Concept 1
Average is total divided by count
For n observations, arithmetic mean = sum of observations ÷ n. Therefore total = average × count. For 8, 11, 14 and 19, the average is 52÷4=13.
Balance interpretation: deviations above and below the mean add to zero. Relative to 13, the deviations −5, −2, +1 and +6 sum to zero.
Concept 2
Recover a missing observation from the total
If five numbers average 24, their total is 120. If four known numbers total 91, the missing number is 120−91=29.
When one new value x joins n items with average A and changes the average to B, use nA+x=(n+1)B.
Concept 3
Use symmetry for equally spaced values
The average of equally spaced values is their middle value, or the mean of the two middle values. Thus the average of 17, 21, 25, 29 and 33 is 25.
For consecutive integers from a through b, average=(a+b)/2. Their count is b−a+1.
Concept 4
Combine totals, not averages
If n₁ items average A₁ and n₂ items average A₂, the combined average is (n₁A₁+n₂A₂)/(n₁+n₂). For 20 learners averaging 60 and 30 averaging 70, the combined mean is (1200+2100)/50=66.
Range check: a combined average must lie between the two group averages when both groups have positive sizes.
Concept 5
A replacement changes the total by one difference
If one observation x is replaced by y in a group of n, the average changes by (y−x)/n. Replacing 42 with 57 in a group of 10 raises the average by 15/10=1.5.
For an incorrect entry, corrected average = old average +(correct value−wrong value)/count.
Concept 6
Weights tell how much each value counts
Weighted average = Σ(wx)/Σw. A course score with 40% coursework at 70 and 60% exam at 80 is 0.4×70+0.6×80=76.
For two groups, the ratio of their sizes is opposite to their distances from the combined mean: n₁:n₂=(A₂−A):(A−A₁), when A₁<A<A₂.
Applications
How exams disguise averages
Age
Track the same group carefully: after one year, every member’s age—and therefore the average—rises by one year.
Runs and scores
Total runs = innings average × counted innings; note whether a not-out innings affects the denominator.
Speed
Average speed is total distance ÷ total time, not usually the simple mean of two speeds.
Expenditure
Average expenditure × number of people gives the group total before additions or departures.
Worked methods
Examples with reasoning
Six values average 18; five total 83
Total=108, so the missing value is 108−83=25.
Average of 12 items rises from 24 to 26
The total rises by 12×2=24.
Ten workers average 32 and fifteen average 42
Combined average=(320+630)/25=38.
One value 36 was recorded as 63 among nine values
Corrected average = old average +(36−63)/9 = old average−3.
Equal distances at 40 km/h and 60 km/h
For equal distances, average speed=2ab/(a+b)=4800/100=48 km/h.
Exam traps
Errors that produce convincing wrong answers
Averaging averages
Use group sizes unless the groups contain equal numbers of items.
Wrong denominator
After adding or removing an item, update the count as well as the total.
Age group changes
A birth, death, joining or leaving means it is no longer the same fixed group.
Average speed
Equal distances use a harmonic mean; equal times use an arithmetic mean.
Replacement sign
Use new minus old; reversing it changes the direction of the average.
Units
Convert hours and minutes, kilometres and metres, or rupees and paise before combining.
Original practice
Check understanding after learning
Original syllabus-aligned questions—not claimed previous-year questions.
Rapid revision
The one-minute sheet
- Average=total/count; total=average×count.
- Use totals to add, remove, combine or correct observations.
- Equally spaced values balance at their middle.
- Combined average is weighted by group size.
- Replacement changes the average by (new−old)/count.
- Average speed always comes from total distance/total time.
Source ledger
Official foundations
- SSC CGL official syllabus: Mathematical Abilities
- NCERT official mathematics exemplar directory
- NCERT official textbook portal
Explanations, worked examples and practice are original. Official references are credited and not reproduced. Drafted 24 September 2026.
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