 + 32×50 + k×70 + 19×90)/(17 + 28 + 32 + k + 19) = 50 (170 + 840 + 1600 + 70k + 1710)/(96 + k) = 50 (4320 + 70k)/(96 + k) = 50 4320 + 70k = 50(96 + k) 4320 + 70k = 4800 + 50k 20k = 480 k = 24

CHAPTER 15: PROBABILITY

1. INTRODUCTION TO PROBABILITY

Probability is the measure of likelihood that an event will occur. It quantifies uncertainty and helps us make predictions about random events.

Key Terms

  • Experiment: An action or process that results in one or more outcomes
  • Trial: Each performance of an experiment
  • Event: A subset of possible outcomes of an experiment
  • Sample Space: Set of all possible outcomes of an experiment
  • Favorable Outcomes: Outcomes that satisfy the given condition
  • Random Experiment: An experiment whose outcome cannot be predicted

2. CLASSICAL DEFINITION OF PROBABILITY

Formula: P(E) = Number of favorable outcomes / Total number of possible outcomes

Where:

  • P(E) = Probability of event E
  • 0 ≤ P(E) ≤ 1
  • P(E) = 0 means event E will never occur (impossible event)
  • P(E) = 1 means event E will always occur (certain event)

Properties of Probability

  1. P(E) + P(Ē) = 1, where Ē is complement of E
  2. P(impossible event) = 0
  3. P(certain event) = 1
  4. 0 ≤ P(E) ≤ 1 for any event E

3. TYPES OF EVENTS

3.1 Simple Event

An event with only one outcome. Example: Getting heads in a coin toss

3.2 Compound Event

An event with more than one outcome. Example: Getting an even number on rolling a die

3.3 Complementary Events

Two events are complementary if one must occur when the other doesn't. Example: Getting heads or tails in coin toss

3.4 Mutually Exclusive Events

Events that cannot occur simultaneously. Example: Getting heads and tails simultaneously in one toss

3.5 Independent Events

Events where occurrence of one doesn't affect the other. Example: Results of two separate coin tosses

4. PROBABILITY WITH CARDS

Standard Deck Information

  • Total cards: 52
  • Suits: 4 (Hearts ♥, Diamonds ♦, Clubs ♣, Spades ♠)
  • Cards per suit: 13
  • Red cards: 26 (Hearts + Diamonds)
  • Black cards: 26 (Clubs + Spades)
  • Face cards: 12 (4 Kings, 4 Queens, 4 Jacks)
  • Number cards: 40 (Ace to 10 in each suit)

Common Probabilities

  • P(Red card) = 26/52 = 1/2
  • P(Face card) = 12/52 = 3/13
  • P(King) = 4/52 = 1/13
  • P(Heart) = 13/52 = 1/4

5. PRACTICE QUESTIONS - PROBABILITY

MCQs (Multiple Choice Questions)

  1. The probability of getting a prime number on throwing a die is: a) 1/6 b) 1/3 c) 1/2 d) 2/3 Answer: c) 1/2 (Prime numbers: 2, 3, 5)
  2. If P(E) = 0.25, then P(not E) is: a) 0.25 b) 0.5 c) 0.75 d) 1 Answer: c) 0.75
  3. The probability of getting a red card from a deck of 52 cards is: a) 1/4 b) 1/2 c) 3/4 d) 1 Answer: b) 1/2
  4. If two coins are tossed, the probability of getting at least one head is: a) 1/4 b) 1/2 c) 3/4 d) 1 Answer: c) 3/4
  5. The sum of probabilities of all possible outcomes of an experiment is: a) 0 b) 1 c) -1 d) Cannot be determined Answer: b) 1

Fill in the Blanks

  1. The probability of an impossible event is _______. Answer: 0
  2. If P(A) = 0.7, then P(not A) = _______. Answer: 0.3
  3. The set of all possible outcomes is called _______. Answer: Sample space
  4. Events that cannot occur together are called _______. Answer: Mutually exclusive
  5. The probability of a certain event is _______. Answer: 1

Very Short Answer Questions (1 Mark)

