Box-and-whisker plots — quartiles, IQR: Question bank

2.7.4 · D5 · Maths › Statistics & Probability — Intermediate › Box-and-whisker plots — quartiles, IQR

Definition Woh objects jinpe ye traps tikhe hain

Questions se pehle, exactly woh objects pin down karo jo test ho rahe hain — neeche har trap sirf inhi ko assume karta hai, isliye hum inhe yahan build karte hain, parent note ko assume karne ki jagah:

== $Q_{2}$ (the median)== — sorted data ki middle value (ya do middle values ka average agar count $n$ even ho). Aadha data neeche hai, aadha upar.
== $Q_{1}$ (lower quartile)== — lower half ka median, kabhi-kabhi sorted queue mein roughly ek-chauthai upar ki value.
== $Q_{3}$ (upper quartile)== — upper half ka median, roughly teen-chauthai upar.
IQR (interquartile range) $= Q_{3} - Q_{1}$ . Ye box ki width hai: data ke central 50% ka span, $Q_{1}$ se $Q_{3}$ tak.
Fences woh cutoff thresholds hain jo kisi point ko outlier kehne ke liye use hote hain, har edge se डेढ़ box-widths aage measure kiye jaate hain: $Lower fence = Q_{1} - 1.5 IQR, Upper fence = Q_{3} + 1.5 IQR$ Jo point strictly fence ke bahar ho use outlier maana jaata hai; whisker ruk jaati hai fence ke andar last real data value par, kabhi fence par nahi.

Ye picture hai jisko neeche har trap refer karta hai — sorted queue char equal-count chunks mein split hai, box, whiskers, aur ek akela outlier dot:

Aur ye dikhata hai ki ek skewed box ek symmetric ke saath kaisi lagti hai — dekho median line har box ke andar kahan hai:

True ya false — justify karo

Recall Reveal set A

Median hamesha box ke bilkul beech mein hoti hai. ::: False. Median $Q_{2}$ middle value hai, lekin box $Q_{1}$ se $Q_{3}$ tak span karta hai; agar data skewed hai toh median ek edge ke paas ho sakti hai (figure mein right-hand box dekho). Iska off-centre position hi Skewness reveal karta hai. Wider box ka matlab hamesha zyaada data points hain. ::: False. Box width IQR hai (values ka spread), count nahi. Har box hamesha data ke same middle 50% ko rakhta hai, chahe wo kitna bhi wide kyun na ho. Agar do data sets ka IQR same hai toh unka range bhi same hai. ::: False. IQR dono sides ke outer 25% ko ignore karta hai; do sets ka IQR same ho sakta hai phir bhi unka min/max alag ho sakta hai, isliye range bhi alag hogi. Outlier hatane se IQR bahut badal jaata hai. ::: False (usually). Yahi toh ek robust measure ka poora point hai — outliers quartiles ke bahar rehte hain, isliye ek delete karne se $Q_{1}$ ya $Q_{3}$ rarely move hote hain. Dekho Outliers and Robust Statistics. Whiskers hamesha data ke minimum aur maximum tak pahunchti hain. ::: False. Whiskers fences ke andar most extreme value par rukti hain. Agar outliers hain, toh min/max alag dots ke roop mein draw kiye jaate hain (figure mein akela dot dekho) aur whisker unse pehle khatam ho jaati hai. Exactly 25% data points strictly $Q_{1}$ se neeche hain. ::: Roughly, exactly nahi. Ek bade tie-free set mein ye 25% ke karib hota hai, lekin ties ke saath bahut saare points $Q_{1}$ ke barabar ho sakte hain (isliye ek-chauthai se kam strictly neeche hote hain), aur chhote $n$ ke saath count ek exact quarter mein divide nahi ho sakta — jaise $n = 7$ mein lower half mein 3 points hain, jo us half ke $3/7 \approx 43%$ hain, aur sirf poore points cut se neeche gir sakte hain. Quartiles woh positions hain jo counts ko jitna data allow kare utna evenly split karte hain, exact-percentage guarantee nahi hai. Ek box plot bata sakta hai ki data symmetric hai ya nahi. ::: True. Equal whisker lengths aur centred median symmetry suggest karte hain; unequal whiskers Skewness suggest karti hain — ye plot ek visual skew detector hai.

