Cumulative frequency — ogive, median from graph: Foundations

2.7.2 · D1 · Maths › Statistics & Probability — Intermediate › Cumulative frequency — ogive, median from graph

Is curve par chadhne se pehle, hume har woh symbol earn karna hoga jo parent topic mein aata hai. Hum inhe ek-ek karke build karenge — koi bhi symbol yahaan tabhi aayega jab usse words aur ek picture mein define kar diya jaaye.

0. Kaccha maal: ek observation aur ek count

Figure 1 dekho: har observation ek dot hai. Hum unhe buckets mein sweep karte hain aur count karte hain. Woh count hi $f$ hai.

Figure — Cumulative frequency — ogive, median from graph

1. Classes aur unki boundaries

Figure 2 mein har bucket number line par ek strip hai. Left edge lower boundary hai; right edge upper boundary hai; beech ka distance $h$ hai.

Topic upper boundary ko itna important kyun maanta hai? Kyunki hum abhi ek line se neeche sab kuch count karne wale hain — aur us line ko rakhne ki natural jagah bucket ka right edge hai.

2. Running total: cumulative frequency

Hum jaante hain $f$ = "is bucket mein kitne." Ab hum alag sawaal poochhte hain: is bucket mein aur iske left ke saare buckets mein kitne hain? Woh running total hi is poore topic ka star hai.

Symbol $\sum$ (ek bada Greek "S", Sum ke liye) bas "inhe jodo" ka shorthand hai:

$C F_{i} = f_{1} + f_{2} + \dots + f_{i} = \sum_{j = 1}^{i} f_{j}$

Right-hand side ko zor se padho: " $j = 1$ se shuru karo, har $f_{j}$ ko $j = i$ tak jodo." Chhota $j$ bas ek counter hai jo buckets mein chalata hai. Kuch mysterious nahi — yeh ek running total hai, jaise shopping receipt ko line by line jodna.

Figure 3 receipt wali idea dikhata hai: har bar ki height $f$ hai; unke peeche ki staircase accumulate hoti $C F$ hai.

3. $N$ aur halfway mark $N /2$

4. Median — jise hum dhoondh rahe hain

Ogive par, median woh x-value hai jo seedha us point ke neeche hai jahan curve height $N /2$ tak pahunchti hai. Upar jo kuch bhi banaya, uska poora payoff yahi hai.

Yeh raha aakhiri tool. Jab halfway height $N /2$ ek bucket ke andar land kare, toh ogive jump nahi karta — woh slope karta hai. Slope se value padhne ke liye hum maan lete hain ki observations bucket mein evenly spread hain, aur proportion mein slide karte hain. Woh even-sharing rule Linear interpolation hai.

$fraction into the bucket = \frac{\frac{N}{2} - C F _{b}}{f}$

jahan $C F_{b}$ is bucket se just pehle ka running total hai aur $f$ is bucket ka apna count hai. Woh fraction ko width $h$ se multiply karo aur lower boundary $L$ mein jodo:

$Median = L + (\frac{\frac{N}{2} - C F _{b}}{f}) \times h$

Yahaan har symbol ab earned hai: $L$ = median class ki lower boundary, $C F_{b}$ = usse pehle ka running total, $f$ = uska count, $h$ = uski width, $N /2$ = halfway height.

Prerequisite map

Har box ek symbol hai jo humne upar banaya; arrows follow karo aur tum ogive se median par pahunch jaoge. Usi curve se tum baad mein Quartiles and percentiles from ogive padh sakte ho, aur median ko Mean and mode of grouped data se compare kar sakte ho.

Equipment checklist

Jawaab chhupao aur khud test karo — agar koi line rokti hai, us section ko dobara padho.

Frequency

f

of a class ka kya matlab hai?

Kitni observations us class mein aati hain (uska count).

Class ka upper boundary kya hai, aur ogive use kyun karta hai?

Bucket ka right edge;

C F

usse neeche sab kuch count karta hai, isliye point honestly us x-value par baitha hota hai.

\sum_{j = 1}^{i} f_{j}

tumhe kya karne ko kehta hai?

f_{1}

f_{i}

tak jodo — ek running total.

C F

ke do anchor facts kya hain?

C F_{1} = f_{1}

aur

C F_{k} = N

N

kya hai?

Observations ki total sankhya, saari frequencies ka sum.

Ogive par height

N /2

tak kyun chadhte hain (

(N + 1) /2

nahi)?

Ogive ek smooth continuous curve hai, isliye hum halfway height use karte hain, discrete position nahi.

Median formula mein

C F_{b}

kya hai?

Median class se just pehle wali class ki cumulative frequency.

Linear interpolation median class ke andar kya assumption karta hai?

Observations uniformly spread hain, isliye curve bucket ke across ek straight ramp hai.

10 - 19

jaisi discontinuous classes ko kaise fix karte ho?

Lower se

0.5

ghataao aur upper mein

0.5

jodao →

9.5 - 19.5