  1. Q: A die is thrown. Find the probability of getting an odd number. A: Odd numbers: 1, 3, 5 P(odd) = 3/6 = 1/2
  2. Q: What is the probability of getting a king from a deck of cards? A: P(king) = 4/52 = 1/13
  3. Q: If P(E) = 3/7, find P(not E). A: P(not E) = 1 - P(E) = 1 - 3/7 = 4/7

Short Answer Questions (2-3 Marks)

  1. Q: Two dice are thrown simultaneously. Find the probability of getting: a) Sum = 7 b) Same number on both dice Solution: Total outcomes = 6 × 6 = 36 a) Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes P(sum = 7) = 6/36 = 1/6 b) Same number: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) = 6 outcomes P(same number) = 6/36 = 1/6
  2. Q: A bag contains 3 red balls, 5 white balls, and 7 black balls. A ball is drawn at random. Find the probability that the ball is: a) Red b) Not black Solution: Total balls = 3 + 5 + 7 = 15 a) P(red) = 3/15 = 1/5 b) Not black = red + white = 3 + 5 = 8 P(not black) = 8/15

Long Answer Questions (4-5 Marks)

  1. Q: A card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting: a) A red king b) A face card c) A red card or a king d) Neither a heart nor a king Solution: a) Red kings = 2 (King of hearts, King of diamonds) P(red king) = 2/52 = 1/26 b) Face cards = 12 (4 kings + 4 queens + 4 jacks) P(face card) = 12/52 = 3/13 c) Red cards = 26, Kings = 4, Red kings = 2 Red cards or king = 26 + 4 - 2 = 28 P(red card or king) = 28/52 = 7/13 d) Hearts = 13, Kings = 4, King of hearts = 1 Heart or king = 13 + 4 - 1 = 16 Neither heart nor king = 52 - 16 = 36 P(neither heart nor king) = 36/52 = 9/13

Higher Order Thinking Skills (HOTS) Questions

  1. Q: A bag contains balls numbered 1 to 45. If a ball is drawn at random, what is the probability that the number on the ball is: a) A perfect square b) Divisible by 5 c) A prime number less than 20 Solution: Total balls = 45 a) Perfect squares from 1 to 45: 1, 4, 9, 16, 25, 36 = 6 numbers P(perfect square) = 6/45 = 2/15 b) Numbers divisible by 5: 5, 10, 15, 20, 25, 30, 35, 40, 45 = 9 numbers P(divisible by 5) = 9/45 = 1/5 c) Prime numbers less than 20: 2, 3, 5, 7, 11, 13, 17, 19 = 8 numbers P(prime less than 20) = 8/45

6. PREVIOUS 5 YEARS CBSE BOARD QUESTIONS

2024 CBSE Questions

Statistics:

  1. Q: Find the median of the following data: Class: 0-10, 10-20, 20-30, 30-40, 40-50 Frequency: 5, 3, 4, 3, 5 Solution: n = 20, n/2 = 10, Median class = 20-30 Median = 20 + [(10-8)/4] × 10 = 25

Probability:

  1. Q: A card is drawn from a deck of 52 cards. Find the probability of getting a red face card. Answer: Red face cards = 6, P = 6/52 = 3/26

2023 CBSE Questions

Statistics:

  1. Q: If the mode of the data 4, 5, 7, k, 8, 9, 7 is 7, find all possible values of k. Answer: For mode to be 7, k can be 7 or any value except 4, 5, 8, 9

Probability:

  1. Q: Two dice are thrown. Find the probability that the sum is greater than 10. Answer: Favorable outcomes: (5,6), (6,5), (6,6) = 3 P = 3/36 = 1/12

2022 CBSE Questions

Statistics:

  1. Q: Find the mean of first 10 multiples of 4. Answer: Multiples: 4, 8, 12, ..., 40 Mean = (4+8+...+40)/10 = 220/10 = 22

Probability:

  1. Q: A box contains 12 balls of which x are black. If one ball is drawn, the probability of drawing a black ball is twice the probability of drawing a white ball. Find x. Answer: P(black) = x/12, P(white) = (12-x)/12 x/12 = 2(12-x)/12, x = 8

2021 CBSE Questions

Statistics:

  1. Q: The median of the distribution is 14.4. Find the value of x. Class: 0-6, 6-12, 12-18, 18-24, 24-30 Frequency: 4, x, 5, 1, 2 Solution: n = 12+x, n/2 = 6+x/2 Median class = 12-18 (since CF crosses n/2 here) 14.4 = 12 + [(6+x/2-4)/5] × 6 Solving: x = 8

