Cumulative frequency — ogive, median from graph

2.7.2 · Maths › Statistics & Probability — Intermediate

Cumulative frequency KYA hai?

Do types hoti hain:

Less-than CF: upper boundary tak ki saari frequencies ka total. Jab aap daayein jaate ho toh badhti hai.
More-than CF: lower boundary se aage ki saari frequencies ka total. Jab aap daayein jaate ho toh ghatti hai.

CF table KAISE banayein (first principles se)

Maano classes ki frequencies $f_{1}, f_{2}, \dots, f_{k}$ hain. Less-than CF accumulation se define hoti hai:

$C F_{1} = f_{1}, C F_{i} = C F_{i - 1} + f_{i}$

Toh is recurrence ko unroll karne par:

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

Last value hamesha $C F_{k} = N$ hoti hai, jo observations ki total sankhya hai. Yahi aapka sanity check hai.

Zaruri rule — upper boundary kyun use karein? Less-than ogive ke liye, CF ko upper class boundary ke against plot karo, midpoint ke against nahi. KYO? Kyunki $C F_{i}$ class $i$ ki upper boundary se neeche ki sabhi cheezein count karta hai — isliye point $(upper boundary, C F_{i})$ yeh saaf kehta hai ki "itne values is x se neeche hain."

Curve pehli class ki lower boundary par CF $= 0$ se shuru hoti hai aur $N$ tak uthti hai.

Figure — Cumulative frequency — ogive, median from graph

Graph se median KAISE nikaalein

Steps:

$N /2$ calculate karo (grouped/continuous data ke liye $N /2$ use karo, $(N + 1) /2$ nahi — neeche mistake dekho).
Y-axis par $N /2$ mark karo. Curve tak ek horizontal line kheencho.
X-axis par ek vertical line chhaado. Value padho → wahi median hai.

Twin-ogive method: Less-than aur more-than dono ogives ko same axes par draw karo. Jahan woh intersect karein, wahan x-axis par ek perpendicular chhaao — woh x-value median hai.

Graph ke peeche ka formula (derivation)

Graph median class ke andar linear interpolation hai. Chaliye derive karte hain.

Median class woh class hai jahan CF pehli baar $N /2$ tak pahunchti ya usse zyada hoti hai. Maano us class ke andar frequency uniformly faili hui hai. Maano:

$L$ = median class ki lower boundary
$C F_{b}$ = median class se pehle wali class ki cumulative frequency
$f$ = median class ki frequency
$h$ = class width

Hume CF $= C F_{b}$ (at $x = L$ ) se $N /2$ tak jaana hai. Extra count ki zarurat hai $(\frac{N}{2} - C F_{b})$ . Kyunki $f$ observations width $h$ mein evenly faile hain, x ka har unit count mein $f / h$ badhata hai. Toh horizontal distance yeh hai:

$Δ x = \frac{\frac{N}{2} - C F _{b}}{f / h} = (\frac{\frac{N}{2} - C F _{b}}{f}) h$

Worked Examples

Worked example Example 1 — CF banao aur median padho

40 students ke marks:

Marks	Frequency $f$
0–10	4
10–20	6
20–30	10
30–40	12
40–50	8

Less-than CF table:

Less than	CF
10	4
20	10
30	20
40	32
50	40

Yeh step kyun? Har CF = pichli CF + current frequency (running total). Final CF $= 40 = N$ . ✓

Median: $N /2 = 20$ . CF 20 se pehle 10 hai (at "less than 20"), aur CF 20–30 class ki top par 20 tak pahunchti hai — toh median class 20–30 hai (yeh woh class hai jahan running total pehli baar $N /2$ tak pahunchti hai). Kyun? Median class woh hai jahan CF pehli baar $N /2 = 20$ ko hit ya cross kare; CF 10 (below 20) se 20 tak 20–30 class ke andar jaati hai, isliye woh class median position contain karti hai.

$L = 20, C F_{b} = 10, f = 10, h = 10$ $Median = 20 + (\frac{20 - 10}{10}) \times 10 = 20 + 1 \times 10 = 30$ Graph par: $y = 20$ par jao, curve tak daayein slide karo, neeche chhaao → $x = 30$ (median yahan bilkul boundary par baith jaata hai). ✓

Common Mistakes (Steel-manned)

Active Recall

Recall Quick self-test (hide and answer)

$x = 30$ par ogive ki y-value ka literally kya matlab hai?
Grouped data ke liye $N /2$ kyun use karte hain aur $(N + 1) /2$ kyun nahi?
Twin-ogive method mein crossing point kya khaas hai?
"Count needed = $N /2 - C F_{b}$ " se median formula derive karo.

Answers: (1) 30 se kam observations ki sankhya. (2) Grouped data → smooth curve, halfway height. (3) less-than = more-than = $N /2$ → median. (4) $Δ x = (N /2 - C F_{b}) / (f / h)$ , $L$ mein jodao.

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho ek queue mein bacche chhaote se lambe sort hain. "Cumulative frequency" sirf yeh count karna hai ki "meri kheechi hui line se kitne bacche chhhote hain." Agar main lines kheenchta rahun aur count karta rahun, aur plot karun, toh mujhe ek curve milta hai jo hamesha upar jaata hai. Ab mujhe beech wala baccha chahiye. Aadhe bacche 20 hain (40 mein se). Toh main height axis par 20 tak jaata hun, right tak chalta hun jab tak curve na mile, phir seedha neeche dekhta hun — woh height beech wale bacche ki height hai. Aasaan!

Flashcards

Cumulative frequency of a class is

us class ki frequency aur usse pehle ki saari frequencies ka sum (running total).

An ogive is a graph of

cumulative frequency (y) vs class boundary (x), smooth curve se join kiya hua.

For a less-than ogive you plot CF against the

upper class boundary.

The final cumulative frequency always equals

N, observations ki total sankhya.

To read the median from an ogive, go up the y-axis to

N/2, curve tak across, phir x-axis par neeche.

The median class is the class

jiske andar cumulative frequency pehli baar N/2 tak pahunchti ya cross karti hai (CF N/2 se neeche se ≥ N/2 tak jaati hai).

Grouped-data median formula

Median = L + ((N/2 − CF_b)/f)·h.

In the twin-ogive method the median is the x-value at

less-than aur more-than ogives ka intersection.

Why N/2 (not (N+1)/2) for grouped data

kyunki ogive ko continuous smooth curve treat kiya jaata hai, isliye total height ko half mein split karte hain.

Meaning of L, CF_b, f, h in the median formula

median class ki lower boundary, usse pehle wali class ki CF, median class ki frequency, class width.

Connections

Median (measures of central tendency)
Frequency distribution table
Histogram and frequency polygon
Quartiles and percentiles from ogive
Mean and mode of grouped data
Linear interpolation

Cumulative frequency — ogive, median from graph

Cumulative frequency KYA hai?

CF table KAISE banayein (first principles se)

Ogive KAISE draw karein

Graph se median KAISE nikaalein

Graph ke peeche ka formula (derivation)

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Flashcards

Connections

Concept Map