Statistics 📊

Data mein se meaningful information nikalna — average toh Class 9 mein seekha tha. Ab Class 11 mein dispersion seekhenge — data kitna spread hai!

🏏 Two Batsmen — Kaun Better?

Rahul aur Virat dono ke 5 matches ke scores dekhte hain: Rahul: 50, 50, 50, 50, 50 (Average = 50). Virat: 10, 20, 80, 90, 50 (Average = 50). Average same hai! Lekin clearly Rahul zyada consistent hai — Virat ka performance bahut vary karta hai. Average akela data describe nahi kar sakta — humhe Dispersion bhi measure karna hoga — data kitna spread hai mean ke aas paas!

1. Measures of Central Tendency — Quick Recap

Class 9-10 mein yeh seekhe the — yahan quick revision:

2. Measures of Dispersion — Class 11 ka Main Topic

Dispersion batata hai ki data points mean ke kitne door hain — spread kitna hai. Kum dispersion = consistent data. Zyada dispersion = variable data.

📏 Range

Sabse simple dispersion. Maximum aur minimum ka difference.

Range = Max − Min

Easy lekin outliers se bahut affect hota hai.

📐 Mean Deviation (MD)

Har data point ki mean se absolute distance ka average.

MD = Σ|xᵢ − x̄| / n

Mean ya Median kisi ke bhi baare mein calculate hota hai.

📊 Variance (σ²)

Squared deviations ka average — zyada accurate lekin unit square ho jaati hai.

σ² = Σ(xᵢ − x̄)² / n

Outliers ko extra weight milta hai (square ki wajah se).

📈 Std Deviation (σ)

Variance ka square root — units original data jaisi wapas aati hain. Sabse important!

σ = √[Σ(xᵢ−x̄)²/n]

Zyada σ = zyada spread. Less σ = data mean ke paas.

3. Variance aur Standard Deviation — Step by Step

✅ Data: 4, 7, 8, 11, 15 ke liye Mean, Variance aur SD nikalo

Step 1 — Mean: x̄ = (4+7+8+11+15)/5 = 45/5 = 9
Step 2 — Deviations from mean: (4−9)=−5, (7−9)=−2, (8−9)=−1, (11−9)=2, (15−9)=6
Step 3 — Squared deviations: 25, 4, 1, 4, 36
Step 4 — Variance σ² = (25+4+1+4+36)/5 = 70/5 = 14
Step 5 — SD σ = √14 ≈ 3.74

Matlab: Average data point mean se approximately 3.74 units door hai!

xᵢxᵢ − x̄(xᵢ − x̄)²
4−525
7−24
8−11
1124
15636
Σ=45070

Note: Deviations ka sum hamesha 0 hota hai (Σ(xᵢ−x̄) = 0) — yeh check ke liye use karo!

4. Shortcut Formula for Variance

Bade data mein har baar mean se subtract karna tedious hota hai. Yeh shortcut formula directly nikalta hai:

σ² = Σxᵢ²/n − (x̄)² = Σxᵢ²/n − (Σxᵢ/n)²

"Mean of squares minus square of mean" — yaad karo yaise! Yeh formula calculation bahut fast karta hai.

✅ Shortcut se check karo: Data 4,7,8,11,15

Σxᵢ² = 16+49+64+121+225 = 475
σ² = 475/5 − (9)² = 95 − 81 = 14 ✓

Same answer! Shortcut formula aur long method dono check ho gaye.

5. Coefficient of Variation (CV)

Sirf SD se do alag datasets compare nahi kar sakte — agar units alag hoon ya means bahut alag hoon. CV ek relative measure hai — percentage mein:

CV = (σ / x̄) × 100 %

Kam CV = zyada consistent (less variable). Zyada CV = zyada variable (less consistent). Jab do datasets compare karne hoon toh CV use karo!

✅ Rahul aur Virat ka CV compare karo

Rahul scores: 50,50,50,50,50. x̄=50, σ=0. CV = 0%
Virat scores: 10,20,80,90,50. x̄=50.
Deviations²: 1600+900+900+1600+0 = 5000. σ²=1000. σ=√1000≈31.6
CV Virat = (31.6/50)×100 = 63.2%
Rahul ka CV=0% — perfectly consistent! Virat ka CV=63.2% — highly variable.

6. Grouped Data ka SD

Jab data groups/classes mein ho (frequency distribution) toh:

x̄ = Σfᵢxᵢ / Σfᵢ (weighted mean)
σ² = Σfᵢ(xᵢ−x̄)² / Σfᵢ

Yahan xᵢ = class midpoint, fᵢ = class frequency. Ek ek class ka midpoint aur frequency use hoti hai.

7. Practice Questions

Q1. 6, 8, 10, 12, 14 ka mean deviation (mean ke baare mein) nikalo.

Solution

Mean = 10. |deviations| = 4,2,0,2,4. MD = 12/5 = 2.4

Q2. 2, 4, 6, 8, 10 ka variance aur SD nikalo.

Solution

x̄=6. (xᵢ−x̄)²: 16,4,0,4,16. σ²=40/5=8. σ=2√2≈2.83

Q3. Ek dataset ka mean 20 aur SD 4 hai. CV nikalo.

Solution

CV = (4/20)×100 = 20%

Q4. Shortcut se 3,6,9,12,15 ka variance nikalo.

Solution

x̄=9. Σxᵢ²=9+36+81+144+225=495. σ²=495/5−81=99−81=18. σ=√18=3√2

Q5. Factory A ka CV=25%, Factory B ka CV=18% hai. Kaun zyada consistent hai?

Solution

Kam CV = zyada consistent. Factory B zyada consistent hai!

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