📚 KS3 CIE Statistics: Formula & Theorem Quick Reference Handbook | KS3 CIE 统计：公式定理速查手册
This quick reference handbook collects every essential formula, theorem and key concept you will meet in the CIE KS3 Statistics curriculum. Use it to revise efficiently, check your working or build confidence before assessments. Each section pairs clear definitions with practical examples, so you can move from ‘I think I know it’ to ‘I can apply it fluently’.
这本速查手册汇集了 CIE KS3 统计课程中所有核心公式、定理和重要概念。你可以用它高效复习、检查解答或提升考试信心。每个小节都将清晰的定义与实用示例配对，帮助你从“好像懂了”顺利过渡到“能流畅应用”。
1. Mean | 平均数
The mean (or arithmetic average) is found by adding all the values together and then dividing by the number of values. It is the most common measure of central tendency.
平均数（算术平均数）是把所有数据相加，再除以数据个数得到的。它是描述数据中心位置最常用的指标。
- For a list of raw data: Mean = (sum of all values) / (number of values)
- 对于原始数据列表：平均数 = (所有数值之和) / (数据个数)
In symbols, if we have values x₁, x₂, …, xₙ, then Mean = (Σx)/n where n is the count of values.
用符号表示，如果有数值 x₁, x₂, …, xₙ，那么平均数 = (Σx)/n，其中 n 是数据的个数。
Mean = (Sum of all data) ÷ (Number of data points)
If a frequency table is given, use: Mean = Σ(f × x) / Σf, where f is the frequency and x is the data value.
若给出频数表，应用公式：平均数 = Σ(f × x) / Σf，其中 f 是频数，x 是对应的数据值。
2. Median | 中位数
The median is the middle value when the data is arranged in order of size. It splits the dataset into two equal halves and is not affected by extreme values (outliers).
中位数是将数据按大小排序后最中间的那个值。它将数据集平分成两半，不受极端值（异常值）影响。
- For an odd number of values: Median = the value at position (n+1)/2.
- 数据个数为奇数时：中位数位于第 (n+1)/2 个位置。
- For an even number of values: Median = the average of the values at positions n/2 and n/2 + 1.
- 数据个数为偶数时：中位数是第 n/2 个和第 n/2+1 个位置数值的平均数。
First, always sort the data from smallest to largest. Then find the position, not the value itself.
务必先将数据从小到大排序，然后找出位置，而不是直接找值本身。
3. Mode | 众数
The mode is the value that appears most frequently in a data set. A set may have one mode, more than one mode (bimodal or multimodal), or no mode at all if all values occur equally often.
众数是数据集中出现次数最多的值。一组数据可能有一个众数、多个众数（双众数或多众数），或者如果没有值重复出现则无众数。
- From a frequency table, the mode is the value with the highest frequency.
- 在频数表中，众数是频数最高的那个数值。
- For grouped data, the modal class is the class interval with the largest frequency.
- 对于分组数据，众数组是频数最大的那个组区间。
The mode is especially useful for categorical data, where mean and median may not make sense.
众数对分类数据尤其有用，因为在这类数据中平均数和位数可能没有意义。
4. Range | 极差
The range is a simple measure of spread. It tells you how far apart the smallest and largest values are.
极差是一种简单的离散程度度量，它告诉你最小值与最大值相距多远。
Range = Largest value − Smallest value
A small range suggests the data is tightly clustered; a large range indicates wide variation. Remember that the range is sensitive to outliers.
极差小说明数据集中紧凑；极差大则说明变异大。注意极差容易受异常值影响。
5. Mean from a Frequency Table | 频数表求平均数
When data is presented in a frequency table, you multiply each data value by its frequency, sum these products, then divide by the total frequency.
当数据以频数表形式呈现时，要将每个数据值乘以其频数，求和后除以总频数。
Example table (test scores):
示例表格（测验分数）：
| Score (x) | Frequency (f) | f × x |
| 2 | 3 | 6 |
| 3 | 5 | 15 |
| 4 | 2 | 8 |
| Total | 10 | 29 |
Mean = 29 ÷ 10 = 2.9
平均数 = 29 ÷ 10 = 2.9
Always add a column for f×x if it is not already provided; this reduces careless errors.
如果表格中没有 f×x 这一列，最好自己添加，这样可以减少粗心错误。
6. Estimated Mean from Grouped Data | 分组数据求估计平均数
For grouped data, we do not know each individual value, so we use the midpoint of each class interval as an estimate.
对于分组数据，我们不知道每个单独的数值，因此用每个组区间的中点值作为估计。
- Find the midpoint of each class: (lower bound + upper bound) / 2.
