📚 Handling Data and Probability Review (Page 308) | 数据处理与概率复习（第308页）
This article covers the key topics from the Handling Data and Probability section of the Cambridge KS3 Mathematics curriculum, corresponding to the review exercises on page 308. We will revisit essential concepts and methods, from calculating averages and interpreting statistical diagrams to using probability scales and Venn diagrams. Each section pairs English explanations with Chinese translations to support bilingual learners.
本文涵盖剑桥 KS3 数学课程中数据处理与概率部分的关键主题，对应第308页的复习练习。我们将回顾基本概念和方法，从计算平均值和解读统计图表，到使用概率尺度和文氏图。每个部分都提供英文解释与中文翻译，以帮助双语学习者。
1. Mean, Median, Mode and Range | 均值、中位数、众数和极差
The three measures of average — mean, median and mode — summarise a data set in different ways. The mean is found by adding all values and dividing by the number of values. The median is the middle value when the data are arranged in order. If there are two middle numbers, take their mean. The mode is the value that appears most frequently. The range measures spread: largest value minus smallest value.
三种平均数——均值、中位数和众数——以不同方式概括一组数据。均值是将所有数值相加后除以数值的个数。中位数是将数据按顺序排列后中间的值；如果有两个中间数，则取它们的均值。众数是出现频率最高的数值。极差衡量数据的分散程度：最大值减去最小值。
For a small data set like 4, 7, 7, 9, 12: mean = (4+7+7+9+12) ÷ 5 = 7.8; median = 7 (the third value); mode = 7; range = 12 − 4 = 8.
对于小数据集如 4, 7, 7, 9, 12：均值 = (4+7+7+9+12) ÷ 5 = 7.8；中位数 = 7（第三个值）；众数 = 7；极差 = 12 − 4 = 8。
2. Frequency Tables and Grouped Data | 频数表与分组数据
When data are presented in a frequency table, you can still find the mean, median, mode and range. To calculate the mean from a frequency table, multiply each value by its frequency, sum these products, then divide by the total frequency. The median is located by finding the position using total frequency (n+1)/2 and identifying the corresponding data value from cumulative frequency. The modal class for grouped data is the interval with the highest frequency.
当数据以频数表呈现时，仍可求出均值、中位数、众数和极差。从频数表计算均值时，将每个值乘以其频数，求和这些乘积，再除以总频数。中位数通过总频数 (n+1)/2 定位，并在累积频数中找出对应的数据值。对于分组数据，众数所在组是频数最高的区间。
An example: scores 1 (freq 3), 2 (freq 5), 3 (freq 2). Mean = (1×3 + 2×5 + 3×2) ÷ (3+5+2) = 19 ÷ 10 = 1.9. Median position = (10+1)/2 = 5.5th, which lies in value 2 (since cumulative frequencies: 3, 8).
例如：分数 1（频数3）, 2（频数5）, 3（频数2）。均值 = (1×3 + 2×5 + 3×2) ÷ (3+5+2) = 19 ÷ 10 = 1.9。中位数位置 = (10+1)/2 = 5.5，位于值 2（累积频数为 3, 8）。
3. Introduction to Probability | 概率入门
Probability measures how likely an event is to happen. It is always a number between 0 and 1. A probability of 0 means the event is impossible; a probability of 1 means it is certain. Probabilities can be expressed as fractions, decimals or percentages. The sum of probabilities of all possible outcomes of an experiment is 1.
概率衡量事件发生的可能性。它总是介于 0 和 1 之间的一个数。概率为 0 表示事件不可能发生；概率为 1 表示事件必然发生。概率可用分数、小数或百分数表示。一次试验所有可能结果的概率之和为 1。
Basic formula: Probability of an event = (number of favourable outcomes) ÷ (total number of possible outcomes), assuming all outcomes are equally likely.
基本公式：事件的概率 =（有利结果的个数）÷（所有可能结果的总数），假设所有结果等可能。
P(A) = n(A) / n(S)
where n(A) is the number of outcomes in event A and n(S) is the total number of outcomes in the sample space.
其中 n(A) 是事件 A 中的结果数，n(S) 是样本空间中的结果总数。
4. The Probability Scale | 概率尺度
The probability scale is a line from 0 to 1 on which you can mark the likelihood of events. Words such as impossible, unlikely, even chance, likely and certain correspond to regions on this scale. An even chance has a probability of 0.5, or ½, or 50%.