Error dhundho

Recall Reveal set B

"Maine $Q_{1}$ position $n /4$ par value lekar nikala." ::: $n /4$ index aksar points ke beech padhta hai aur convention par depend karta hai. School ka sahi method hai lower half ka median (Tukey's hinges), raw quarter-position count nahi. " $n = 7$ ke liye maine $Q_{1}$ aur $Q_{3}$ nikalte waqt median ko dono halveso mein include kiya." ::: Jab $n$ odd ho toh single middle value kisi bhi half mein nahi aati. Isko include karna ek point ko double-count karta hai aur dono quartiles ko andar shift kar deta hai. "Upper fence $Q_{3} + 1.5 \times Q_{3}$ par hai." ::: Measurement stick IQR $= Q_{3} - Q_{1}$ hai, $Q_{3}$ khud nahi: upper fence $= Q_{3} + 1.5 IQR$ . $Q_{3}$ use karne se fence spread ki jagah location ke saath scale karne lagti hai. "Upper fence par exactly baithne wala point outlier hai." ::: Rule woh points flag karta hai jo fences ke strictly bahar hain. Fence ke barabar value boundary case hai aur flag nahi hoti. "Maine whisker fence tak puri tarah kheechi." ::: Fences invisible cutoffs hain, plotted lines nahi. Whisker fence ke andar last real data value par khatam hoti hai, jo usually fence se pehle hoti hai. "Data $9, 2, 5, 1$ tha isliye median $5$ hai." ::: Pehle sort karna zaroori hai: $1, 2, 5, 9$ se median $\frac{2 + 5}{2} = 3.5$ milta hai. Quartiles position se define hote hain, unsorted data par meaningless hain. "IQR negative aaya, toh maine galat subtract kiya — chaliye isko $Q_{1} - Q_{3}$ mein flip karte hain." ::: Sahi sorted data par $Q_{3} \geq Q_{1}$ hamesha hota hai, isliye IQR $= Q_{3} - Q_{1} \geq 0$ . Negative result matlab computation slip hai, na ki koi formula jo flip karna ho. "Mere calculator ne $Q_{1} = 3.25$ diya lekin textbook mein $3$ hai — hamse se koi galat hai." ::: Zaroori nahi. Software alag quartile-interpolation conventions use karta hai (Why-questions set dekho); is tarah ke mismatches convention differences hain, errors nahi, jab tak har method consistently apply kiya jaaye.

Why questions

Recall Reveal set C

Hum quartiles lene se pehle median par kyun split karte hain, count ko directly quarter karne ki jagah? ::: Median-of-halves rule convention-independent hai aur hamesha well-defined values par land karta hai, jabki raw quarter-indices textbooks aur software mein ek-doosre se alag hote hain. Alag software packages same data ke liye alag quartiles kyun report karte hain? ::: Woh do choices par differ karte hain: (1) kya median khud har half mein include hoti hai ("inclusive", Tukey's hinges) ya exclude hoti hai ("exclusive") jab $n$ odd ho, aur (2) jab quarter-position ek whole index na ho tab points ke beech interpolate kaise karein. Excel ka QUARTILE.INC, QUARTILE.EXC, aur R ke nau "types" sab legitimate hain; school syllabi median-of-halves method ko standardise karti hain ambiguity se bachne ke liye. IQR plain range se spread ka better measure kyun hai? ::: Range sirf do sabse extreme (sabse kam reliable) points use karta hai, isliye ek wild value use inflate kar deta hai. IQR dono sides se outer 25% discard karta hai, use robust banata hai — median ka spread-sibling. Fence rule mein multiplier $1.5$ kyun hai, $1$ ya $3$ nahi? ::: Ye ek tuned convention hai. Roughly normal data par quartiles lagbhag $\pm 0.6745 σ$ par hote hain, isliye $Q_{3} + 1.5 IQR$ par fence centre se lagbhag $0.6745 σ + 1.5 (2 \times 0.6745) σ \approx 2.698 σ$ par land karti hai. Ek side par $2.698 σ$ ke baad ki tail lagbhag $0.35%$ hai, isliye dono tails milaakar roughly $0.7%$ points flag hote hain — notable hone ke liye kaafi rare, par itna tight nahi ki sab kuch trip kare. $3.0$ ka multiplier "extreme" outliers ko far-out mark karta hai. Box ke andar off-centre median hamein skew ke baare mein kyun batata hai? ::: Agar median $Q_{1}$ ke paas hai, toh values low side par pile up hoti hain aur high taraf stretch karti hain (right skew); box ki geometry distribution ki shape mirror karti hai — Skewness aur opening figure mein right-hand box dekho. Incomes ke liye median mean se "fairer centre" kyun hai? ::: Mean ek billionaire se khich jaata hai; median sirf queue ke beech mein khadhne wale ki parwah karta hai, isliye extreme values use distort nahi kar sakti. Dekho Median and Measures of Central Tendency. Quartiles specific percentiles se kyun correspond karte hain? ::: $Q_{1}, Q_{2}, Q_{3}$ exactly 25th, 50th, aur 75th percentiles hain — quartiles simply woh special percentiles hain jo data ko chaar mein cut karte hain.

