🔹 1. Introduction

Statistics deals with the collection, classification, analysis, and interpretation of numerical data.

👉 Used in:

• Economics
• Science
• Research
• Data analysis

---

🔹 2. Types of Data

✔️ 1. Ungrouped Data

Data given individually
Example: 2, 5, 7, 10

---

✔️ 2. Grouped Data

Data arranged in classes

---

✔️ Types of Grouped Data

• Discrete Frequency Distribution
• Continuous Frequency Distribution

---

🔹 3. Measures of Central Tendency

These measures represent the central value of data.

---

🔸 (A) Mean (Average)

---

✔️ 1. Arithmetic Mean (Ungrouped Data)

$\bar{x} = \frac{\sum x}{n}$xˉ=n∑x​

---

✔️ 2. Mean (Discrete Data)

$\bar{x} = \frac{\sum f x}{\sum f}$xˉ=∑f∑fx​

---

✔️ 3. Mean (Continuous Data)

(i) Direct Method

$\bar{x} = \frac{\sum f x}{\sum f}$xˉ=∑f∑fx​

---

(ii) Assumed Mean Method

$\bar{x} = a + \frac{\sum f d}{\sum f}$xˉ=a+∑f∑fd​

---

(iii) Step-Deviation Method

$\bar{x} = a + h \frac{\sum f u}{\sum f}$xˉ=a+h∑f∑fu​

---

🔸 (B) Median

Middle value of data.

---

✔️ Ungrouped Data

• Arrange data
• Middle value

---

✔️ Continuous Data

$\text{Median} = l + \frac{\left(\frac{n}{2} – cf\right)}{f} \times h$Median=l+f(2n​−cf)​×h

---

Where:

• l = lower boundary of median class
• cf = cumulative frequency
• f = frequency of median class
• h = class width

---

🔸 (C) Mode

Most frequent value.

---

✔️ Formula (Continuous Data)

$\text{Mode} = l + \frac{(f_1 – f_0)}{2f_1 – f_0 – f_2} \times h$Mode=l+2f1​−f0​−f2​(f1​−f0​)​×h

---

Where:

• f₁ = modal class frequency
• f₀ = preceding frequency
• f₂ = succeeding frequency

---

🔹 4. Empirical Relation

Relation between mean, median, and mode:

$\text{Mode} = 3 \times \text{Median} – 2 \times \text{Mean}$Mode=3×Median−2×Mean

---

🔹 5. Measures of Dispersion

These show how data is spread.

---

🔸 (A) Range

$\text{Range} = \text{Max} – \text{Min}$Range=Max−Min

---

🔸 (B) Mean Deviation

---

✔️ About Mean

$\text{MD} = \frac{\sum |x – \bar{x}|}{n}$MD=n∑∣x−xˉ∣​

---

✔️ For Frequency Data

$\text{MD} = \frac{\sum f |x – \bar{x}|}{\sum f}$MD=∑f∑f∣x−xˉ∣​

---

🔸 (C) Variance

$\sigma^2 = \frac{\sum (x – \bar{x})^2}{n}$σ2=n∑(x−xˉ)2​

---

🔸 (D) Standard Deviation

$\sigma = \sqrt{\frac{\sum (x – \bar{x})^2}{n}}$σ=n∑(x−xˉ)2​​

---

✔️ For Frequency Distribution

$\sigma = \sqrt{\frac{\sum f(x – \bar{x})^2}{\sum f}}$σ=∑f∑f(x−xˉ)2​​

---

🔹 6. Shortcut Formula for Standard Deviation

$\sigma = \sqrt{\frac{\sum f x^2}{\sum f} – \bar{x}^2}$σ=∑f∑fx2​−xˉ2​

---

🔹 7. Coefficient of Variation (CV)

Used for comparison:

$CV = \frac{\sigma}{\bar{x}} \times 100$CV=xˉσ​×100

---

🔹 8. Important Terms

• Class Interval
• Class Mark
• Frequency
• Cumulative Frequency

---

🔹 9. Steps to Solve Questions

✔️ Identify data type
✔️ Choose correct formula
✔️ Compute frequencies carefully
✔️ Use shortcut methods for speed

---

🔹 10. Applications

✔️ Data analysis
✔️ Economics
✔️ Business decisions
✔️ Research studies

---

🔹 11. JEE & CBSE Important Points

✔️ Practice all three methods of mean
✔️ Median & mode formulas important
✔️ Standard deviation is high-weightage
✔️ Learn shortcut formulas
✔️ Be careful with calculations

---

🔹 12. Common Mistakes

❌ Wrong class interval
❌ Calculation errors
❌ Using wrong formula
❌ Ignoring frequencies

---

🔹 13. Practice Questions

1. Find mean using direct method
2. Calculate median for grouped data
3. Find mode using formula
4. Compute standard deviation
5. Compare datasets using CV

---

🔹 14. Quick Revision Sheet

• Mean = Σx / n
• Median formula (grouped)
• Mode formula
• σ = √(Σ(x−x̄)² / n)
• CV = (σ / x̄) × 100

---

🔹 15. Conclusion

Statistics is a practical and scoring chapter in Class 11 Maths.

👉 Strong understanding of:

• Mean
• Median
• Mode
• Standard Deviation

ensures excellent performance in CBSE & competitive exams.