概率尺度是一条从 0 到 1 的直线，你可以在上面标记事件的可能性。词语如不可能、不太可能、等可能、很可能和必然对应尺度上的区域。等可能事件的概率为 0.5，即 ½，或 50%。
When rolling a fair six-sided die, the probability of rolling a 3 is 1/6 ≈ 0.167, which is ‘unlikely’. The probability of rolling an even number is 3/6 = 0.5, an ‘even chance’. The probability of rolling a number less than 7 is 1, ‘certain’.
掷一枚均匀六面骰子时，掷出 3 的概率是 1/6 ≈ 0.167，属于“不太可能”。掷出偶数的概率是 3/6 = 0.5，属于“等可能”。掷出小于 7 的数字的概率是 1，属于“必然”。
5. Sample Space Diagrams | 样本空间图
A sample space is the set of all possible outcomes of an experiment. For two events, a sample space diagram (or possibility space diagram) helps list all outcomes systematically. A common method is a two-way table showing all combinations of the outcomes of each event.
样本空间是一次试验所有可能结果的集合。对于两个事件，样本空间图（或可能性空间图）有助于系统地列出所有结果。常用方法是双向表，显示每个事件结果的所有组合。
For example, when flipping a coin and rolling a fair die, the sample space has 2 × 6 = 12 outcomes. The table has coin outcomes (H, T) as columns and die outcomes (1–6) as rows. You can then calculate probabilities: P(heads and even) = number of favourable outcomes (3) / 12 = ¼.
例如，抛一枚硬币并掷一枚均匀骰子时，样本空间有 2 × 6 = 12 个结果。双向表以硬币结果（正、反）为列，骰子结果（1–6）为行。然后可以计算概率：P(正面且偶数) = 有利结果数 (3) / 12 = ¼。
6. Probability of Combined Events | 复合事件的概率
When two events are independent, the outcome of one does not affect the other. To find the probability that both independent events occur, multiply their individual probabilities. To find the probability that either event A or event B occurs, add the probabilities only if they are mutually exclusive (cannot happen at the same time).
当两个事件独立时，一个事件的结果不影响另一个。求两个独立事件同时发生的概率，将各自的概率相乘。求事件 A 或事件 B 发生的概率，只有在它们互斥（不能同时发生）时才将概率相加。
For non-mutually exclusive events, use the addition rule: P(A or B) = P(A) + P(B) − P(A and B). This avoids double-counting the overlap.
对于非互斥事件，使用加法规则：P(A 或 B) = P(A) + P(B) − P(A 且 B)。这避免了重叠部分的重复计算。
Example: drawing a card from a standard pack, P(King) = 4/52, P(Heart) = 13/52, P(King and Heart) = 1/52. Therefore P(King or Heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.
示例：从一副标准扑克牌中抽一张牌，P(国王) = 4/52, P(红心) = 13/52, P(国王且红心) = 1/52。因此 P(国王或红心) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13。
7. Venn Diagrams and Set Notation | 文氏图与集合符号
A Venn diagram shows relationships between sets. The universal set E (or U) contains all elements under consideration. Sets are represented by circles. The intersection (A ∩ B) is the region where sets overlap; the union (A ∪ B) includes all elements in either set. The complement of A (A’) is everything in the universal set not in A.
文氏图展示集合之间的关系。全集 E（或 U）包含所考虑的所有元素。集合用圆圈表示。交集（A ∩ B）是集合重叠的区域；并集（A ∪ B）包含任一集合中的所有元素。A 的补集（A’）是全集中不在 A 内的所有元素。
When filling a Venn diagram, start by placing the intersection values, then the remaining parts of each set, and finally the elements outside all sets but inside the universal set. This method ensures accuracy when calculating probabilities from Venn diagrams.
填写文氏图时，先放置交集部分的值，然后是每个集合的其余部分，最后是在全集内但不在任何集合中的元素。此方法可确保从文氏图计算概率时的准确性。
8. Using Venn Diagrams to Find Probability | 使用文氏图求概率
Given a Venn diagram with numbers representing frequencies or probabilities, you can find P(A), P(B), P(A ∩ B), P(A ∪ B) and conditional probabilities. Probability is simply the number in the region of interest divided by the total number in the universal set (if frequencies are used) or read directly if probabilities are given.
给定一个文氏图，其中的数字代表频数或概率，你可以求出 P(A)、P(B)、P(A ∩ B)、P(A ∪ B) 和条件概率。概率就是感兴趣区域的数字除以全集中的总数（如果使用频数），或者如果直接给出概率，则直接读取。
For a survey of 30 students, 12 study Art (A), 15 study Biology (B), 5 study both. The Venn diagram shows 5 in the intersection, 7 in A only, 10 in B only, and 8 outside. Then P(A) = 12/30 = 0.4; P(A ∩ B) = 5/30 = 1/6; P(A ∪ B) = (7+5+10)/30 = 22/30 = 11/15. The probability a student chosen at random studies neither subject is 8/30 = 4/15.
