Visual walkthrough — Measures of central tendency — mean (grouped - ungrouped), median (grouped), mode (grouped)

2.7.1 · D2 · Maths › Statistics & Probability — Intermediate › Measures of central tendency — mean (grouped - ungrouped), m

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Step 1 — Hum actually dhundh kya rahe hain

KYA. Median woh value hai jiske neeche aadha data ho aur aadha upar. Agar mere paas $n$ log hain, toh mujhe dhundhna hai ki beech wala banda kahan khada hai.

KYU. Ungrouped data mein main simply sort karke beech wale ki taraf point kar sakta tha. Lekin grouped data individuals ko chhupa deta hai — mujhe sirf itna pata hai ki kitne log har price/height/mark band mein aate hain, unki exact values nahi. Toh "beech wale ki taraf point karna" ek trick maangta hai.

PICTURE. Imagine karo sab log sabse chote se sabse lambe tak line mein khade hain. Median us insaan ki height hai jo line ke halfway mark par khada hai.

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Step 2 — Line ko class-boxes mein kaat lo

KYA. Group data humein ek lamba ordered line nahi deta; yeh humein boxes deta hai. Har box (class) value-axis par width $h$ ka ek stretch hai, jisme $f_i$ log hain.

KYU. Bas itni hi information hamare paas hai. Box kehta hai "value $l$ aur value $l+h$ ke beech kahin $f_i$ log hain" — lekin kahan box ke andar har ek khada hai, yeh nahi batata. Woh missing detail exactly wahi hai jo hum baad mein assume karenge.

PICTURE. Wahi line, ab value-axis ke saath labelled boxes mein kati hui.

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Step 3 — Running total banao (cumulative frequency)

KYA. Boxes ko left se right walk karo, track rakhte hue ki running total mein kitne log past ho gaye hain. Woh running total cumulative frequency $cf$ hai.

KYU. Running total ek position counter hai. Jab yeh pehli baar $\frac{n}{2}$ tak pahunchta hai, main beech wale insaan ko past kar chuka hun. Jo bhi box main us waqt khada hun wahan median rehti hai.

PICTURE. Boxes jiske upar cumulative counts ki ek urti seedhi hai. Dashed line $\frac{n}{2}$ mark karti hai.

Us box ke pieces ko naam do, aur uske predecessor ko:

$$\underset{\begin{array}{c}\text{lower boundary}\\ \text{of median box}\end{array}}{\underbrace{l}}\underset{\begin{array}{c}\text{people inside}\\ \text{median box}\end{array}}{\underbrace{f}}\underset{\begin{array}{c}\text{running total}\\ \text{just BEFORE median box}\end{array}}{\underbrace{cf}}\underset{\text{box width}}{\underbrace{h}}$$

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Step 4 — Beech tak pahunchne ke liye kitne aur log chahiye?

KYA. Jab main median box ke left edge par pahunchta hun, mera counter $cf$ read karta hai. Main abhi middle tak nahi pahuncha. Jo log abhi bhi chahiye unki sankhya hai $\frac{n}{2}-cf.$

KYU. $cf$ pehle ke boxes mein sabko count karta hai. Beech wala insaan $\frac{n}{2}$ par baitha hai. Unke beech ka gap, $\frac{n}{2}-cf$, yeh hai ki mujhe is box ke andar kitna aur chalna hai.

PICTURE. Median box par zoom in: left edge wahan hai jahan counter $cf$ read karta tha; ek chhota red arrow jiski length $\frac{n}{2}-cf$ hai woh box ke andar age beech wale insaan ki taraf point karta hai.

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Step 5 — Ek assumption: box ko evenly spread karo

KYA. Median box ke andar hum nahi jaante ki $f$ logon mein se har ek kahan khada hai. Toh hum assume karte hain ki woh evenly spaced hain poori width $h$ mein.

KYU. Yeh wahi honest guess hai jab detail missing ho — crowd ko uniform treat karo. Yeh "kaunsi value?" ko simple proportion mein badal deta hai. Yahi assumption median formula ko linear interpolation banati hai.

PICTURE. Median box mein $f$ evenly spaced tick-people. Har insaan width $\frac{h}{f}$ ka ek slice leta hai.

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Step 6 — Andar chalo, aur value padho

KYA. Beech wale insaan tak pahunchne ke liye main $\left(\frac{n}{2}-cf\right)$ logon ko step karta hun, jisme se har ek width $\frac{h}{f}$ rakhta hai. Toh main advance karta hun $\left(\frac{n}{2}-cf\right)\times \frac{h}{f}$ left edge $l$ se.

KYU. Distance = (logon ki sankhya) × (width jo har insaan owns karta hai). Woh distance starting value $l$ mein add karo aur tum exactly median value par land karte ho.

PICTURE. Left edge $l$ par; ek green advance arrow jitni lambayi upar di gayi hai; tip median value par land karta hai.

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Step 7 — Edge aur degenerate cases

KYA. Corners check karo taaki koi scenario surprise na kare.

KYU. Ek formula jis par tum trust karte ho use apne extremes survive karne chahiye.

PICTURE. Median box teen baar dikhaya gaya: middle-person left door par (a), right wall par (b), aur odd $n$ ke liye fractional position (c).

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Ek-picture summary

Upar sab kuch ek single ogive-style picture mein collapse hota hai: staircase ko $\frac{n}{2}$ height tak chadhao, median box ke andar neeche utaro, aur axis se value padho. Woh drop hi formula hai.

Recall Feynman retelling — plain words mein khud bolke batao

Sabko chote se lambe tak line mein khado aur pucho, "queue ke exact middle mein kaun khada hai?" Grouped data mujhe har insaan ki spot nahi batata — sirf itna batata hai ki har price-band box mein kitne hain. Toh main boxes ko left se right walk karta hun, running count tally karta hun. Jis moment mera count halfway cross karta hai, $n/2$, main jaanta hun beech wala insaan is box mein hai. Main check karta hun ki box mein enter karne ke baad mujhe kitne aur log chahiye the, $n/2-cf$. Mujhe nahi pata ki woh andar kaise arranged hain, toh main fairly assume karta hun ki woh evenly spread hain — har ek width $h/f$ ka slice le raha hai. Main exactly utne logon ko step karta hun jitni mujhe abhi bhi zarurat thi, har step ek slice wide, aur jahan main rukta hun woh median value hai. Likhne par, woh stroll hai $l+\frac{n/2-cf}{f}h$ — darwaze $l$ se shuru, box ke andar fraction of the way chalo, box width $h$ se scale kiya.

Recall Quick self-check

Grouped data ke liye hum $n/2$ kyun use karte hain na ki $(n+1)/2$? ::: Grouped data ko continuous crowd treat kiya jata hai, toh halfway position $n/2$ hai; fractional positions natural hain, integer rounding ki zarurat nahi. Numerator $\frac{n}{2}-cf$ physically kya represent karta hai? ::: Un logon ki sankhya jinhe tumhe middle tak pahunchne se pehle median class mein enter karne ke baad bhi walk karna hai. Agar $\frac{n}{2}=cf$ exactly ho, toh median kya hai? ::: $l$ — middle bilkul median class ki left boundary par baitha hai.

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Yahan se aage: median ka cousin, distribution ka skew, tumhe batata hai ki mean aur median kab alag ho jaate hain; spread batata hai ki crowd is centre ke around kitna tight hai.