Example: First 12 observations ka average 36 hai. Remaining 9 observations ka average 15 hai. Total 21 observations ka average kya hai?

Sum of first 12 = $12 \times 36 = 432$
Sum of remaining 9 = $9 \times 15 = 135$
Total Sum = $432 + 135 = 567$
Overall Average = $\frac{567}{12+9} = \frac{567}{21} = 27$

Concept 03: Jab Data Mein Miscalculation Ho Jaye (When data gets messed up)

Kabhi-kabhi data entry mein gadbad ho jaati hai. Jaise, ek number ko galat padh liya. Isko theek kaise karein?

Method: $Correct~Average = Old~Average + \frac{Sum~of~corrections}{Total~numbers}$
Where $Sum~of~corrections = (Correct~value_1 - Incorrect~value_1) + (Correct~value_2 - Incorrect~value_2) + ...$

Example 1: 10 numbers ka average 40 hai. Ek number 64 ko galati se 46 padh liya. Correct average kya hai?

Old Average = 40
Incorrect value = 46, Correct value = 64
Correction needed = $64 - 46 = +18$ (Sum mein 18 add karna hai)
Correct Average = $40 + \frac{18}{10} = 40 + 1.8 = 41.8$

Example 2: 'n' numbers ka average 'm' hai. Ek number 'p' ko galati se 'q' padh liya aur ek number 'x' ko galati se 'y' padh liya. Correct average?

Sum of Corrections = $(p-q) + (x-y)$
Correct Average = $m + \frac{(p-q) + (x-y)}{n}$

Concept 04: Age Related Average (Umra Ka Hisaab)

Jab groups mein log add ya remove hote hain, toh average age kaise badalti hai?

Example: Ek class mein 40 students ka average 15 saal hai. Agar 8 new students join karte hain toh average age 0.2 saal badh jaati hai. New students ki average age kya hai?

Old total age = $40 \times 15 = 600$ years
New total students = $40 + 8 = 48$
New average age = $15 + 0.2 = 15.2$ years
New total age = $48 \times 15.2 = 729.6$ years
Age of 8 new students = $729.6 - 600 = 129.6$ years
Average age of 8 new students = $\frac{129.6}{8} = 16.2$ years

Concept 05: Replacement, Inclusion, Exclusion (Aadmi Aaye, Aadmi Gaye!)

Jab group mein koi naya member aata hai, ya koi purana member chala jata hai, ya koi replace ho jata hai, toh average kaise change hota hai?

1. Inclusion (Naya Banda Aaya)

Example: Ek class mein 50 students ka average weight 45 kg hai. Jab ek naya student aata hai, toh average weight 100g badh jata hai. New student ka weight kya hai?

Old total weight = $50 \times 45 = 2250$ kg
New number of students = $50+1 = 51$
Increase in average = 100g = 0.1 kg
New average weight = $45 + 0.1 = 45.1$ kg
New total weight = $51 \times 45.1 = 2299.1$ kg
Weight of new student = $2299.1 - 2250 = 49.1$ kg

2. Exclusion (Ek Banda Chala Gaya)

Example: 34 students ka average weight 42 kg hai. Agar teacher ka weight bhi include karein toh average 400g badh jata hai. Teacher ka weight kya hai?

Total weight of students = $34 \times 42 = 1428$ kg
Increase in average = 400g = 0.4 kg
New average (students + teacher) = $42 + 0.4 = 42.4$ kg
Total number of people = $34 + 1 = 35$
Total weight (students + teacher) = $35 \times 42.4 = 1484$ kg
Teacher's weight = $1484 - 1428 = 56$ kg

3. Replacement (Ek Gaya, Doosra Aaya)

Example: 25 boys ka average height 1.5 m hai. Agar 5 boys group chhod dete hain aur 5 naye boys join karte hain, toh average height 0.15 m badh jati hai. Naye 5 boys ka average height kya hai?

Initial total height of 25 boys = $25 \times 1.5 = 37.5$ m