Lesson 1 of 2 | Totals, Weights and Units
Averages: Rebuild the Total Before Changing the Group
The quantity an average hides
An average compresses a total and a count into one number. Recover those two quantities before combining groups, correcting a reading or removing a member. The basic relationship is total = mean × count. Use a positive count and consistent units.
Worked example: unequal groups
A group of 15 candidates averages 48 marks. Another 10 candidates average 63. The first total is 15 × 48 = 720; the second is 10 × 63 = 630. Together they score 1350 across 25 candidates, so the combined mean is 54.
The midpoint 55.5 would give the groups equal weight even though their sizes differ. The correct mean lies between 48 and 63 and is closer to 48 because the first group is larger. This bound is a useful check, not a substitute for calculation.
Worked example: correct a reading
Eight measurements have a recorded mean of 17. A recorded value of 29 should have been 13. The recorded total is 8 × 17 = 136. Replace the wrong value: 136 − 29 + 13 = 120. The number of measurements remains eight, so the corrected mean is 15. Do not change the count when you only replace a value.
Worked example: a member leaves
Six people have mean age 24 years. Five remaining people average 23 after one person leaves. The original total age is 144, and the remaining total is 115. The departing person is 29 years old. Both the total and count changed. Check by adding 29 back to 115 and dividing by six.
Choose the correct update
| Situation | New total | New count |
|---|---|---|
| Add one observation x | Old total + x | Old count + 1 |
| Remove one observation x | Old total − x | Old count − 1 |
| Replace wrong value w with c | Old total − w + c | Unchanged |
| Combine two groups | Sum of their totals | Sum of their counts |
Try it, then check
- Five scores average 16. Add a score of 28. What is the new mean?
- A group of 12 averages 30 and a group of 18 averages 40. Find the combined mean.
- Ten values average 21, but 34 should be 14. Find the corrected mean.
Answers: 1. (80 + 28) / 6 = 18. 2. (360 + 720) / 30 = 36. 3. (210 − 34 + 14) / 10 = 19.
These are original foundation exercises, not official CDS previous-year questions. In the next lesson, identify the correct totals for average speed.