综合练习题 - 第3章 Representations of Data
以下是6道综合练习题，涵盖茎叶图、直方图、箱线图、异常值、偏度、数据比较等所有核心知识点。
The stem and leaf diagram below shows the heights (in cm) of 25 students.
| Stem | Leaf | Key: \( 15 \mid 2 = 152 \, \text{cm} \) |
|---|---|---|
| 15 | 2 3 5 6 8 | (6) |
| 16 | 0 1 1 3 4 7 8 | (7) |
| 17 | 0 2 2 5 6 9 | (6) |
| 18 | 1 3 4 8 9 | (5) |
| 19 | 0 2 5 | (3) |
a) Draw the stem and leaf diagram.
b) Find \( Q_1, Q_2, Q_3 \).
c) Calculate the interquartile range (IQR).
解答过程：
b) 四分位数计算：总数据量25
c) 四分位距计算：
\( \text{IQR} = Q_3 - Q_1 = 184 - 161 = 23 \, \text{cm} \)
A histogram shows the time (in minutes) spent on homework by 60 students. The table below summarises the data:
| Time (min) | \( 10 \leq t < 20 \) | \( 20 \leq t < 30 \) | \( 30 \leq t < 40 \) | \( 40 \leq t < 60 \) |
|---|---|---|---|---|
| Frequency | 10 | 20 | 15 | 15 |
a) Calculate the frequency density for each class.
b) Draw the histogram and frequency polygon.
解答过程：
a) 频率密度计算：频率密度 = 频率 ÷ 组宽
| 组别 | 频率 | 组宽 | 频率密度 |
|---|---|---|---|
| 10-20 | 10 | 10 | 1.0 |
| 20-30 | 20 | 10 | 2.0 |
| 30-40 | 15 | 10 | 1.5 |
| 40-60 | 15 | 20 | 0.75 |
b) 直方图绘制要点：
The following data show the number of books read by 15 students in a month:
8, 12, 15, 18, 20, 22, 25, 28, 30, 32, 35, 40, 45, 50, 80
a) Find \( Q_1, Q_2, Q_3 \) and IQR.
b) Determine if there are any outliers using \( k = 1.5 \).
解答过程：
a) 四分位数计算：数据已排序
b) 异常值判断：使用公式 \( Q_3 + k \times \text{IQR} \) 和 \( Q_1 - k \times \text{IQR} \)
Two box plots below show the test scores of students in Class A and Class B.
| Class A | Class B | |
|---|---|---|
| \( Q_1 \) | 55 | 60 |
| \( Q_2 \) | 65 | 70 |
| \( Q_3 \) | 75 | 80 |
| Min | 40 | 50 |
| Max | 85 | 90 |
a) Draw box plots for both classes.
b) Compare the test scores of Class A and Class B in terms of location and spread.
解答过程：
b) 数据对比分析：
位置度量（Location）：
离散度量（Spread）：
整体分布：
For a data set, the mean is 50, the median is 45, and the standard deviation is 10.
a) Calculate the skewness using the formula \( \frac{3(\text{mean} - \text{median})}{\text{standard deviation}} \).
b) Describe the skewness and justify using the relationship between mean, median, and the skewness formula.
解答过程：
a) 偏度计算：
\[ \text{偏度} = \frac{3(\text{mean} - \text{median})}{\text{standard deviation}} = \frac{3(50 - 45)}{10} = \frac{3 \times 5}{10} = 1.5 \]
b) 偏度描述与解释：
The back to back stem and leaf diagram below shows the number of goals scored by two football teams in a season.
| Team A | Stem | Team B |
|---|---|---|
| 8 6 5 | 0 | 7 9 |
| 9 7 5 3 | 1 | 8 6 4 2 |
| 8 6 4 2 | 2 | 9 7 5 3 |
| 5 3 1 | 3 | 8 6 4 |
| 2 | 4 | 7 5 |
Key: \( 1 \mid 5 = 15 \) goals
a) Find the median, \( Q_1, Q_3 \) for each team.
b) Compare the goal-scoring performance of Team A and Team B.
解答过程：
a) 各队统计量计算：
Team A数据：5,6,8,13,15,17,19,22,24,26,28,31,33,35,42（共15个）
Team B数据：7,9,12,14,16,18,23,25,27,29,34,36,38,45,47（共15个）
b) 进球表现对比：