一项针对 30 名学生的调查，12 人选修艺术（A），15 人选修生物（B），5 人两者都选。文氏图显示交集为 5，仅 A 为 7，仅 B 为 10，外部为 8。那么 P(A) = 12/30 = 0.4；P(A ∩ B) = 5/30 = 1/6；P(A ∪ B) = (7+5+10)/30 = 22/30 = 11/15。随机选一名学生，其两科都不修的概率为 8/30 = 4/15。
9. Two-Way Tables and Probability | 双向表与概率
Two-way tables organise data according to two categorical variables. They are especially useful for finding conditional probabilities and checking whether events are independent. Each cell shows the frequency of a specific combination. The totals in the margins are important for probability calculations.
双向表根据两个分类变量组织数据。它们对于求条件概率和检验事件是否独立特别有用。每个单元格显示特定组合的频数。边栏的总计对于概率计算很重要。
You can find the probability of an event directly from the table by taking the relevant total and dividing by the overall total. For example, a table showing gender and favourite sport can answer: P(random student is male and likes football) = cell frequency / total. For conditional probability, restrict the sample space to the given condition.
你可以直接从表中通过相关总计除以总体总计来求一个事件的概率。例如，一张显示性别和喜好运动的表格可以回答：P(随机学生为男性且喜欢足球) = 单元格频数 / 总计。对于条件概率，将样本空间限制在给定条件下。
10. Interpreting Statistical Diagrams | 解读统计图表
KS3 includes a variety of diagrams: bar charts, pie charts, line graphs, scatter graphs and stem-and-leaf diagrams. Each type displays data differently and suits particular purposes. A bar chart compares frequencies of categories. A pie chart shows proportions. A line graph describes trends over time. A scatter graph reveals correlation between two variables. A stem-and-leaf diagram retains original data while showing distribution.
KS3 涵盖多种图表：条形图、饼图、折线图、散点图和茎叶图。每种图表以不同方式显示数据，适用于特定目的。条形图比较各类别的频数。饼图展示比例。折线图描述随时间变化的趋势。散点图揭示两个变量之间的相关性。茎叶图在展示分布的同时保留原始数据。
When interpreting diagrams, always read titles, axis labels and scales carefully. For pie charts, remember that the total angle is 360°; the frequency for a sector is (angle/360) × total frequency. For scatter graphs, describe the relationship as positive, negative or no correlation, and note any outliers.
解读图表时，务必仔细阅读标题、坐标轴标签和刻度。对于饼图，记住总角度为 360°；一个扇区的频数为 (角度/360) × 总频数。对于散点图，将关系描述为正相关、负相关或无相关，并注意任何异常值。
11. Avoiding Common Mistakes | 避免常见错误
When working with averages, remember to order the data before finding the median. Confusing mean with median is a frequent error. In probability, ensure that all outcomes are equally likely before using ‘favourable over total’. When adding probabilities for combined events, always check for mutual exclusivity; otherwise use the correct formula. In Venn diagrams, the intersection is often counted twice if you are not careful.
处理平均数时，记住在求中位数之前要排序。混淆均值和标准差是常见错误。在概率中，确保所有结果等可能存在才使用“有利/总数”。为复合事件相加概率时，务必检查是否互斥；否则要用正确公式。在文氏图中，如果不小心，交集部分常常会被计算两次。
Another tip: when a question says ‘estimate the probability’, you may need to use relative frequency from an experiment. Recall the law of large numbers: as the number of trials increases, the relative frequency gets closer to the theoretical probability.
另一个提示：当问题要求“估计概率”时，你可能需要使用来自实验的相对频率。记住大数定律：随着试验次数增加，相对频率逐渐接近理论概率。
12. Summary and Practice Advice | 总结与练习建议
The topics on page 308 bring together essential KS3 statistics and probability skills. Make sure you can switch comfortably between fractions, decimals and percentages when expressing probability. Practise constructing sample spaces and Venn diagrams from word problems. Regular practice with past paper questions will build speed and confidence.
第308页的主题汇集了 KS3 统计与概率的基本技能。确保你能在表达概率时自如切换分数、小数和百分数。练习根据文字题构建样本空间和文氏图。经常练习往年试题将提升速度和自信心。
For further revision, revisit earlier chapters in your Cambridge textbook and complete the end-of-unit assessments. Use the bilingual summaries here to reinforce your understanding of key terms in both English and Chinese.
进一步复习时，请重温剑桥教材前面的章节，并完成单元末评估。使用本文的双语总结巩固对关键术语的英中理解。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply