Mean, median, mode — calculation for raw and grouped data

1.3.4 · Maths › Basic Data & Probability

Overview

Teen measures of central tendency humein batate hain ki ek dataset ka "center" kahan hai. Har ek alag sawaal ka jawaab deta hai: mean arithmetic average hai, median sorted hone par beech ki value hai, aur mode sabse zyada baar aane wali value hai. Kab kaunsa use karna hai aur raw vs. grouped data mein farq samajhna real-world datasets interpret karne ke liye bahut zaroori hai.

Core Intuition

Raw Data (Ungrouped)

Mean (Arithmetic Average)

Median (Middle Value)

Mode (Most Frequent)

Grouped Data (Frequency Distribution)

Jab data class intervals mein organize hota hai (jaise 0-10, 10-20), toh hum individual values nahi dekh sakte. Hum class marks (midpoints) aur frequencies (counts) use karte hain.

Figure — Mean, median, mode — calculation for raw and grouped data

Grouped Data ke liye Mean

Worked example Example 4: Student test scores (grouped)

Class Interval	Frequency ( $f$ )	Class Mark ( $x$ )	$f \cdot x$
0-20	5	10	50
20-40	8	30	240
40-60	12	50	600
60-80	10	70	700
80-100	5	90	450
Total	$N = 40$	—	2040

Step 1: Class marks calculate karo

0-20 ke liye: $x = \frac{0 + 20}{2} = 10$ Midpoint kyun? Hum maan rahe hain ki class ke andar uniform distribution hai.

Step 2: Har class ke liye $f \times x$ multiply karo

0-20 ke liye: $5 \times 10 = 50$ Kyun? 10 par maane gaye 5 students ka contribution $5 \times 10$ hai.

Step 3: Sabhi $f \cdot x = 2040$ ka sum karo Step 4: Mean = $\frac{2040}{40} = 51$ marks Interpretation: Average score 51 hai.

Grouped Data ke liye Median

Worked example Example 5: Grouped data se Median

Example 4 ki same table use karke, cumulative frequency add karo:

Class	Frequency ( $f$ )	Cumulative Frequency (CF)
0-20	5	5
20-40	8	13
40-60	12	25 ← median class
60-80	10	35
80-100	5	40

Step 1: $\frac{N}{2} = \frac{40}{2} = 20$ nikalo Kyun? Median 20th observation hai.

Step 2: Median class locate karo

20-40 tak CF 13 hai (< 20)
40-60 tak CF 25 hai (≥ 20) ← median class 40-60 hai Kyun? 20th value is class mein aati hai.

Step 3: Formula apply karo

$L = 40$ (40-60 ki lower boundary)
$C F = 13$ (median class se pehle ki cumulative frequency)
$f = 12$ (median class ki frequency)
$h = 60 - 40 = 20$

$Median = 40 + (\frac{20 - 13}{12}) \times 20 = 40 + \frac{7}{12} \times 20 = 40 + 11.67 = 51.67$

Yeh step kyun? Humein 12 mein se 7 aur observations chahiye (20 − 13). Yeh class width (20) ka $\frac{7}{12}$ hai, isliye lower boundary mein $11.67$ add karte hain.

Median ≈ 51.67 marks

Grouped Data ke liye Mode

Common Mistakes

Step-by-Step Algorithms

Raw Data

Mean: Sabhi values ka sum karo → count se divide karo
Median: Data sort karo → agar odd hai toh middle pick karo; agar even hai toh do middle values ka average lo
Mode: Frequencies count karo → sabse frequent pick karo (ya none/multiple modes identify karo)

Grouped Data

Mean: Class marks compute karo → frequencies se multiply karo → sum karo → $N$ se divide karo
Median: Cumulative frequency compute karo → median class dhundho (jahan $C F \geq \frac{N}{2}$ ) → interpolation formula apply karo
Mode: Modal class dhundho (sabse zyada $f$ ) → neighbors use karke interpolation formula apply karo

When to Use Each Measure

Measure	Best For	Limitation
Mean	Symmetric data, jab total/sum chahiye	Outliers se sensitive
Median	Skewed data, outliers present hon	Actual values ignore karta hai, sirf position
Mode	Categorical data, "typical" value dhundna	Ho sakta hai exist hi na kare ya multiple hon

Feynman Explanation

Recall Ek 12-saal ke bachche ko samjhao

Imagine karo tum aur 4 doston ko yeh chocolate bars mile: 2, 3, 5, 6 aur 9 bars.

