An average speed is almost never the simple average of two speeds — unless the TIME spent at each speed happens to be equal.
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A simple average treats every value equally. A weighted average gives bigger groups more influence, in proportion to their size.
📌 Two kinds of average
Simple average: sum of all values ÷ number of values.
Weighted average: multiply each group by its own size, add them, then divide by the TOTAL count of all groups combined.
Never average two group averages directly unless both groups are exactly the same size.
📊 Worked example
| Group | Count | Average weight |
|---|---|---|
| Group A | 30 students | 40 kg |
| Group B | 20 students | 50 kg |
| Combined weighted average | 50 students | (30×40 + 20×50) ÷ 50 = 44 kg |
When a value joins or leaves a group, work with TOTALS, not averages, to find the missing value.
Inclusion: the average of 10 numbers is 20 (total 200). An 11th number is added, and the new average is 21 (new total 231). The 11th number is 231 − 200 = 31.
Exclusion: the average of 20 numbers is 30 (total 600). One number is removed, leaving an average of 29 for 19 numbers (new total 551). The removed number is 600 − 551 = 49.
Batting averages and speed averages both hide the same trap: never average the averages directly.
✨ Two classic traps
Batting average rises by 4 after scoring 92 in the 17th innings
find the old average
16×A + 92 = 17×(A+4) → A = 24 (old), 28 (new).
Speed: equal DISTANCE at two speeds
average speed formula
Average speed = 2ab ÷ (a+b), not (a+b)/2.
When to use which speed formula: if the TIME is equal at each speed, average = (a+b)/2 is correct. If the DISTANCE is equal at each speed, you must use 2ab/(a+b) instead.
12 questions on averages, weighted means and inclusion/exclusion.
