Box-and-whisker plots — quartiles, IQR — Hinglish notes

2.7.4 · Maths › Statistics & Probability — Intermediate

Quartiles ki ZAROORAT kyun hai?

Mean tumhe ek "centre" batata hai, lekin jab data skewed ho ya outliers hon toh woh tumse jhooth bolta hai (ek billionaire average income ko barbad kar deta hai). Hum chahte hain ek aisi description jo robust ho — kuch extreme values se prabhavit na ho.

Idea: average nikaalane ki jagah, bas data ko sort karo aur positions mark karo. Agar tum incomes ki ek queue mein chhote se bade ki taraf khade ho, toh "bilkul beech wala aadmi" (the median) ek fair centre hai chahe sabse ameer kitna bhi ameer kyun na ho.

Idea ko aage badhao: 1/4 raaste par aur 3/4 raaste par khade vyakti ko mark karo. Ab tumhare paas teen cut points hain jo queue ko chaar equal parts mein divide karte hain. Woh cut points hi quartiles hain.

Quartiles compute karne ka TARIKA (scratch se)

Step 0 — sort karo data ko ascending order mein. (Neeche sab kuch isi assumption par hai.)

Step 1 — median $Q_{2}$ :

$n$ odd ho → middle value, position $\frac{n + 1}{2}$ .
$n$ even ho → do middle values ka average.

Step 2 — halves mein split karo. Lower half = median position ke neeche ki saari values; upper half = median position ke upar ki saari values.

Agar $n$ odd hai, toh median khud kisi bhi half mein nahi rakha jaata.
Agar $n$ even hai, toh dono halves mein exactly $n /2$ values aati hain.

Step 3 — $Q_{1}$ = lower half ka median, $Q_{3}$ = upper half ka median.

IQR — beech ke 50% ka spread

Outliers detect karna — 1.5 × IQR fences

Hume ek rule chahiye "pack se unusually door" ke liye. Tukey ka rule IQR ko ek natural yardstick ki tarah use karke ek fence banata hai.

1.5 kyun? Yah ek convention hai jo isliye chuna gaya taaki roughly normal data ke liye, sirf lagbhag 0.7% points flag hon — itna rare ki interesting lage, itna strict bhi nahi ki kuch naa dikhe. (Multiplier 3.0 "far out" extreme outliers mark karta hai.)

Plot ki Anatomy

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

Left whisker end → sabse chhota non-outlier
Box ka left edge → $Q_{1}$
Box mein line → median $Q_{2}$
Box ka right edge → $Q_{3}$
Right whisker end → sabse bada non-outlier
Whiskers ke baad dots → outliers

Worked Example 1 — odd $n$

Data: $7, 2, 9, 4, 5, 12, 3$ ( $n = 7$ )

Sort karo: $2, 3, 4, 5, 7, 9, 12$ — Kyun? Quartiles position se define hote hain, unsorted mein meaningless hain.

Median $Q_{2}$ : position $\frac{7 + 1}{2} = 4$ → value $5$ . Kyun? 7 items ka middle.

Lower half (median ke neeche, ise exclude karo): $2, 3, 4$ . → $Q_{1} = 3$ . Upper half: $7, 9, 12$ . → $Q_{3} = 9$ . Median ko exclude kyun karein? $n$ odd hai, toh woh kisi bhi half mein nahi aata.

IQR $= 9 - 3 = 6$ .

Fences: $3 - 1.5 (6) = - 6$ ; $9 + 1.5 (6) = 18$ . Koi value bahar nahi → koi outlier nahi.

Worked Example 2 — even $n$ + ek outlier

Data (sorted): $1, 3, 3, 4, 5, 6, 6, 7, 8, 30$ ( $n = 10$ )

Median: 5th aur 6th values ka average $= \frac{5 + 6}{2} = 5.5$ . Kyun? Even $n$ , koi single middle nahi.

Lower half (pehle 5): $1, 3, 3, 4, 5$ → $Q_{1} = 3$ (iska median). Upper half (aakhri 5): $6, 6, 7, 8, 30$ → $Q_{3} = 7$ . 5/5 split kyun? Even $n$ cleanly divide hota hai.

IQR $= 7 - 3 = 4$ .

Upper fence $= 7 + 1.5 (4) = 13$ . Value $30 > 13$ → outlier! Yah matter kyun karta hai? Right whisker $8$ par rukti hai (sabse bada value ≤ 13), aur $30$ ek akele dot ki tarah draw hota hai.

Forecast-then-Verify

Recall Feynman: ek 12-saal ke bachche ko samjhao

Apne saare doston ko chhote se bade ki taraf line mein khada karo. Bilkul beech wala bachcha median hai. Ab chhote wale half ka middle aur bade wale half ka middle dhundho — yeh dono sab ko chaar equal groups mein split karte hain. Box plot sirf ek box hai jo pehle split se teesre split tak draw ki gayi hai, beech mein ek line ke saath. Whiskers baahon ki tarah sabse chhote aur sabse bade bachche tak pahunchti hain jo weirdly extreme nahi hain. Agar koi bahut zyada door ho (jaise ek giant), toh hum use ek akele dot ki tarah draw karte hain — woh outlier hai.

Recall checklist

Connections

Median and Measures of Central Tendency — $Q_{2}$ median hai.
Range and Spread — IQR, range ka robust cousin hai.
Outliers and Robust Statistics — 1.5×IQR rule yahaan rahta hai.
Percentiles and Quantiles — quartiles, 25/50/75 percentiles hain.
Skewness — box ki asymmetry skew reveal karti hai.
Normal Distribution — 1.5 kyun ≈0.7% flag karta hai.

Ek data set ka median

Q_{2}

kya hota hai?

Sorted data ki middle value (ya do middle values ka average agar

n

even ho).

Q_{1}

kaise dhundhte hain?

Data sort karo, median dhundho, phir lower half ka median lo (median khud ko exclude karo agar

n

odd ho).

IQR define karo.

IQR = Q_{3} - Q_{1}

— data ke central 50% ka range.

Spread ke liye IQR ko range par kyun prefer kiya jaata hai?

Yah dono taraf ke extreme outer 25% ko ignore karta hai, isliye outliers ke liye robust hai, jabki range sirf max aur min use karta hai.

Outlier fences batao.

Lower

= Q_{1} - 1.5 IQR

; Upper

= Q_{3} + 1.5 IQR

; bahar ke points outliers hain.

Whisker kahaan khatam hoti hai?

Sabse extreme data value par jo abhi bhi fences ke andar ho — fence par nahi.

2, 3, 4, 5, 7, 9, 12

ke liye

Q_{1}, Q_{2}, Q_{3}

dhundho.

Q_{2} = 5

Q_{1} = 3

Q_{3} = 9

, IQR

= 6

Outlier rule mein multiplier 1.5 kyun?

Convention: roughly normal data ke liye yah sirf ~0.7% points flag karta hai — itna rare ki notable ho lekin over-sensitive na ho.