Average
What you must know
Average (arithmetic mean) = sum of observations ÷ number of observations. Rearranged, sum = average × count, which is the workhorse of most exam questions. For an arithmetic progression, the average equals the middle term (or the mean of the first and last terms). Weighted averages use the formula (n₁x₁ + n₂x₂)/(n₁ + n₂).
A very frequent pattern: when one item is replaced or a new member joins, compare the old total with the new total. If the average of n items rises by d when one item is replaced, the replacement is larger by n × d. Similarly, the age of a new person joining a group of n whose average rises by d is old average + (n+1)d.
Traps
- Averaging averages without weighting by group size.
- Forgetting that adding a member increases the count as well as the sum.
- In cricket-style problems, "average after n innings" includes the new innings in the divisor.
- Consecutive-number averages: the average of n consecutive integers is the middle one, not the first or last.
- Replacement problems: the change applies to the whole group, so multiply by n.
Rule of thumb and example
Convert every average question into a statement about totals; totals add, averages do not.
Worked example: The average age of 10 students is 15. A teacher aged 45 joins. New total = 150 + 45 = 195, new count = 11, new average = 195/11 ≈ 17.73 years. Note the rise is (45 − 15)/11 ≈ 2.73, not 30/10.