Average shortcuts: assumed mean, middle term and replacements
औसत के शॉर्टकट: कल्पित माध्य, मध्य पद और प्रतिस्थापन
Stop adding long lists. Guess an average, correct it with the deviations, use the middle term for evenly spaced numbers, and solve “one person joins or is replaced” questions in one line.
एक औसत मान लें और विचलन से सुधारें; समान अंतर वाली संख्याओं में बीच वाला पद ही औसत है।
Assumed mean (deviation method)
- Pick a round number near the middle as a guess.
- Write how far each value is from it (+ or −).
- Average of the deviations, added to the guess, is the answer.
Example Average of 47, 52, 49, 55, 57
Guess 50. Deviations: −3, +2, −1, +5, +7 → total +10 → 10 ÷ 5 = 2 → 50 + 2 = 52
Evenly spaced numbers
If numbers go up by the same step, the average is the middle term (or the average of the first and last).
Example 12, 15, 18, 21, 24
Middle term → 18
Example Average of 1 to 50
(1 + 50) ÷ 2 = 25.5
Example Average of the first 50 even numbers
(2 + 100) ÷ 2 = 51
Someone joins the group
New average = old average + (new value − old average) ÷ new count.
Example 10 players average 25 runs; a player with 47 joins
25 + (47 − 25) ÷ 11 = 25 + 2 = 27
Someone is replaced
If replacing one member changes the average of n members by d, the newcomer differs from the person leaving by n × d.
Example The average weight of 8 people rises by 2 kg when a 40 kg person is replaced. Weight of the new person?
40 + 8 × 2 = 56 kg
Try it yourself · अभ्यास
- Average of 98, 103, 101, 96, 102?
Show answer
Guess 100: −2, +3, +1, −4, +2 = 0 → 100
- Average of the first 20 natural numbers?
Show answer
(1 + 20) ÷ 2 = 10.5