📚 Interpreting Data: Averages, Spread and Statistical Diagrams | 数据解读：平均数、离散程度与统计图表
In KS3 Cambridge mathematics, data handling is not just about drawing charts; it is about choosing suitable averages, measuring spread and explaining what the data actually shows. This article revises the essential statistical tools you need for Checkpoint-style questions, including mean, median, mode, range, frequency tables, bar charts, pie charts and line graphs. Work through the paired English-Chinese notes, then test yourself on the exam-style tips at the end.
在 KS3 剑桥数学中，数据处理不仅是画图，更要会选择恰当的平均数、衡量数据分散程度并解释数据真正说明了什么。本文帮助你复习 Checkpoint 常考的核心统计工具，包括平均数、中位数、众数、极差、频数表、条形图、饼图和折线图。请按中英文对照笔记学习，最后用考试型提示检查自己。
1. Three Averages and Range | 三种平均数与极差
The three averages summarise the centre of a data set in different ways. The mean is calculated by adding all values and dividing by the number of values. The median is the middle value when the data are ordered from smallest to largest. The mode is the value that appears most often. The range measures the spread of the data: largest value minus smallest value.
三种平均数以不同方式概括数据集的中心。平均数的计算方法是将所有数值相加再除以数值的个数。中位数是将数据从小到大排序后位于中间位置的数值。众数是出现次数最多的数值。极差衡量数据分散程度，等于最大值减最小值。
Mean = (Sum of all values) ÷ (Number of values) | 平均数 = 所有数值之和 ÷ 数值个数
Median position = (n + 1) ÷ 2 for an ordered list | 中位数位置 = 排序后第 (n + 1) ÷ 2 个
Range = Highest value − Lowest value | 极差 = 最高值 − 最低值
For an even number of values, the median is the average of the two middle values. If no value repeats, the mode does not exist, or a data set can have more than one mode.
当数据个数为偶数时，中位数是中间两个数值的平均数。如果没有数值重复，众数不存在；一个数据集也可以有多个众数。
2. Frequency Tables for Discrete Data | 离散数据的频数表
A frequency table shows how many times each value occurs. It helps you find the mode quickly because the mode is the value with the highest frequency. To find the mean from a frequency table, multiply each value by its frequency, add these products, then divide by the total frequency.
频数表展示每个数值出现的次数。它能帮助你快速找到众数，因为众数是频数最高的数值。要从频数表求平均数，先将每个数值乘以其频数，把这些乘积相加，再除以总频数。
| Score | 得分 | Frequency | 频数 | Score × Frequency | 得分 × 频数 |
|---|---|---|
| 5 | 3 | 15 |
| 6 | 5 | 30 |
| 7 | 2 | 14 |
Mean = (15 + 30 + 14) ÷ (3 + 5 + 2) = 59 ÷ 10 = 5.9
+ 1) ÷ 2 = 5.5th value. This means the median lies between the 5th and 6th ordered values.
如果表格列出了所有数据值，平均数和中位数都可以准确求出。中位数位置使用总频数：总频数为 10，因此中位数位置是 (10 + 1) ÷ 2 = 5.5，表示中位数位于排序后第 5 和第 6 个数值之间。
3. Grouped Frequency Tables | 分组频数表
ean from a grouped table, but you can estimate it by using the midpoint of each class interval.
当数据是连续的或范围较大时，数据会被归入组区间，例如 0 ≤ x < 10 或 10 ≤ x < 20。你无法从分组表中求出精确的平均数，但可以用每个组区间的组中点来估计平均数。
The modal class is the group with the highest frequency, not a single value. The class with the median is found by working out the cumulative frequency until you reach the middle position.
众数组是频数最高的组，而不是单个数值。中位数所在的组通过累加频数找到，直到达到中间位置。
Estimated mean = (Sum of midpoint × frequency) ÷ Total frequency | 估计平均数 = (组中点 × 频数之和) ÷ 总频数
- English: A grouped table loses some accuracy because individual values inside each interval are unknown.
- 中文：分组表会损失一些精确度，因为每个区间内的具体数值未知。
- English: Checkpoint questions usually ask for an estimate, the modal class, or the median interval.
- 中文：Checkpoint 考题通常要求求估计值、众数组或中位数所在区间。
4. Bar Charts and Dual Bar Charts | 条形图与双柱条形图
A bar chart is used for discrete data or categories. The height of each bar represents the frequency. Bars must have equal width, and there should be equal gaps between bars. A dual bar chart compares two related data sets side by side, which is useful for comparing boys and girls or two years.
条形图用于离散数据或类别。每个条形的高度代表频数。条形的宽度必须相等，条形之间应留有相等间隔。双柱条形图将两组相关数据并排比较，适合比较男生与女生或两个年份的数据。
- English: Label both axes clearly: the horizontal axis shows categories, and the vertical axis shows frequency.
- 中文：清楚地标记两个坐标轴：横轴表示类别，纵轴表示频数。
- English: Use a sensible scale so the tallest bar fits comfortably on the grid.
- 中文：使用合理的刻度，让最高的条形能舒适地落在网格内。
- English: In a dual bar chart, provide a key to identify the two sets of bars.
- 中文：在双柱条形图中，要提供图例以识别两组条形。
5. Pie Charts and Calculating Sector Angles | 饼图与圆心角计算
A pie chart shows proportions of a whole. The entire circle represents 360°. To find the angle for each category, multiply the fraction of the total frequency by 360°. This angle is called the sector angle.
饼图展示整体中各部分的比例。整个圆代表 360°。要计算每个类别的圆心角，将该类别频数占总频数的分数乘以 360°。这个角称为扇形圆心角。
Sector angle = (Frequency ÷ Total frequency) × 360° | 圆心角 = (频数 ÷ 总频数) × 360°
12 ÷ 30 × 360° = 144°. Always check that all sector angles add up to 360° before drawing.
