📚 Year 7 CIE Statistics: Quick Reference Formula & Theorem Handbook | CIE Year 7 统计公式定理速查手册
Welcome to your go-to revision guide for Year 7 CIE Statistics. This handbook brings together every formula, definition, and key theorem you will meet in the course, explained in clear, paired English and Chinese paragraphs. Use it to test your recall, build confidence, and make sure no mark is lost because you forgot the exact wording of a rule. The sections follow the typical teaching order from data collection right through to basic probability and Venn diagrams, each one accompanied by a worked example or a table you can lean on in the final minutes before the exam.
欢迎使用这本 Year 7 CIE 统计速查手册。它汇集了课程中出现的每一个公式、定义和重要定理，并用清晰对照的英文和中文段落加以解释。你可以用它来检测记忆效果、建立自信，确保不会因为忘记某个规则的准确表述而丢分。各个小节大致按照从数据收集到基础概率和维恩图的教学顺序排列，每一节都配有一个例题或一张表格，方便你在考试前的最后几分钟紧急回顾。
1. Mean (Arithmetic Average) | 平均数
The mean is the most common measure of central tendency. It is found by adding together every value in the data set and then dividing by the total number of values. The formula is written as:
平均数是最常用的集中趋势量数。计算方法是把数据集中的每一个数值相加，再除以数值的总个数。公式写作：
Mean = (Sum of all values) ÷ (Number of values)
If you have five test scores: 12, 15, 18, 21, 14, the sum is 80 and the number of values is 5, so the mean is 80 ÷ 5 = 16. Always check whether the question asks you to round your answer; unless told otherwise, give the exact decimal or fraction.
假如你有五个考试分数：12、15、18、21、14，总和是 80，数值个数是 5，因此平均数为 80 ÷ 5 = 16。始终注意题目是否要求对答案进行取整；除非另有说明，应给出精确的小数或分数。
2. Median | 中位数
The median is the middle value when the data are arranged in order from smallest to largest. For an odd number of values, the median is simply the value sitting in the exact middle position. For an even number of values, the median is the mean of the two middle values.
中位数是将数据按从小到大的顺序排列后处于中间位置的数值。当数据个数为奇数时，中位数就是正中间的那个值；当数据个数为偶数时，中位数是中间两个数值的平均数。
Example: Data set 7, 9, 11, 14, 15 has 5 values, so the median is the 3rd value = 11. Data set 8, 10, 12, 16 has 4 values; the two middle values are 10 and 12, so median = (10 + 12) ÷ 2 = 11.
例题：数据集 7, 9, 11, 14, 15 有 5 个数值，中位数为第 3 个数值 = 11。数据集 8, 10, 12, 16 有 4 个数值；中间两个值是 10 和 12，因此中位数 = (10 + 12) ÷ 2 = 11。
3. Mode | 众数
The mode is the value that appears most often in a data set. There can be one mode, more than one mode (bimodal or multimodal), or no mode at all if every value appears the same number of times. The mode is the only average that can be used with non-numerical data, such as colours or favourite subjects.
众数是数据集中出现次数最多的数值。可能只有一个众数，也可能有多个众数（双众数或多重众数），如果每个数值都出现同样的次数则可能没有众数。众数是唯一可以用于非数值数据的平均数，比如颜色或最喜欢的科目。
For the data 4, 5, 7, 5, 6, 5, 9, the mode is 5 because it appears three times while all others appear fewer times. If you have shoe sizes 3, 4, 4, 5, 5, 6, both 4 and 5 are modes (bimodal).
对于数据 4, 5, 7, 5, 6, 5, 9，众数是 5，因为它出现了三次而其他数的出现次数都更少。如果你有鞋码 3, 4, 4, 5, 5, 6，那么 4 和 5 都是众数（双众数）。
4. Range | 极差
The range measures how spread out the data are. It is calculated by subtracting the smallest value from the largest value. A larger range indicates greater variability.
极差用来衡量数据的散布程度。计算方法是用最大值减去最小值。极差越大说明数据的变动幅度越大。
Range = Largest value − Smallest value
For the data set 23, 45, 12, 38, 56, the largest is 56 and the smallest is 12, so the range is 56 − 12 = 44. Be careful with negative numbers: for −5, −2, 0, 3, the largest is 3 and the smallest is −5, so range = 3 − (−5) = 8.