Mean (average): Agar sab chocolates pool karo (2+3+5+6+9 = 25) aur equally share karo, toh sabko 25÷5 = 5 bars milenge. Yeh fair hai, lekin agar ek dost ko 100 bars mile (lucky!), toh suddenly average 20 ho jaata hai, chahe tumhara zyataar ke paas 10 se kam hi ho. Mean extremes se kheencha jaata hai.

Median (middle): Order mein khade ho jaao: {2, 3, 5, 6, 9}. Beech wale insaan ke paas 5 bars hain. Chahe aakhiri dost ke paas 1000 bars hon, beech wale ke paas phir bhi 5 hi rahenge. Median ko extremes ki parwah nahi!

Mode (most popular): Agar 3 doston ke paas 5 bars hain, 1 ke paas 2, 1 ke paas 9, toh 5 mode hai—yeh sabse zyada aata hai. Yeh batata hai ki tumhare group mein kya "normal" hai.

Grouped data ke liye (jaise "5 doston ke paas 0-5 bars hain, 3 doston ke paas 5-10 bars hain"), exact numbers nahi dikh sakte. Toh pretend karo ki 0-5 group mein sabke paas 2.5 bars hain (0 aur 5 ka middle) aur calculate karo. Yeh estimating jaisa hai!

Mnemonic

Connections Measures of Dispersion — center dhundne ke baad, hum spread measure karte hain (range, variance, standard deviation)

Cumulative Frequency Graphs — ogives median aur quartiles visualize karne mein help karti hain
Frequency Distributions — grouped data calculations ki foundation
Box Plots — median aur quartiles use karke visual summary
Skewness — skewed distributions mein mean, median, mode ka relationship (right skew ke liye mean > median > mode)
Weighted Mean — generalization jahan har value ka ek weight hota hai (grouped data mein $f \cdot x$ jaisa)

Flashcards

#flashcards/maths

Raw data ke mean ka formula kya hai? :: $\overset{x}{ˉ} = \frac{\sum x _{i}}{n}$ jahan $n$ observations ki count hai.

Odd number of raw data values ke liye median kaise nikaalte hain?

Data sort karo, phir

(\frac{n + 1}{2})^{th}

term pick karo.

Even number of raw data values ke liye median kaise nikaalte hain?

Data sort karo, phir

(\frac{n}{2})^{th}

aur

(\frac{n}{2} + 1)^{th}

terms ka average lo.

Ek dataset ka mode kya hota hai?

Woh value jo sabse zyada baar aati hai. Ek dataset mein no mode, one mode, ya multiple modes ho sakte hain (jaise bimodal agar do values highest frequency ke liye tie karte hain).

Class interval 30-40 ka class mark (midpoint) kya hai?

\frac{30 + 40}{2} = 35

Grouped data ke mean ka formula?

\overset{x}{ˉ} = \frac{\sum f \cdot x}{N}

jahan

f

frequency hai,

x

class mark hai,

N

total frequency hai.

Grouped data ke median ka formula?

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

jahan

L

median class ki lower boundary hai,

C F

usse pehle ki cumulative frequency hai,

f

uski frequency hai,

h

class width hai.

Grouped data mein median class kaise identify karte hain?

Woh class dhundho jahan cumulative frequency pehli baar

\frac{N}{2}

ke barabar ya usse zyada ho.

Grouped data ke mode ka formula?

Mode = L + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h

jahan

L

modal class ki lower boundary hai,

f_{1}

uski frequency hai,

f_{0}

pehle wali hai,

f_{2}

baad wali class ki frequency hai,

h

class width hai.

Modal class kaise identify karte hain?

Sabse zyada frequency wali class.

Median nikalne se pehle data sort kyun karna zaroori hai?

Median ordered data mein beech ki value ke roop mein defined hai; sort kiye bina sahi middle position identify nahi kar sakte.

Grouped data mein mean ke liye class boundaries ki jagah class marks kyun use karte hain?

Hum assume karte hain ki class ke sabhi values uniformly distributed hain, isliye midpoint (class mark) us class ki typical value ka best estimate hai.

Agar mean < median < mode, toh distribution __ skewed hai.

Left-skewed (negatively skewed).

Agar mean > median > mode, toh distribution ___ skewed hai.

Right-skewed (positively skewed).

Outliers wale data ke liye median mean se better kyun hai?

Median sirf middle position(s) par depend karta hai, actual values par nahi, isliye extreme outliers use distort nahi karte.