Box-and-whisker plots — quartiles, IQR: Foundations

2.7.4 · D1 · Maths › Statistics & Probability — Intermediate › Box-and-whisker plots — quartiles, IQR

Is page mein assume kiya gaya hai ki aapko kuch bhi pata nahi. Parent note Box-and-whisker plots — quartiles, IQR padhne se pehle, aapko ideas ka ek chhota sa toolkit chahiye. Hum har ek ko zero se banate hain, usse ek picture se jodte hain, aur batate hain kyun is topic ko yeh chahiye. Baad ka har idea sirf pehle waale ideas pe tikta hai.

0. "Data" aur "ek value" — hum hold kya kar rahe hain?

Symbols se pehle, raw material.

Picture: sochiye har value ek manka (bead) hai. Aapka poora data set muththi bhar loose mankyon ka hai.

$n$ hamara sabse pehla symbol hai.

1. Sorting — ek dheer ko queue mein badalna

Is poore topic ka sabse important move.

Figure — Box-and-whisker plots — quartiles, IQR

Figure dekho. Upar, manke ek ulajhi hui dheer mein hain (unsorted). Neeche, wahi manke ek queue mein khade hain, sabse chhota left pe, sabse bada right pe. Us rearrangement ko ascending sorting kehte hain.

2. Position vs value — har manke ke paas yeh do cheezein hoti hain

Yeh subtle hai aur iske liye apni picture worth hai.

Sorted queue mein har manke ke paas do alag numbers hote hain:

uski position — woh line mein kahan khada hai: 1st, 2nd, 3rd, … (neeche line ke yellow mein dikhaya gaya).
uski value — woh kaunsa number hai (manke ke andar blue mein dikhaya gaya).

3. Queue mein kitne fraction aage — "aadha", "ek-chauthai", "teen-chauthai"

Ek box plot queue ko $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{3}{4}$ pe mark karta hai. Isliye hume bilkul clear hona chahiye ki queue ka ek fraction matlab kya hota hai.

Picture: sorted queue mein kharo. Tab tak chalo jab tak theek aadhe log aapke peechhe hon — woh jagah halfway mark hai. Sirf ek-chauthai bheed tak chalo — woh quarter mark hai.

4. Median $Q_{2}$ — halfway value

Ab hum halfway mark ka naam de sakte hain.

Isko kaise dhundhein yeh depend karta hai ki $n$ odd hai ya even — kyunki "bilkul beech" tabhi exist karta hai jab count odd ho.

$n$ odd (upar wali row, 7 manke): ek akela beech wala manka hai. Uski position hai $\frac{n + 1}{2} = \frac{7 + 1}{2} = 4$ , isliye median 4th value hai.
$n$ even (neeche wali row, 6 manke): koi single manka beech mein nahi hai — centre do mankyon ke beech padata hai (3rd aur 4th). Median unka average hai.

5. Quartiles $Q_{1}$ aur $Q_{3}$ — quarter aur three-quarter values

"Quart" matlab chaar. Quartiles woh marks hain jo queue ko chaar equal-count groups mein kaatte hain.

Yeh teeno symbols exactly 25th, 50th, aur 75th Percentiles and Quantiles hain — quartiles sirf nice quarter-marks pe percentiles hain.

6. Subtraction "distance" ke roop mein — IQR banana

Parent note ka spread measure hai $Q_{3} - Q_{1}$ . Subtract kyun karte hain?

7. Multiplier — " $1.5 \times IQR$ " aur fences

Aakhri symbol unpack karna: number $1.5$ aur word "fence".

Yeh topic ko kaise feed karte hain

Equipment checklist

Recall Kya aap bina dekhe har ek ka jawab de sakte hain?

Symbol $n$ kya represent karta hai? ::: Data set mein kitni values hain unka count. Quartiles dhundhne se pehle kisi bhi data ke saath aapko pehle (Step 0) kya karna chahiye? ::: Use ascending sort karo (chhote se bade tak). Kisi value ki position aur uski value mein kya farq hai? ::: Position woh hai jahan woh sorted queue mein khada hai (1st, 2nd, …); value woh actual number hai jo woh hai. Queue mein "ek-chauthai raasta" ka matlab kya hai? ::: Har chaar mein se ek manka aapke left mein hai, teen right mein. Median $Q_{2}$ kya hai aur odd vs even $n$ ke liye isko dhundhna kaise alag hota hai? ::: Halfway value; odd $n$ ke liye yeh position $(n + 1) /2$ pe single middle value hai, even $n$ ke liye yeh do middle values ka average hai. $Q_{1}$ aur $Q_{3}$ ek-ek phrase mein kya hain? ::: $Q_{1}$ = lower half ka median; $Q_{3}$ = upper half ka median. $b - a$ spread kyun measure karta hai? ::: Yeh number line pe distance/gap hai — bada gap matlab zyada spread out. IQR formula likhो. ::: $IQR = Q_{3} - Q_{1}$ . Outlier fences set karne ke liye IQR ko (fixed number ki jagah) multiply kyun karte hain? ::: Taaki "door" wala yardstick data ke apne spread ke saath scale kare; har edge se $1.5 IQR$ door.