📚 Year 7 CIE Statistics: Formula & Theorem Quick Reference Guide | 七年级 CIE 统计：公式定理速查手册
This handbook provides a clear and concise summary of the essential formulas, rules and concepts covered in the Year 7 CIE Statistics curriculum. Keep it close for quick revision and problem-solving support.
本手册清晰简明地总结了七年级 CIE 统计课程所涵盖的核心公式、规则和概念。随身携带以便快速复习和辅助解题。
1. Mean | 平均数
The mean is the average of a set of numbers. To find the mean, add all the values together and then divide by how many values there are.
平均数是一组数据的平均值。计算平均数时，先将所有数值相加，再除以数值的个数。
Mean = (Sum of all values) / (Number of values)
For example, to calculate the mean of 3, 7 and 8: (3 + 7 + 8) / 3 = 18 / 3 = 6.
例如，计算 3、7、8 的平均数：(3 + 7 + 8) / 3 = 18 / 3 = 6。
2. Median | 中位数
The median is the middle value when a data set is ordered from smallest to largest. If there is an odd number of values, the median is the exact middle number. If there is an even number of values, the median is the mean of the two middle numbers.
中位数是将数据集从小到大排序后位于中间的值。如果数据个数是奇数，中位数就是正中间的那个数；如果数据个数是偶数，中位数是中间两个数的平均数。
Example with odd count: 2, 5, 6, 8, 9 — median is 6.
奇数例子：2, 5, 6, 8, 9 — 中位数为 6。
Example with even count: 3, 4, 6, 9 — median = (4 + 6) / 2 = 5.
偶数例子：3, 4, 6, 9 — 中位数 = (4 + 6) / 2 = 5。
3. Mode | 众数
The mode is the value that appears most often in a data set. A set may have one mode, more than one mode, or no mode at all if all values appear equally often.
众数是数据集中出现次数最多的数值。一个数据集可能有一个众数、多个众数，或者当所有数值出现次数相同时没有众数。
Example: in 2, 3, 4, 4, 5, 6, 6, 6, 7, the mode is 6 because it occurs three times.
例如：在 2, 3, 4, 4, 5, 6, 6, 6, 7 中，众数是 6，因为它出现了三次。
If a set is 1, 2, 3, 4, all numbers appear once, so there is no mode.
如果数据集为 1, 2, 3, 4，所有数字各出现一次，因此没有众数。
4. Range | 极差
The range measures how spread out the data is. It is the difference between the largest and smallest values.
极差衡量数据的分散程度。它是最大值与最小值之间的差值。
Range = Largest value − Smallest value
For the data set 14, 21, 8, 32, 9, the largest is 32, the smallest is 8, so the range = 32 − 8 = 24.
对于数据集 14, 21, 8, 32, 9，最大值是 32，最小值是 8，因此极差 = 32 − 8 = 24。
5. Frequency Tables | 频率表
A frequency table shows how often each value or category occurs. The total of the frequencies equals the total number of data items. To calculate the mean from a frequency table, multiply each value by its frequency, add the products and divide by the total frequency.
频率表显示每个数值或类别出现的次数。频率的总和等于数据的总个数。通过频率表计算平均数时，将每个数值乘以其频率，相加后再除以总频率。
Mean = (Sum of (Value × Frequency)) / (Total Frequency)
Example table:
示例表格：
| Value (数值) | Frequency (频率) |
|---|---|
| 1 | 4 |
| 3 | 2 |
| 5 | 4 |
Sum of (value × freq) = (1×4) + (3×2) + (5×4) = 4 + 6 + 20 = 30. Total frequency = 4+2+4 = 10. Mean = 30/10 = 3.
值乘频率之和 = (1×4) + (3×2) + (5×4) = 4 + 6 + 20 = 30。总频率 = 4+2+4 = 10。平均数 = 30/10 = 3。
6. Bar Charts | 条形图
A bar chart uses rectangular bars to represent data. The length or height of each bar shows the frequency or value of the category. Bars can be drawn vertically or horizontally, and they must all have the same width with equal spaces between them.
条形图使用矩形条表示数据。每个条形的高度或长度代表该类别的频率或数值。条形可以垂直或水平绘制，且所有条形的宽度必须相同，间距相等。
Key rules: label both axes, give the chart a title, and use a suitable scale. The vertical axis usually shows frequency, and the horizontal axis shows the category.
关键规则：标注两条坐标轴，给图表添加标题，并使用合适的刻度。纵轴通常显示频率，横轴显示类别。
7. Pie Charts | 饼图
A pie chart displays data as slices of a circle. The size of each slice is proportional to the frequency of the category it represents. The angle for each slice is calculated by:
饼图以圆形切片的形式展示数据。每个切片的大小与其代表的类别频率成正比。每片的角度通过以下公式计算：
Angle = (Category Frequency / Total Frequency) × 360°
For example, if there are 30 students and 12 walk to school, the angle for ‘walk’ is (12/30) × 360° = 144°.
