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

2.7.1 · Maths › Statistics & Probability — Intermediate

1. Mean (average)

1.1 Ungrouped mean — derive karo

KYA HAI: Hum "fair share" chahte hain. Agar total $S$ hai aur use $n$ logon mein equally baant dein: $\overset{x}{ˉ} = \frac{S}{n} = \frac{\sum x _{i}}{n}$

YEH balance point kyun hai: Mean woh value $m$ hai jo deviations ko cancel kara deti hai: $\sum (x_{i} - m) = 0 \Rightarrow \sum x_{i} - nm = 0 \Rightarrow m = \frac{\sum x _{i}}{n}$ Yeh step kyun? Total deviation ko zero set karna hi balance point ki definition hai — upar ke pulls neeche ke pulls ke barabar hote hain.

1.2 Grouped mean (frequency data)

Jab data classes mein hota hai, har class ka ek midpoint $x_{i}$ hota hai (assume karo ki us class ki har value ≈ uska midpoint hai) aur ek frequency $f_{i}$ hoti hai.

2. Median (beech wali value)

2.1 Ungrouped median

Data ko order karo. Phir:

Agar $n$ odd hai: median = $(\frac{n + 1}{2})^{th}$ value.
Agar $n$ even hai: median = $\frac{n}{2}^{th}$ aur $(\frac{n}{2} + 1)^{th}$ values ka mean.

2.2 Grouped median — derive karo

Hum individual values nahi dekh sakte, isliye hum cumulative frequency (cf) use karke beech wala banda kahan padta hai dhundhte hain.

Derivation (linear interpolation), har piece kyun hai: Hum $\frac{n}{2}$ -th item ki position chahte hain. Median class se pehle, $c f$ items already use ho chuke hain. Hume abhi $\frac{n}{2} - c f$ aur items chahiye. Assume karo ki median class ke $f$ items uski width $h$ mein evenly spread hain. Toh har item $\frac{h}{f}$ width occupy karta hai. Class mein $(\frac{n}{2} - c f)$ items tak pahunchne ke liye hum move karte hain: $(\frac{n}{2} - c f) \times \frac{h}{f}$ lower boundary $l$ se aage. $l$ mein add karo → ho gaya.

3. Mode (peak)

3.1 Grouped mode — derive karo

Modal class = sabse zyada frequency $f_{1}$ wali class. Lekin asli peak neighbouring classes ke hisaab se centre ke left ya right mein ho sakti hai.

Derivation intuition, KYUN: Dekho ki peak pehli class se kitni zyada upar uthti hai, $(f_{1} - f_{0})$ , aur dono neighbours pe total "excess" kitna hai, $(f_{1} - f_{0}) + (f_{1} - f_{2}) = 2 f_{1} - f_{0} - f_{2}$ . Yeh fraction batata hai ki modal class mein asli peak kitni door hai. Agar pehle wali class zyada heavy hai, toh mode left ki taraf jhukta hai; agar baad wali class heavy hai, toh right ki taraf. Width $h$ se multiply karo, $l$ mein add karo.

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

\sum (x_{i} - m) = 0

ho; upar aur neeche ke pulls cancel ho jaate hain.

Grouped mean direct formula?

\overset{x}{ˉ} = \frac{\sum f _{i} x _{i}}{\sum f _{i}}

class midpoints use karke.

Step-deviation mean formula aur

u_{i}

kya hota hai?

\overset{x}{ˉ} = a + h \frac{\sum f _{i} u _{i}}{\sum f _{i}}

, jahaan

u_{i} = \frac{x _{i} - a}{h}

Grouped median formula?

l + (\frac{n /2 - c f}{f}) h

Median formula mein

c f

kya hai?

Median class se pehle wali class ki cumulative frequency.

Median class kaise identify karte hain?

Pehli class jiska cumulative frequency

\geq n /2

ho.

Grouped mode formula?

l + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) h

Mode mein

f_{0}, f_{1}, f_{2}

kya hain?

f_{1}

=modal class freq,

f_{0}

=preceding class freq,

f_{2}

=following class freq.

Teeno ke beech empirical relation?

Mode

\approx 3

Median

- 2

Mean.

Extreme values se sabse zyada kaun sa measure affect hota hai?

Mean (median aur mode outliers ignore karte hain).

Even

n

data ka median?

(n /2)

-th aur

(n /2 + 1)

-th ordered values ka average.

Assumed-mean method kyun use karte hain?

Same mean milta hai, lekin chote deviation numbers arithmetic aasaan bana dete hain.

Recall Feynman: 12-saal ke bachche ko samjhao

Socho kuch bachche hain jinke paas alag-alag number of candies hain.

Mean: SAARI candies ek pile mein daalo aur equally baanto — woh equal share hi mean hai.
Median: bachhon ko sabse kam se sabse zyada candies ke order mein line mein khada karo; bilkul beech mein khada bacha — wahi median hai.
Mode: woh candy-count jo sabse ZYADA bachhon ke paas hai — wahi mode hai. Jab candies groups mein di jaati hain ("5 se 10 candies"), hum har bacha nahi dekh sakte, toh hum pretend karte hain ki group mein har kisi ke paas beech wali amount hai, aur hum bheed wale group mein slide karte hain yeh guess karne ke liye ki beech wala bacha ya peak actually kahan hai. Wahi sliding $l$ , $f$ , $h$ wala chhota formula hai.

Connections

Cumulative Frequency & Ogives — median graphically $n /2$ ordinate pe padhna.
Histograms & Frequency Distributions — modal class sabse tall bar hoti hai.
Measures of Dispersion — Variance & Standard Deviation — mean ke aas-paas spread.
Skewness — Mean, Median, Mode kyun alag hote hain; empirical relation.
Weighted Mean — grouped mean idea ka generalisation.
Probability Distributions — Expected Value — mean hi expectation $E [X]$ hai.

Class	Freq $f_{i}$	Midpoint $x_{i}$
0–10	5	5
10–20	8	15
20–30	12	25
30–40	7	35
40–50	3	45