对于数据集 23, 45, 12, 38, 56，最大值是 56，最小值是 12，因此极差 = 56 − 12 = 44。遇到负数要小心：对于 −5, −2, 0, 3，最大值是 3，最小值是 −5，因此极差 = 3 − (−5) = 8。
5. Frequency Tables | 频率表
A frequency table organises raw data into categories or values alongside the number of times each appears (frequency). Tally marks are often used when collecting data. From a frequency table you can easily find the mode (the value with the highest frequency) and total number of data items (sum of frequencies).
频率表把原始数据按类别或数值进行整理，并列出每个数值出现的次数（频数）。收集数据时常用画正字计数。借助频率表，你可以轻松找到众数（频数最高的数值）以及数据总个数（频数之和）。
| Score / 分数 | Tally / 计数 | Frequency / 频数 |
|---|---|---|
| 4 | || | 2 |
| 5 | |||| | 4 |
| 6 | ||| | 3 |
The total frequency is 2+4+3 = 9, so there are 9 data points. The mode is 5 because it has the highest frequency (4).
总频数为 2+4+3 = 9，因此共有 9 个数据点。众数是 5，因为它的频数最高（4）。
6. Bar Charts | 条形图
Bar charts display categorical data using rectangular bars whose heights (or lengths in a horizontal bar chart) are proportional to the values they represent. Gaps between the bars emphasise that the categories are separate. A bar-line chart is a variation where thin lines replace bars, often used when drawing by hand.
条形图用矩形条来表示分类数据，条的高度（在水平条形图中为长度）与所代表的数值成比例。条与条之间的空隙强调各个类别是相互独立的。条形线图是一种变体，用细线代替条形，通常用于手绘图表。
Key features: label both axes clearly, write the category names under the bars, give the chart a title, and choose a scale that fits the graph paper. For a bar-line chart, place a cross or dot at the top of each frequency and join them with vertical lines.
主要特征：清晰地标注两个坐标轴，在条形下方写出类别名称，给图表加上标题，并选择适合图纸的刻度。对于条形线图，在每个频数顶端打叉或画点，再用竖线将它们连接起来。
7. Pictograms | 象形图
A pictogram uses identical small pictures or symbols to represent a fixed number of items. It makes data visually appealing and easy to compare. A key must be provided to show how many units one symbol stands for. A half or part of a symbol can be used to represent smaller amounts.
象形图用相同的小图画或符号来表示固定数量的项目。它让数据看起来直观且易于比较。必须提供一个图例，说明一个符号代表多少个单位。可以使用半个或部分符号来表示较小的数量。
If one smiley face represents 4 students, then 3 smiley faces mean 12 students, and half a smiley face means 2 students. Always check the key carefully and draw symbols of identical size when constructing a pictogram.
如果一个笑脸代表 4 名学生，那么 3 个笑脸就意味着 12 名学生，半个笑脸则表示 2 名学生。绘制象形图时一定要仔细查看图例，并画出大小相同的符号。
8. Pie Charts | 饼图
Pie charts show proportions of a whole. The whole circle (360°) represents the total frequency. Each category is represented by a sector whose angle is calculated using the formula:
饼图用来显示整体中的各部分比例。整个圆（360°）代表总频数。每个类别用一个扇形表示，扇形的角度通过下面的公式计算：
Sector angle = (Category frequency ÷ Total frequency) × 360°
After calculating all angles, check they sum to 360°. Use a protractor to draw each sector, label each one with the category name, and give the chart a title. If percentages are given, the angle formula becomes (Percentage ÷ 100) × 360°.
计算完所有角度后，检查它们的总和是否为 360°。用量角器画出每个扇形，标注每个类别的名称，并给图表加上标题。如果给出的是百分比，角度公式变为 (百分比 ÷ 100) × 360°。
9. Probability Scale | 概率尺度
Probability describes how likely an event is to happen. It is measured on a scale from 0 to 1, where 0 means impossible and 1 means certain. Probabilities can be written as fractions, decimals, or percentages. An event that is equally likely as not has a probability of ½, 0.5, or 50%.
