Averages

The Core Truth: Average is the point where Net Deviation is ZERO.

The Method: Instead of summing huge numbers, pick an 'Assumed Average' (A).

If numbers are 48, 52, 55. Let A = 50.

• 48 is -2 (below 50).
• 52 is +2 (above 50).
• 55 is +5 (above 50).
• Net Deviation = -2 + 2 + 5 = +5.
• Average = 50 + 53 = 51.66.

When a person joins a group, they bring a 'Surplus' or 'Deficit'.

New Avg = Old Avg + Total IncreaseTotal People

Example: Avg age of 10 people increases by 2 years when a new person replaces a 40-year-old.

Mental Step: New person must cover the old person (40) PLUS bring extra for everyone (10 × 2 = 20).

Age = 40 + 20 = 60 years. (No X required).

If Group A has N1 items with Avg A1, and Group B has N2 items with Avg A2:

Use Alligation:

The gap between A1 and A2 is shared in ratio N2:N1.

Example: Class A (20 students, Avg 30), Class B (30 students, Avg 50).

• Weighted Avg = 20(30) + 30(50)50 = 42.

For any Arithmetic Progression (AP) like 12, 14, 16... or 7, 14, 21...

Average = Middle Number

If terms are even (e.g., 2, 4, 6, 8), Average is mid-point of 4 and 6 (i.e., 5).

Master Formula: Average = First Term + Last Term2.

If distance is constant:

Avg Speed = 2xyx+y

Why? Time is variable. Speed x Time = Distance.

If 3 equal distances at speeds x, y, z: Avg = 3xyzxy + yz + zx.

Batting Avg: Total RunsTotal Innings [Out].

Bowling Avg: Total Runs GivenTotal Wickets Taken.

Trick: For Bowling, a LOWER average is better.

If a bowler takes wickets cheaply, his average drops.