Problem 5 · 2015 AMC 8
Easy
Arithmetic & Operations
careful-counting
Billy's basketball team scored the following points over the course of the first 11 games of the season: 42, 47, 53, 53, 58, 58, 58, 61, 64, 65, 73. If his team scores 40 in the 12th game, which of the following statistics will show an increase?
Show answer
Answer: A — Range increases.
Show hints
Hint 1 of 2
The one new fact is that 40 sits below the old minimum of 42 — it's a new rock-bottom score. Adding a low value can't pull any average or middle up; ask which statistic actually grows when the bottom drops.
Still stuck? Show hint 2 →
Hint 2 of 2
Sort the five statistics by what they depend on: median/mean/mode react to where the bulk of the data sits, mid-range = (max+min)/2 averages the two ends, but range = max − min is the only one that widens when the minimum falls. Reason about each, don't recompute all of them.
Show solution
Approach: reason about which statistic a new low value can push up
- 40 is below the old low of 42, so it's the new minimum; the maximum (73) is unchanged.
- Range = max − min: was 73 − 42 = 31, now 73 − 40 = 33 → increases. A wider gap is exactly what a new low creates.
- Quick rule-out for the rest: mean drops (a below-average score lowers the average), median can only stay or fall (adding a small value never lifts the middle), mode stays 58, and mid-range = (max+min)/2 falls because min fell while max held.
- Only the range increases. Why this transfers: before crunching numbers, sort statistics by what each one 'feels' — spread (range, mid-range) vs. center (mean, median, mode) — and a single new value usually moves only the ones it touches.
Mark:
· log in to save