概率描述一个事件发生的可能性有多大。它用 0 到 1 之间的尺度来衡量，其中 0 表示不可能，1 表示必然发生。概率可以用分数、小数或百分数表示。一个发生与不发生的可能性相同的事件，其概率为 ½、0.5 或 50%。
A probability of 0.2 is closer to impossible; a probability of 0.85 is closer to certain. Marking events on a probability scale helps build intuition. Words like ‘likely’, ‘unlikely’, ‘even chance’ and ‘very likely’ are often used in Year 7 questions.
概率为 0.2 的事件更接近不可能；概率为 0.85 的事件更接近必然。把事件标在概率尺度上有助于培养直观理解。Year 7 的题目中常出现“很可能”、“不太可能”、“可能性各半”和“极有可能”等词语。
10. Calculating Probability | 概率计算
For equally likely outcomes, the probability of an event E is given by:
对于等可能结果，事件 E 的概率由以下公式给出：
P(E) = Number of favourable outcomes ÷ Total number of possible outcomes
When rolling a fair six‑sided die, P(rolling a 4) = 1/6. P(rolling an even number) = 3/6 = 1/2. The sum of probabilities of all possible outcomes is always 1. The probability of an event not happening is 1 minus the probability that it does happen: P(not E) = 1 − P(E).
掷一个均匀的六面骰子时，P(掷出 4) = 1/6。P(掷出偶数) = 3/6 = 1/2。所有可能结果的概率之和始终为 1。某事件不发生的概率等于 1 减去它发生的概率：P(并非 E) = 1 − P(E)。
11. Venn Diagrams | 维恩图
A Venn diagram uses overlapping circles to show relationships between sets. The rectangle represents the universal set. Each circle represents a set. The overlapping region (intersection) contains elements common to both sets. The union of two sets consists of all elements in either set (or both).
维恩图用相互交叠的圆圈来表示集合之间的关系。矩形代表全集，每个圆代表一个集合。重叠区域（交集）包含两个集合共有的元素。两个集合的并集则包含属于两个集合中任意一个（或同时属于两者）的所有元素。
When given numbers, place them in the correct regions: first fill the intersection, then the parts unique to each set, and finally any numbers outside both sets but inside the rectangle. From the completed diagram you can find probabilities such as P(A ∩ B) and P(A ∪ B).
当给出数字时，把它们放在正确的区域：先填写交集部分，再填写每个集合独有的部分，最后填写在两个集合之外但在矩形之内的数字。根据完成的维恩图，你可以求出 P(A ∩ B) 和 P(A ∪ B) 等概率。
12. Mean from a Frequency Table | 从频率表求平均数
When data are grouped in a frequency table, you cannot simply add all values. Instead, multiply each value by its frequency to get the total for that group, sum those products, and then divide by the total frequency.
当数据以频率表的形式分组列出时，不能简单地把所有数值相加。应该先将每个数值乘以其频数，得到该组的总和，再把这些乘积相加，最后除以总频数。
Mean = (Σ (value × frequency)) ÷ (Total frequency)
The symbol Σ (capital sigma) means ‘sum of’. Create an extra column in the table for ‘value × frequency’. For example, if the table shows score 2 with frequency 5, score 3 with frequency 7 and score 4 with frequency 3, then Σ(value × freq) = (2×5) + (3×7) + (4×3) = 10 + 21 + 12 = 43. Total frequency = 5+7+3 = 15. Mean = 43 ÷ 15 ≈ 2.87 (rounded to 3 s.f. or as instructed).
符号 Σ（大写西格玛）表示“求和”。在表格中增加一列“数值 × 频数”。例如，如果表格显示分数 2（频数 5）、分数 3（频数 7）和分数 4（频数 3），那么 Σ(数值 × 频数) = (2×5) + (3×7) + (4×3) = 10 + 21 + 12 = 43。总频数 = 5+7+3 = 15。平均数 = 43 ÷ 15 ≈ 2.87（按题目要求取三位有效数字或按其他要求）。
Published by TutorHao | Statistics Revision Series | aleveler.com
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