Complete Guide to CBSE Class 10 Maths: Statistics & Probability (2025–26 Syllabus)

📘 CBSE Class 10 Maths (Chapter: Statistics & Probability)

✨ 1. Key Concepts of the Chapter

✅ Statistics

• Statistics is the study of collecting, organizing, analyzing, and interpreting numerical data.

• In this chapter, we focus on statistical measures:

  • Mean (Average)

  • Median (Middle value)

  • Mode (Most frequent value)

• Data dealt with here is grouped data (continuous classes).

✅ Probability

• Probability measures the likelihood of an event.

• Classical definition:

  $$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$

• Deals with simple experiments: coin toss, dice roll, card pick, etc.

📌 2. Important Formulae

🔷 For Statistics (Grouped Data)

👉 Mean:

$$\text{Mean},\ \overline{x} = \frac{\sum f_ix_i}{\sum f_i}$$

Where:

• $f_i$ = frequency

• $x_i$ = mid-point of each class

👉 Median:

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

Where:

• $l$ = lower boundary of median class

• $N$ = total frequency

• $F$ = cumulative frequency before median class

• $f$ = frequency of median class

• $h$ = class size

👉 Mode:

$$\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$$

Where:

• $f_1$ = frequency of modal class

• $f_0$ and $f_2$ = frequencies of the classes before and after modal class

🔷 For Probability

$$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total outcomes}} \quad ;\quad 0 \leq P(E) \leq 1$$

📚 3. NCERT Exercise Questions & Answers

📝 Statistics

✅ Ex 13.1 (Grouped Data)

Q1. Find the mean for the following data:

Class Interval	Frequency
0–10	5
10–20	8
20–30	15
30–40	16
40–50	6

Solution:

Class	f	x (Midpoint)	fx
0-10	5	5	25
10-20	8	15	120
20-30	15	25	375
30-40	16	35	560
40-50	6	45	270

• Total $f = 5+8+15+16+6 = 50$

• Total $fx = 1350$

$$\text{Mean} = \frac{1350}{50} = \boxed{27}$$

Ex 13.2 (Median + Mode)

Q2. Find median of the data:

C.I.	Frequency
0–10	2
10–20	5
20–30	8
30–40	12
40–50	3

Solution:

• $N = 2+5+8+12+3 = 30$ ⇒ $\frac{N}{2} = 15$

• Cumulative Frequencies:
  2, 7, 15, 27, 30

• Median Class = 30–40

  • $l = 30$, $F = 15$, $f = 12$, $h = 10$

$$\text{Median} = 30 + \left( \frac{15 - 15}{12} \right) \times 10 = 30$$

Answer: $\boxed{30}$

Q3. Find the mode:

C.I.	Frequency
0–10	4
10–20	6
20–30	10
30–40	12
40–50	5

• Modal class = 30–40 (highest freq = 12)

• $l = 30$, $f_1 = 12$, $f_0 = 10$, $f_2 = 5$, $h = 10$

$$\text{Mode} = 30 + \left( \frac{12 - 10}{2 \times 12 - 10 - 5} \right) \times 10 = 30 + \left( \frac{2}{9} \right) \times 10 = \boxed{32.22}$$

📝 Probability (Ex 15.1)

Q1. A coin is tossed once. What is the probability of getting (a) a head, (b) a tail?

• S = {H, T}; favorable for head = 1, total = 2

• (a) P(H) = $\frac{1}{2}$

• (b) P(T) = $\frac{1}{2}$

Answer:

• $\boxed{P(H) = \frac{1}{2};\quad P(T) = \frac{1}{2}}$

Q2. A number is randomly chosen from 1 to 10. What is the probability that it is (i) prime, (ii) divisible by 3?

• Numbers 1–10 ⇒ total = 10

• (i) Prime numbers = {2, 3, 5, 7} ⇒ 4 favorable

$$P(\text{prime}) = \frac{4}{10} = \boxed{\frac{2}{5}}$$

• (ii) Div by 3 = {3, 6, 9} ⇒ 3 favorable

$$P = \frac{3}{10}$$

Answer:

• (i) $\frac{2}{5}$

• (ii) $\frac{3}{10}$

🧾 4. CBSE Previous Year Questions (PYQs) + Model Answers

✳️ Q1 (CBSE 2023):

Find the mean for the following data:

C.I.	Freq
10–20	5
20–30	8
30–40	15
40–50	16
50–60	6

Answer:

• Mid-points: 15, 25, 35, 45, 55

• fx: 75, 200, 525, 720, 330 ⇒ $\sum fx = 1850$

• $\sum f = 50$

$$\text{Mean} = \frac{1850}{50} = \boxed{37}$$

✳️ Q2 (CBSE 2022): A dice is thrown once. Find the probability that the number is (i) odd, (ii) multiple of 3.

• Total = 6 outcomes = {1,2,3,4,5,6}

• (i) Odd = 1,3,5 ⇒ P = 3/6 = 1/2

• (ii) Multiple of 3 = 3,6 ⇒ P = 2/6 = 1/3

Answer:

• $\boxed{(i)\ \frac{1}{2};\ (ii)\ \frac{1}{3}}$

✳️ Q3 (CBSE 2020):

In a group of 100 students, 20 are left-handed. If one is chosen at random, what is the probability the student is right-handed?

• Right-handed = 100 − 20 = 80

$$P = \frac{80}{100} = \boxed{\frac{4}{5}}$$

🎯 5. Tips for Full Marks in Board Exams

✅ Always write the formula before substitution.
✅ Show all intermediate steps.
✅ Box the final answer.
✅ Use correct units when needed.
✅ Don’t skip cumulative frequency steps in median.

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