有五个数字。关于这些数字的一些信息如下所示。 极差 = 6 平均数 = 7 中位数 = 8 众数 = 9 求这五个数字。 答：___, ___, ___, ___, ___ 5名男生的平均体重是 68 kg，8名女生的平均体重是 55 kg。为了求出所有孩子的平均体重，Caleb 计算了 68 kg 和 55 kg 的平均值。解释为什么 Caleb 的方法是错误的，并求出正确答案。 答

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1. Use the median and mode to fix the middle and repeated largest values.

Arrange the five numbers in ascending order.

Since the median is $8$, the middle number is $8$:

$\_,\ \_,\ 8,\ \_,\ \_$

Since the mode is $9$, $9$ must occur more often than any other number. As $9>8$, the two $9$s must be after $8$:

$\_,\ \_,\ 8,\ 9,\ 9$

2. Use the range and mean to find the two missing numbers.

The largest number is $9$. Since the range is $6$,

$\text{smallest number}=9-6=3$

So the numbers are

$3,\ \_,\ 8,\ 9,\ 9$

The mean is $7$, so the total is

$5\times 7=35$

Let the missing number be $x$.

$3+x+8+9+9=35$

$x+29=35$

$x=6$

Hence the five numbers are

$3,\ 6,\ 8,\ 9,\ 9$

1. Explain why taking the simple average of the two given averages is incorrect.

Caleb is wrong because there are $5$ boys but $8$ girls. The two averages, $68$ kg and $55$ kg, are based on different numbers of children, so they must be weighted by the group sizes.

2. Find the total mass of all children and divide by the total number of children.

Total mass of boys $=5\times68=340$ kg.

Total mass of girls $=8\times55=440$ kg.

Combined average mass
$=\frac{340+440}{5+8}$
$=\frac{780}{13}=60\text{ kg}.$