Probability:

  1. Q: Find the probability that a leap year has 53 Sundays. Answer: Leap year has 366 days = 52 weeks + 2 days Extra 2 days can be: (Sat,Sun), (Sun,Mon), (Mon,Tue), (Tue,Wed), (Wed,Thu), (Thu,Fri), (Fri,Sat) Favorable outcomes with Sunday: 2 P = 2/7

2020 CBSE Questions

Statistics:

  1. Q: Find the mode of the following frequency distribution: Class: 10-20, 20-30, 30-40, 40-50, 50-60 Frequency: 8, 16, 36, 34, 6 Solution: Modal class = 30-40 (highest frequency = 36) Mode = 30 + [(36-16)/(2×36-16-34)] × 10 = 30 + 20/22 × 10 = 39.09

Probability:

  1. Q: A card is drawn from a pack of 52 cards. Find the probability of getting a king or a heart. Answer: Kings = 4, Hearts = 13, King of hearts = 1 Favorable = 4 + 13 - 1 = 16 P = 16/52 = 4/13

7. TIPS, TRICKS AND MISTAKE ALERTS

Statistics Tips:

  1. Always check if data is grouped or ungrouped before applying formulas
  2. In assumed mean method, choose assumed mean close to middle value
  3. For median, always find n/2 first, then locate median class
  4. In mode formula, be careful about f₀ and f₂ identification
  5. Draw cumulative frequency table systematically

Common Mistakes in Statistics:

  • Using wrong median formula for odd/even n
  • Confusing less than and more than cumulative frequency
  • Not finding median class correctly
  • Using frequency instead of cumulative frequency in median formula
  • Forgetting to multiply by class width in final calculation

Probability Tips:

  1. Always identify sample space first
  2. Count favorable outcomes carefully
  3. Use complement when it's easier: P(at least one) = 1 - P(none)
  4. Remember card deck composition
  5. Check if answer lies between 0 and 1

Common Mistakes in Probability:

  • Confusing "at least" with "exactly"
  • Not considering all possible outcomes
  • Adding probabilities of non-mutually exclusive events incorrectly
  • Forgetting to reduce fractions to lowest terms
  • Misunderstanding "without replacement" vs. "with replacement"

8. SUMMARY TABLE

Statistics Formulas

MeasureUngrouped DataGrouped Data
MeanΣx/nDirect: Σfx/Σf
Assumed: a + Σfd/Σf
Step: a + h×Σfu/Σf
MedianMiddle value when arrangedl + [(n/2-CF)/f] × h
ModeMost frequent valuel + [(f₁-f₀)/(2f₁-f₀-f₂)] × h

Key Values to Remember

ItemValue
Cards in deck52
Red cards26
Face cards12
Cards per suit13
Outcomes on die6
Outcomes on coin2
Prime numbers on die3 (2,3,5)

Probability Formulas

Event TypeFormula
Basic ProbabilityP(E) = Favorable outcomes/Total outcomes
ComplementP(not E) = 1 - P(E)
Certain EventP(E) = 1
Impossible EventP(E) = 0

Important Relationships

RelationshipFormula
Empirical RelationMode = 3 Median - 2 Mean
Probability Range0 ≤ P(E) ≤ 1
Complementary EventsP(E) + P(not E) = 1

9. FINAL EXAM PREPARATION CHECKLIST

Must Know Topics:

✓ Mean calculation by all three methods ✓ Median for grouped data ✓ Mode for grouped data ✓ Cumulative frequency and ogives ✓ Basic probability concepts ✓ Card probability problems ✓ Dice probability problems

Practice Focus Areas:

✓ Word problems on central tendency ✓ Finding missing frequency when mean/median is given ✓ Combined probability events ✓ Complementary events ✓ At least/at most probability problems

Quick Revision Points:

✓ All formulas with proper notation ✓ Standard deck composition ✓ Common probability values ✓ Relationship between mean, median, mode ✓ Properties of probability

This comprehensive study material covers all aspects of Statistics and Probability for CBSE Class 10 Mathematics, providing you with the foundation needed for excellent exam performance.

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