Mean (ausat) kya hai? Ek santulan bindu
Short mein jawab
Mean wo akela bindu hai jahaan neeche ki values tak ki kul doori upar ki values tak ki kul doori ke barabar hoti hai. Mean se badi value jodne par mean badhta hai aur chhoti se ghatta hai; har value mein ek sankhya jodne par mean utna badhta hai, aur har value do guni karne par mean do guna.
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Do sankhyaon se kai sankhyaon tak
Koi do sankhyaayein lekar unka mean nikaaliye: 3 aur 7 ka (3 + 7) ÷ 2 = 5 hai, aur 8 aur 9 ka 8.5. Dot plot par mean hamesha dono ke theek beech aata hai. Teen sankhyaon mein bhi wo ‘kendra’ hai, par hamesha chhoron ka beech nahi. Book ke dot plot dikhaate hain: 6, 7, 8 ka mean 7; 3, 6, 9 ka 6; 2, 4, 9 ka 5; aur 4, 11, 15 ka 10. Chaar values mein 11, 13, 17, 19 ka mean 15 hai aur 5, 6, 15, 16 ka 10.5.
Dooriyon ka santulan
Mean ko kendra kya banaata hai? Har value ki mean se doori naapiye. 2, 4 aur 9 mein, mean 5 se, neeche ki values 3 aur 1 door hain, kul 4, aur upar ki value 4 door hai. 10, 10, 11 aur 17 mein, mean 12 se, neeche ki dooriyaan 2 + 2 + 1 = 5 hain aur upar 5. Baayin (LHS) ki kul doori hamesha daayin (RHS) ki kul doori ke barabar hoti hai: mean santulan bindu hai.
Kya koi aur bindu bhi santulan kar sakta hai? 10, 10, 11, 17 ke liye 13 aazmaaiye: baayin ki har doori badhti hai aur daayin ki doori ghatti hai, isliye jod barabar nahi rehte. 12 se neeche ka koi bhi bindu ulti taraf se fail hota hai. Isliye santulan bindu ek hi hai.
Value jodna aur hataana
Agar mean se badi nayi value judti hai, to daayin taraf doori badhti hai, isliye santulan bindu ko upar khisakna padta hai: mean badhta hai. Mean se chhoti value use neeche kheenchti hai, aur mean ke barabar value koi doori nahi jodti, isliye kuch nahi badalta. Value hataane mein ulta hota hai: mean se badi value hataane par mean ghatta hai. Barabar hisse ki nazar se, jab barabar hisse se zyada laane waale log judte hain, to sabka hissa badh jaata hai.
Kai values jodne par bhi mean tay reh sakta hai, agar unki dooriyaan santulit hon. Mean 9 ho to 7 aur 11 (2 neeche, 2 upar) jodne par wo 9 hi rehta hai, aur 7, 7 aur 13 (2 + 2 neeche, 4 upar) jodne par bhi. Do upar aur ek neeche bhi chalta hai, jaise 10 aur 11 ke saath 6: upar 1 + 2, neeche 3.
Har value badalna
Agar har value ek hi maatra se badhe to? 8, 3, 10, 13, 4, 6, 7, 7, 8, 8, 5 ka mean 79 ÷ 11 ≈ 7.18 hai. Har value mein 10 jodiye to poora dot plot 10 daayin khisak jaata hai, aur naya mean lagbhag 17.18 hota hai; 1 ghataaiye to lagbhag 6.18. Algebra wajah dikhaata hai: agar n values ka jod n × a hai, to har mein 3 jodne se jod mein 3n judta hai, aur n se bhaag dene par a + 3 milta hai. Har value do guni ho to mean bhi do guna hota hai: har value ko 5 se guna karne par jod, aur isliye mean, 5 guna ho jaata hai.
Mean se anjaani value nikaalna
Kyonki jod = mean × values ki sankhya, mean chhupi sankhya bata sakta hai. Coach Balwan ke das pehalwaanon ka ausat vazan 39.2 kg hai, yaani kul 392 kg; padhne laayak nau vazan 349 kg jodte hain, to mita hua vazan 43 kg hai. Venkayya ke 15 naariyal ke pedon ka ausat 25.6 hai, kul 384; ek ginti 3 zyada thi, to sahi kul 381 aur sahi mean 25.4 hai.
Aur padhein: line graph kaise padhein?, frequency table se mean aur median, chapter ke notes aur NCERT solutions.
Practice karo
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3, 6 aur 9 ka mean hai:
Jawab dekho
Jawab: (b) 6
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Mean 9 wale data mein nayi value 13 judti hai. Naya mean hoga:
Jawab dekho
Jawab: (c) 9 se zyada
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8, 3, 10, 13, 4, 6, 7, 7, 8, 8, 5 ka mean lagbhag 7.18 hai. Har value do guni karne par mean lagbhag hoga:
Jawab dekho
Jawab: (c) 14.36
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Kya do alag bindu dooriyon ko santulit kar sakte hain? Samjhaaiye.
Jawab dekho
Jawab: Nahi. Bindu ko upar khiskaayein to neeche ki har doori badhti hai aur upar ki har doori ghatti hai, isliye dono jod barabar nahi reh sakte; neeche khiskaane par ulta hota hai.
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6, 8, 9, 10, 12 mein teen aisi values bataaiye jinhe jodne se mean na badle.
Jawab dekho
Jawab: Jaise 7, 7 aur 13. Mean 9 hai; 7 aur 7 usse 2 + 2 = 4 neeche hain aur 13 usse 4 upar, isliye santulan bana rehta hai.
Students ke sawal
Mean ko data ka kendra kyon kehte hain?
Kyonki ye data ko santulit karta hai: mean se chhoti values tak ki kul doori badi values tak ki kul doori ke barabar hoti hai, jaise ek bindu par tika see-saw.
Value hataane par mean ka kya hota hai?
Mean se badi value hataane par mean ghatta hai, chhoti value hataane par badhta hai, aur mean ke barabar value hataane par waisa hi rehta hai.
Har value ko ek hi sankhya se guna karne par mean ka kya hota hai?
Mean bhi usi sankhya se guna ho jaata hai. Agar values ka jod n × a hai, to har value ko 5 se guna karne par jod 5 × n × a ho jaata hai, isliye naya mean 5a hai.
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