统计与数据图表

考什么 · What it tests

需要先会 · Prerequisites

• 平均数 = 总和 ÷ 个数 / Mean = total ÷ count
• 分数与百分比 / Fractions and percentages
• 排序（找中位数前要先排队） / Sorting (order the data before finding the median)

常见套路 · Common moves

1. 反推先还原总和：已知 $n$ 个数平均是 $m$，就先算出总和 $n\times m$，缺的那个数 = 总和减去其余——「均值反推」题几乎都靠这一步。

   To work backwards, first rebuild the total: if $n$ numbers average $m$, the total is $n\times m$, and the missing number equals the total minus the rest — nearly every reverse-mean problem hinges on this.
2. 中位数先排序再数位置：奇数个取正中间，偶数个取中间两个的平均；没排序就报数一定错。

   For the median, sort first then locate: the middle one if the count is odd, the average of the middle two if even; report without sorting and you will be wrong.
3. 加权平均别直接平均：各组人数不同时要按人数加权，用 \frac{\text{各组总和}}{\text{总人数}}，不是几个平均数再取平均。

   Weighted average is not a plain average: when groups differ in size, weight by size using $\displaystyle \frac{\text{sum of all groups}}{\text{total count}}$, not the average of the group averages.

易错点 · Common pitfalls

• 中位数忘了排序，直接取原始数据里居中那个——出题人专门把数据打乱顺序给你。

  Forgetting to sort and taking the physically middle value of the raw list — the writers hand you the data out of order on purpose.
• 把「加权平均」当成「几个平均数的平均」：两个班的平均分不能直接相加除以 2，人数不同结果就不同。

  Treating a weighted average as the average of averages: two classes' mean scores can't just be added and halved — different class sizes change the result.