Foundations — Measures of dispersion — variance, standard deviation

2.7.3 · D1 · Maths › Statistics & Probability — Intermediate › Measures of dispersion — variance, standard deviation

Variance ko samajhne se pehle, usse padhna aana chahiye. Parent note kaafi saare chhote symbols bahut tezi se throw karta hai: $\sum$, $\bar{x}$, $x_i$, $n$, ${\sigma }^2$, $\mu$, $\overline{x^2}$. Agar inme se ek bhi blur hai, toh derivations spellcasting lagne lagti hain. Yeh page har ek ko absolute zero se define karta hai, use ek picture se anchor karta hai, aur batata hai kyun topic uske bina nahi chal sakta.

Raw material: ek dataset

Sab kuch ek un numbers ki list se shuru hota hai jo tumne actually measure ki hain — test marks, dart distances, heights, kuch bhi.

Figure dekho. Number line par har dot ek $x_i$ hai. Subscript mein kuch bhi darne wali baat nahi hai — yeh bilkul waise hi hai jaise ek row mein students ko number karo: "student 1, student 2, ..." $x$ batata hai kis cheez ka, subscript batata hai kaun sa specific wala.

Sab kuch jodna: sigma symbol $\sum$

$x_1+x_2+x_3+x_4+x_5$ likhna paanch numbers ke liye theek hai, lekin paanch sau ke liye bahut mushkil hai. Mathematicians ne ek single symbol invent kiya jiska matlab hai "puri sequence ko jodo."

Figure dekho. Counter $i$ neeche chalta hai: $1,2,3,\dots$ Har step par woh $x_i$ ki value uthata hai aur ek running total mein daal deta hai. Jab counter $n$ cross kar leta hai, ruk jaata hai aur final sum deta hai.

Centre: mean $\bar{x}$ (aur uska cousin $\mu$)

"Numbers middle se kitne door bhatak jaate hain?" poochne ke liye pehle middle chahiye.

Figure dekho. Socho number line ek see-saw hai aur har data value ek weight hai jo jahan land hoti hai wahan baith jaati hai. Mean bilkul woh balance point hai — woh pivot jahan plank na left jhuke na right. Right door ke values left door ke values se counterweight hoti hain.

Gap: deviation $d_i=x_i-\bar{x}$

Ab jab centre mil gaya, hum har value se pooch sakte hain: tum usse kitni door ho?

Figure dekho. Har colored arrow mean se ek data point tak jaata hai. Right taraf point karne wale arrows positive deviations hain (orange), left taraf point karne wale negative hain (blue). Arrow ki length batati hai woh value kitni door hai; direction sign hai.

Do easily-confuse-hone wale averages: $\overline{x^2}$ vs $(\bar{x}{)}^2$

Parent note mein shortcut formula inhe alag karne par jeeti-haari hai.

Final symbols: ${\sigma }^2$ aur $\sigma$

Sab kuch topic ko kaise feed karta hai

Upar se neeche padho: raw data ko mean banane ke liye count aur sum chahiye; mean se deviations measure kar sakte hain; unhe square aur sum karne se variance milta hai; ek square root se standard deviation milta hai, jo phir Coefficient of Variation aur Normal Distribution ko power karta hai.

Equipment checklist

Har ek reveal karo aur check karo ki parent note tackle karne se pehle instantly answer de sako.