例如，如果有 30 名学生，其中 12 人步行上学，则“步行”的角度为 (12/30) × 360° = 144°。
Always check that all angles add up to 360°, and label each slice clearly.
始终检查所有角度加起来是否为 360°，并清楚地标记每个切片。
8. Pictograms | 象形图
A pictogram uses symbols or pictures to represent data. Each symbol stands for a certain number of items. It is essential to include a key to show what one symbol represents.
象形图使用符号或图片来表示数据。每个符号代表一定数量的项目。必须提供图例来说明一个符号代表什么。
Example: If 1 smiley face represents 4 books, then 5 smiley faces represent 5 × 4 = 20 books. Halved symbols may be used to show smaller quantities.
例如：如果 1 个笑脸代表 4 本书，那么 5 个笑脸代表 5 × 4 = 20 本书。可以用半个符号表示更小的数量。
Pictograms are useful for comparing data visually and making information easy to understand.
象形图有助于直观比较数据，使信息易于理解。
9. Collecting Data | 数据收集
Data can be collected in different ways. Primary data is gathered first-hand by the researcher for a specific purpose (e.g. surveys, experiments). Secondary data is information that has already been collected and published by someone else (e.g. internet, books).
数据可以通过不同方式收集。一手数据是研究者为特定目的亲自收集的（例如调查、实验）。二手数据是他人已经收集并公布的信息（例如网络、书籍）。
Good data collection requires clear questions, a suitable sample, and careful recording. A tally chart is often used to organise data as it is collected. Each tally mark ‘|’ counts one, and groups of five are shown by crossing through four marks.
良好的数据收集需要清晰的问题、合适的样本以及仔细的记录。收集数据时常使用记数表来整理。每个计数符号“|”代表一次，每满五个就用一条斜线划掉四个竖线表示。
Example of a tally: |||| represents 4; with a slash across it becomes a group of 5. This makes counting frequencies faster.
记数示例：|||| 表示 4，划上斜线后成为一组 5。这样可以更快地统计频率。
10. Introduction to Probability | 概率入门
Probability measures the chance that an event will happen. It is always a number between 0 and 1, where 0 means impossible and 1 means certain. Probability can also be expressed as a fraction, decimal or percentage.
概率衡量一个事件发生的可能性。它总是一个介于 0 和 1 之间的数，0 表示不可能，1 表示必然发生。概率也可以用分数、小数或百分比表示。
Probability of an event = (Number of favourable outcomes) / (Total number of possible outcomes)
Example: When you roll a fair six-sided die, the probability of rolling an even number is 3/6 = 1/2, because there are 3 even outcomes (2, 4, 6) out of 6 possible outcomes.
例如：掷一个均匀的六面骰子时，掷出偶数的概率为 3/6 = 1/2，因为在 6 种可能结果中有 3 种是偶数（2, 4, 6）。
The sum of probabilities of all possible outcomes is always 1. If an event is impossible, like rolling a 7 on a standard die, its probability is 0.
所有可能结果的概率之和总是 1。如果某个事件不可能发生，例如在标准骰子上掷出 7，其概率为 0。
11. Comparing Data | 数据比较
When two or more sets of data are compared, measures such as mean, median and range help describe differences. The mean tells us about the average value, while the range shows consistency or spread. A smaller range often indicates that the data is more consistent.
当比较两个或多组数据时，平均数、中位数和极差等度量有助于描述差异。平均数反映平均水平，而极差则体现数据的一致性或分散程度。较小的极差通常意味着数据更稳定。
Example: Data set A: 5, 6, 5, 6, 5 (mean 5.4, range 1). Data set B: 10, 2, 9, 1, 6 (mean 5.6, range 9). Even though means are similar, set A is much less spread out.
示例：数据集 A：5, 6, 5, 6, 5（平均数 5.4，极差 1）。数据集 B：10, 2, 9, 1, 6（平均数 5.6，极差 9）。尽管平均数相似，但 A 的分散程度要小得多。
Always use the same measures when comparing data to make fair conclusions.
比较数据时，要始终使用相同的度量标准，才能得出公平的结论。
12. Statistical Enquiry Cycle | 统计探究循环
The statistical enquiry cycle helps you plan and carry out a statistics project. The main stages are: Pose a question, Collect data, Organise and represent data, Analyse and interpret results, and Reach a conclusion.
统计探究循环可以帮助你规划和实施一个统计项目。主要阶段包括：提出问题、收集数据、整理和呈现数据、分析和解读结果、得出结论。
Each stage is linked. For instance, the kind of question influences what data to collect, and the way data is organised affects how it can be analysed.
每个阶段相互关联。例如，所提问题的类型会影响收集什么数据，而数据整理的方式也会影响分析的方法。
Reflecting on the cycle helps you improve your investigation and spot any possible bias in data collection or analysis.
反思整个循环有助于你改进调查，并发现数据收集或分析中可能存在的偏差。
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