Measures of central tendency — mean (grouped - ungrouped), median (grouped), mode (grouped): Foundations

2.7.1 · D1 · Maths › Statistics & Probability — Intermediate › Measures of central tendency — mean (grouped - ungrouped), m

Parent note mein koi bhi formula padhne se pehle, tumhe usmein har symbol ko samajhna hoga. Neeche, har symbol ko yeh milega: plain words → the picture → topic ko kyun chahiye. Yeh iss order mein hain ki har ek apne pehle wale par depend karta hai.

1. The raw ingredients: data, observations, and $n$

Ek table par pebbles ki line ki picture socho. Har pebble ek observation hai. Unhe gino: woh count $n$ hai.

Topic ko kyun chahiye: har centre-measure kitni cheezein hain usse divide ya position karta hai. $n$ ke bina tum fair share ( $\div n$ ) nahi le sakte ya halfway point (position $n /2$ ) nahi dhundh sakte.

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

2. Subscripts: $x_{i}$ and the "for-each" idea

Pebbles ki picture dobara socho, ab left se right numbered: pebble #1, pebble #2, pebble #3. Pebble #3 ke neeche likhi value $x_{3}$ hai.

Topic ko kyun chahiye: mean, median aur mode ke formulas sab "har value" ya "har class" ki baat karte hain. Subscript usi ki language hai.

3. The summation sign $\sum$

Pebbles ki poori row par haath pherne ki picture socho — pebble $1$ se shuru, pebble $n$ par khatam — aur har value ek bucket mein daalo. Bucket ka total $i = 1 \sum n x_{i}$ hai.

Topic ko kyun chahiye: pehla formula, $\overset{x}{ˉ} = \frac{\sum _{i = 1}^{n} x _{i}}{n}$ , "bucket mein total ÷ kitne hain" hai. $\sum$ ke bina tum ise padh nahi sakte.

4. The mean symbol $\overset{x}{ˉ}$ and "balance point"

Topic ko kyun chahiye: $\overset{x}{ˉ}$ parent note ke Section 1 ka headline output hai, aur yeh doosre do centres ke saath empirical relation mein team up karta hai (jo hum sirf tab state kar sakte hain jab teeno symbols define ho jaayein — Section 9 dekho).

5. Frequency $f_{i}$ and grouped data

Pebbles ko labelled jars mein sort karne ki picture socho. Jar " $10 - 20$ " mein 8 pebbles hain → uski frequency $f = 8$ hai.

Topic ko kyun chahiye: jab data jars mein hota hai toh hum individual pebbles nahi dekhte, isliye har grouped formula (mean, median, mode) raw values ki jagah $f_{i}$ ke terms mein likha jaata hai.

6. Class midpoint $m_{i}$ , lower boundary $l$ , and width $h$

Topic ko kyun chahiye: grouped mean $m_{i}$ (midpoint) use karta hai, grouped median aur mode dono $l$ (start) aur $h$ (width) use karte hain class ke andar slide karne ke liye.

7. Ordering data, and the median symbol $\tilde{x}$ (ungrouped first)

8. Cumulative frequency $c f$ and the grouped median

Ek bade tank mein jar 1 daalne ki, phir jar 2, phir jar 3 daalne ki picture socho — har pour ke baad tank ka level read karo. Woh running levels $c f$ values hain. Ek curve ki tarah draw karne par, yeh exactly ek ogive hai.

Topic ko kyun chahiye: median ek position hai ( $n /2$ -th item), aur $c f$ batata hai woh position kis class mein land karti hai; phir ramp us class ke andar kahan interpolate karta hai.

9. The mode: raw data, ties, and the grouped symbols $f_{0}, f_{1}, f_{2}$

Teen bars side by side ki picture socho, beech wala ( $f_{1}$ ) sabse lamba, $f_{0}$ uske left par aur $f_{2}$ uske right par. Mode formula poochta hai: kya lamba bar apne chhote-left ya chhote-right neighbour ki taraf jhuk raha hai? Woh lean decide karta hai ki true peak target class ke andar kahan baithti hai — ek histogram ka visual heart.

10. The fraction / ratio bar and the empirical relation

Har grouped formula ek ratio par khatam hoti hai — ek quantity doosri se divided: $\overset{x}{ˉ} = \frac{total}{count}, median position fraction \frac{n /2 - c f}{f}, mode lean fraction \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} .$

Prerequisite concept map

Har arrow ko "feeds into" padho. Left par chaar building blocks mile karke right par teen centre-measures banate hain, jo milke topic banate hain.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle jawab do.

i = 1 \sum n

par chhote numbers ka kya matlab hai?

i = 1

(bottom) slot se add karna shuru karo aur

i = n

(top) slot par roko; beech ke har

x_{i}

ko add karo.

x_{i}

mein subscript

i

kya represent karta hai?

Ek position name-tag:

x_{i}

i

-va raw observation hai.

Class midpoint ke liye hum kaunsa symbol use karte hain, aur

x_{i}

kyun nahi?

m_{i}

— ek alag letter taaki class ki stand-in value kabhi raw observation se confuse na ho.

\sum

kaunsa single operation command karta hai?

Saari listed values ko ek total mein add karo — pure addition, bottom index se top index tak.

\overset{x}{ˉ}

kya hai aur iska picture kya hai?

The mean; woh pivot/balance point jahan upar aur neeche ke pulls cancel ho jaate hain, toh

\sum (x_{i} - \overset{x}{ˉ}) = 0

Frequency

f_{i}

kya hai?

Kitne observations us class mein aaye (jar-count).

Grouped data se total count

n

kaise milta hai?

Saari frequencies add karo:

n = \sum_{i = 1}^{k} f_{i}

Class

20 - 30

ke liye

l

h

, aur midpoint

m_{i}

bolo.

l = 20

h = 10

m_{i} = 25

Ungrouped median jab

n

odd ho vs even ho?

Odd: the

\frac{n + 1}{2}

-th sorted value. Even: average of the

\frac{n}{2}

-th aur

\frac{n}{2} + 1

-th sorted values.

Median dhundhne se pehle sort kyun karna padta hai?

"Middle" tabhi exist karta hai jab values smallest-to-largest line up hon.

Cumulative frequency

c f

kya track karta hai?

Is class aur usse pehle ki saari classes mein observations ka running total.

Grouped-median formula geometrically kyun kaam karta hai?

Ogive median class ke across ek straight ramp ki tarah climb karti hai; median wahan hai jahan ramp height

n /2

reach karta hai — similar triangles fraction dete hain.

Grouped median formula mein kaunsa

c f

aur kaunsa

f

use karte hain?

c f

= median class se pehle wali class ka cf;

f

= median class ki apni frequency.

Ungrouped mode mein ties ke saath — kya report karte hain?

Saari tied values (unimodal / bimodal); agar har value equally frequent ho, toh koi mode nahi.

Grouped-mode formula kab toot jaata hai, aur iska kya matlab hai?

Jab

2 f_{1} - f_{0} - f_{2} = 0

(flat top) ya do classes highest ke liye tie karein — koi single clear peak nahi; formula force mat karo.

Kaunsi frequency

f_{1}

hai, aur

f_{0}, f_{2}

kya hain?

f_{1}

= modal (target) class ki frequency;

f_{0}

= thoda pehle wali class,

f_{2}

= thoda baad wali class.

Teeno centres ke beech empirical relation state karo.

Mode \approx 3 \tilde{x} - 2 \overset{x}{ˉ}