例如，如果某类别的频数是 12，总频数是 30，则圆心角为 12 ÷ 30 × 360° = 144°。画图前一定要检查所有圆心角之和是否等于 360°。
- English: Use a protractor and compass for accurate Checkpoint drawings.
- 中文：使用量角器和圆规进行准确的 Checkpoint 绘图。
- English: Label each sector or provide a clear key.
- 中文：给每个扇形标注名称，或提供清晰的图例。
6. Line Graphs and Time Series | 折线图与时间序列
A line graph is used to show how data change over time. Time is always plotted on the horizontal axis. Consecutive points are joined by straight lines, which makes trends easier to see. A time series is a line graph where the data are recorded at regular time intervals.
折线图用于展示数据随时间的变化。时间总是画在横轴上。相邻的点用直线连接，这使得趋势更容易观察。时间序列是一种以固定时间间隔记录数据的折线图。
When interpreting a line graph, describe the overall trend using words such as increasing, decreasing, steady, peak or trough. Do not just list individual points. If the vertical axis does not start at zero, small changes can appear exaggerated, so read the scale carefully.
解读折线图时，用 increasing、decreasing、steady、peak 或 trough 等词语描述总体趋势。不要只列出个别点。如果纵轴不是从零开始，微小的变化可能被放大，因此要仔细读取刻度。
7. Comparing Data Sets Using Averages and Range | 用平均数与极差比较数据集
To compare two data sets, give the mean or median and the range. The average tells you which set is generally higher or lower. The range tells you which set is more spread out or more consistent. A smaller range means the data are more consistent.
要比较两组数据，应给出平均数或中位数以及极差。平均数说明哪一组数据总体上更高或更低。极差说明哪一组数据更分散或更稳定。极差越小，数据越稳定。
- English: Use the mean when there are no extreme outliers; use the median when outliers skew the mean.
- 中文：当没有极端异常值时使用平均数；当异常值使平均数产生偏差时使用中位数。
- English: Write a comparison sentence such as "Boys scored a higher mean of 7.2 than girls at 6.4, but girls had a smaller range of 3, so their scores were more consistent."
- 中文：写比较句，例如 "男生的平均分为 7.2，高于女生的 6.4，但女生的极差仅为 3，因此女生的成绩更稳定。"
8. Worked Example: Choosing the Correct Average | 例题解析：选择正确的平均数
A shop records the number of trainers sold in nine days: 4, 5, 5, 6, 6, 6, 7, 8, 42. The mean is 9.9, but the median is 6 and the mode is 6. The unusually high value 42 pulls the mean up, so the median or mode is a better summary of the typical daily sales.
某商店记录九天运动鞋的销量：4、5、5、6、6、6、7、8、42。平均数为 9.9，但中位数和众数都是 6。异常高的数值 42 拉高了平均数，因此中位数或众数更能代表典型日销量。
Mean = (4 + 5 + 5 + 6 + 6 + 6 + 7 + 8 + 42) ÷ 9 = 89 ÷ 9 ≈ 9.9
Median = 6 | Mode = 6 | Range = 42 − 4 = 38
The large range of 38 shows that daily sales were very spread out. In an exam answer, you should explain that the atypical value 42 distorts the mean, so the median or mode is more useful here.
极差高达 38，说明日销量非常分散。在考试答案中，你应该解释非典型值 42 扭曲了平均数，因此这里中位数或众数更有用。
9. Common Misconceptions and Exam Tips | 常见误区与考试提示
Many students confuse the median with the mean, or forget to order the data before finding the median. Another common mistake is dividing the total frequency incorrectly when calculating the mean from a frequency table. Always divide by the sum of the frequencies, not by the number of rows in the table.
许多学生混淆中位数和平均数，或在找中位数前忘记对数据排序。另一个常见错误是在从频数表求平均数时除以错误的总频数。一定要除以频数之和，而不是除以表格的行数。
- English: Order the data before finding the median; do not pick the middle number from an unordered list.
- 中文：求中位数前先将数据排序；不要从未排序列表中直接挑中间数。
- English: For pie charts, label angles and make sure they total 360°.
- 中文：画饼图时标出圆心角，并确保角度之和为 360°。
- English: In comparisons, always mention both an average and a measure of spread.
- 中文：进行比较时，始终同时提到平均数和离散程度的度量。
- English: Read bar chart scales carefully, especially when the scale does not start at zero.
- 中文：仔细读取条形图的刻度，尤其是刻度不是从零开始的时候。
10. Checkpoint Practice Prompts | Checkpoint 练习提示
Use these prompts to check your understanding. Try writing two or three full sentences for each one, using correct statistical vocabulary.
使用以下提示检查你的理解。尝试为每个提示写两到三个完整句子，并使用正确的统计术语。
- English: Explain how to find the median of 12, 7, 9, 15 and 10.
- 中文：说明如何求 12、7、9、15 和 10 的中位数。
- English: A pie chart has total frequency 40. Find the sector angle for a frequency of 8.
- 中文：某饼图的总频数为 40。求频数为 8 的扇形圆心角。
- English: Compare two classes whose test scores have means of 62 and 65, and ranges of 18 and 9.
- 中文：比较两个班级的测试成绩，其平均数分别为 62 和 65，极差分别为 18 和 9。
When you can answer these without hesitation, you are well prepared for the data handling section of the Cambridge KS3 checkpoint paper.
当你能够毫不犹豫地回答这些问题时，你就为 Cambridge KS3 checkpoint 试卷中的数据处理部分做好了充分准备。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图