Edge cases

Recall Reveal set D

Agar data set ki har value identical hai (jaise sab $5$ ) toh $Q_{1}$ aur $Q_{3}$ ka kya hoga? ::: Saare quartiles us value ke barabar ho jaate hain, IQR $= 0$ , aur box ek single line mein collapse ho jaata hai. Plot sahi taur par "no spread" dikhata hai. Kya $Q_{1} = Q_{2} = Q_{3}$ ho sakta hai jab data sab identical nahi hai? ::: Haan. Agar bahut saari values centre par tied hain (jaise $1, 5, 5, 5, 5, 5, 9$ ) toh teeno quartiles coincide ho sakte hain jabki min aur max phir bhi alag ho sakte hain, jo zero-width box aur long whiskers deta hai. Ek single data point ( $n = 1$ ) ke saath quartiles kya hain? ::: Har quartile us ek value par collapse ho jaata hai: $Q_{1} = Q_{2} = Q_{3} =$ woh point, IQR $= 0$ , aur describe karne ke liye koi spread nahi hai. "Box plot" ek single dot hai — degenerate par consistent case. Sirf $n = 2$ data points ke saath quartiles kya hain? ::: Median unka average hai; har "half" ek single value hai, isliye $Q_{1}$ aur $Q_{3}$ simply chhota aur bada point hain. Box poore (chhote) data set ko span karta hai. Agar lower fence negative aaye lekin saara data positive hai, kya iska matlab koi error hai? ::: Nahi. Negative fence simply matlab hai ki koi value itni low nahi hai ki low-outlier ban sake; fence ek threshold hai, data value nahi, aur legitimately minimum se neeche ja sakti hai. Kya ek single extreme value ek side par outlier ho sakti hai jabki doosri side ki whisker bilkul normal ho? ::: Haan. Fences independently har side ke liye compute hoti hain; ek right-skewed set sirf high end par outliers dot kar sakta hai jabki low whisker normally apne minimum tak pahunche. Agar exactly aadha data maximum value par hai, toh median kahan padega? ::: Median us maximum ke barabar ho sakta hai, isliye $Q_{2}$ aur $Q_{3}$ dono right edge par baith sakte hain — box bahut heavily lopsided hai, strong left-to-right pile-up signal karta hai.

Connections

Box-and-whisker plots — quartiles, IQR — parent topic jise ye traps stress-test karte hain.
Median and Measures of Central Tendency — kyun median outliers resist karta hai.
Range and Spread — woh range jise ye traps IQR se contrast karte hain.
Outliers and Robust Statistics — fence rule aur robustness yahan rehti hai.
Percentiles and Quantiles — quartiles as 25/50/75 percentiles.
Skewness — box asymmetry padhna.
Normal Distribution — $1.5$ multiplier ki tail percentage ka origin.