AMC8 2019

AMC8 2019 · Q10

AMC8 2019 · Q10. It mainly tests Averages (mean), Patterns & sequences (misc).

The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?

图表显示了上周工作日足球练习的学生人数。计算均值和中位数后，教练发现周三实际有 $21$ 名参与者。修正后，均值和中位数的变化是哪种情况？

(A) The mean increases by $1$ and the median does not change. The mean increases by $1$ and the median does not change.

(B) The mean increases by $1$ and the median increases by $1$. The mean increases by $1$ and the median increases by $1$.

(C) The mean increases by $1$ and the median increases by $5$. The mean increases by $1$ and the median increases by $5$.

(D) The mean increases by $5$ and the median increases by $1$. The mean increases by $5$ and the median increases by $1$.

(E) The mean increases by $5$ and the median increases by $5$. The mean increases by $5$ and the median increases by $5$.

Answer

Correct choice: (B)

正确答案：（B）

Solution

On Monday, $20$ people come. On Tuesday, $26$ people come. On Wednesday, $16$ people come. On Thursday, $22$ people come. Finally, on Friday, $16$ people come. $20+26+16+22+16=100$, so the mean is $20$. The median is $(16, 16, 20, 22, 26)$ $20$. The coach figures out that actually $21$ people come on Wednesday. The new mean is $21$, while the new median is $(16, 20, 21, 22, 26)$ $21$. Also, the median increases by $1$ because now the median is $21$ instead of $20$. The median and mean both change, so the answer is $\boxed{\textbf{(B)}}$. Another way to compute the change in mean is to notice that the sum increased by $5$ with the correction. So, the average increased by $5/5 = 1$. Then, the median is computed the same way.

周一 $20$ 人，周二 $26$ 人，周三 $16$ 人，周四 $22$ 人，周五 $16$ 人。总和 $100$，均值为 $20$。中位数为 $(16,16,20,22,26)$ 中的 $20$。修正周三为 $21$ 人后，新均值为 $21$，新中位数为 $(16,20,21,22,26)$ 中的 $21$。均值和中位数都增加 $1$，答案为 $\boxed{\textbf{(B)}}$。另一种计算均值变化的方法：总和增加 $5$，平均增加 $5/5=1$。中位数同样计算。

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