MathsHiranyagarbha · Class 11-12
Average same, khiladi alag: variance aur standard deviation
Do batsman ka 5 match ka average 50 hai. Ek har match mein 50 banata hai, doosra 0, 100, 0, 100, 50. Team kis par bharosa kare?
Socho pehle… phir dekho jawab
Mean beech batata hai, phailaav (spread) nahi. Spread ke liye har value ki mean se doori ka square lekar average karte hain — variance; uska square root standard deviation (σ). Pehle batsman ka σ = 0, doosre ka σ ≈ 44.7.
School team ki captain Ritu ke saamne yahi sawaal tha. Doston ne kaha, 'Jisne 100 maara, wahi lo.' Ritu ne pehle dono ka phailaav naapa, phir team ki zaroorat dekh kar dono ki bhoomika tay ki — ek sthir opener, ek bade shot wala finisher.
Formula: σ² = (1/n) Σ (xᵢ − x̄)², σ = √σ². Doosre batsman ke deviations: −50, 50, −50, 50, 0; squares ka jod 10,000; ÷ 5 = 2,000; σ = √2000 ≈ 44.7. Square kyun? Taaki + aur − deviations ek-doosre ko kaat na dein, aur bade bhatkaav ko zyada vajan mile.
Alag mean wale data ki tulna: coefficient of variation CV = (σ / x̄) × 100. Doosre batsman ka CV ≈ 89.4%, pehle ka 0%. Exam ke niyam: har value mein constant jodo toh σ nahi badalta; har value ko k se guna karo toh σ bhi |k| guna.
Trade-off aur kahan chhupa hai: kabhi jokhim bhara bada khiladi chahiye, kabhi sthir. Investment mein σ ko volatility kehte hain; factory ka quality control, mausam ka anuman aur machine learning mein data ko scale karna — sab σ par tike hain. Bharat ka yogdaan: Indian Statistical Institute (1931) ke sansthaapak P.C. Mahalanobis ne 'Mahalanobis distance' diya, jo spread ko dhyaan mein rakh kar batata hai ki koi point data ke jhund se kitna door hai.
Ghar pe try karo
7 din tak roz ek cheez note karo — jaise ghar se school pahunchne ka samay (minute) ya akhbaar mein chhapa shehar ka adhiktam taapmaan. Mean, variance, σ aur CV khud nikaalo. Phir likho: kya akela average poori kahani bata raha tha?
Wow!
Do data sets ka mean aur median bilkul same ho sakta hai, phir bhi unki duniya alag — spread hi asli kahani batata hai!
3 sawaal
- Har value mein 10 joda jaaye toh standard deviation ka kya hoga?
Jawab dekho
Wahi rahega. Sab values ek saath khisakti hain, mean se doori nahi badalti.
- Values 0, 100, 0, 100, 50 ka variance kitna hai?
Jawab dekho
2,000. Squares ka jod 10,000 ÷ 5 = 2,000; σ ≈ 44.7 hai, variance nahi.
- Alag mean wale do data sets ki sthirta (consistency) ki tulna ke liye kya behtar hai?
Jawab dekho
Coefficient of variation. CV = σ/mean × 100, jo spread ko mean ke anupaat mein naapta hai.
NCERT se jodo: NCERT Class 11 Maths — statistics: variance, standard deviation aur coefficient of variation