- 求每组的组中值：(下界 + 上界) / 2。
- Multiply each midpoint by its frequency, sum these, then divide by total frequency.
- 用每个组中值乘以对应的频数，求和后除以总频数。
Estimated Mean = Σ(f × midpoint) / Σf
Example: class 0 ≤ h < 10, midpoint 5, frequency 4 → f × midpoint = 20. Continue for all classes.
示例：组别 0 ≤ h < 10，组中值 5，频数 4 → f × 组中值 = 20。所有组都这样处理。
7. Probability Scale and Basic Rules | 概率尺度与基本法则
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). It can be expressed as a fraction, decimal or percentage.
概率衡量事件发生的可能性大小，尺度从 0（不可能）到 1（必然）。可以用分数、小数或百分数表示。
Probability of an event = (Number of favourable outcomes) / (Total number of equally likely outcomes)
The sum of probabilities of all possible outcomes is always 1. Complementary events: P(not A) = 1 − P(A).
所有可能结果的概率之和总为 1。对立事件：P(非 A) = 1 − P(A)。
8. Venn Diagrams and Probability | 维恩图与概率
A Venn diagram shows sets and their overlaps. In KS3, you will use simple Venn diagrams (usually 2 sets) to solve probability problems.
维恩图显示集合及其交集。在 KS3 阶段，你会使用简单的维恩图（通常两个集合）解决概率问题。
- To find the probability that an item belongs to set A: P(A) = Number in A / Total number of items.
- 求某元素属于集合 A 的概率：P(A) = A 中的元素数 / 总元素数。
- P(A ∪ B) = probability of A or B or both: add all numbers inside either circle, then divide by total.
- P(A ∪ B) 表示 A 或 B 或两者都发生的概率：把在两个圆圈内的所有数字加起来，再除以总数。
- P(A ∩ B) = probability of both A and B: only the overlap region.
- P(A ∩ B) 表示 A 和 B 同时发生的概率：只取重叠区域。
Always write the total number of items outside the rectangle. The sum of all regions inside the rectangle equals the total.
记得在矩形外标出总元素数。矩形内所有区域的数字之和等于总数。
9. Tree Diagrams for Combined Events | 树状图处理复合事件
Tree diagrams display all possible outcomes of two or more events. Multiply along branches for ‘and’ probabilities, add separate branch outcomes for ‘or’ probabilities.
树状图展示两个或多个事件的所有可能结果。顺着分支相乘求“且”的概率，把不同分支结果的概率相加求“或”的概率。
- For independent events: P(A then B) = P(A) × P(B).
- 对于独立事件：P(A 然后 B) = P(A) × P(B)。
- If an object is not replaced, probabilities change on the second branch (conditional probability at a basic level).
- 如果不放回，第二次的概率会改变（基础层面的条件概率）。
Always check that probabilities from a single point sum to 1. Label branches clearly with outcomes and probabilities.
务必检查从同一点出发的所有分支概率之和为 1。清晰地标出每条分支的结果和概率。
10. Scatter Graphs and Correlation | 散点图与相关关系
A scatter graph shows the relationship between two numerical variables. The strength and direction of a linear pattern is called correlation.
散点图显示两个数值变量之间的关系。线性模式的强度和方向称为相关性。
- Positive correlation: as one variable increases, the other tends to increase.
- 正相关：一个变量增加时，另一个变量也倾向于增加。
- Negative correlation: as one variable increases, the other tends to decrease.
- 负相关：一个变量增加时，另一个变量倾向于减少。
- No correlation: no clear pattern.
- 无相关：没有明显的模式。
Correlation does not imply causation. A line of best fit can be drawn by eye, passing through the middle of the points for predictions (interpolation).
相关关系并不意味着因果关系。可以用目测画一条最佳拟合线，使其穿过点的中间以进行预测（内插）。
11. Pie Charts and Bar Charts – Interpreting Proportions | 饼图与条形图——解读比例
Pie charts show proportions of a whole. The angle for a category = (Category frequency / Total frequency) × 360°.
饼图展示各部分在整体中的比例。某一类的扇形角度 = (该类别频数 / 总频数) × 360°。
Bar charts display frequencies with gaps between bars for categorical or discrete data. Height of bar = frequency.
条形图用有间隔的条形展示频数，适用于分类或离散数据。条形高度 = 频数。
When comparing pie charts, check if the total frequencies are the same; percentages or angles alone can be misleading.
比较饼图时，检查总频数是否相同；单独看百分比或角度可能会产生误导。
12. Choosing the Right Average and Measure of Spread | 选择合适的平均数和离散程度指标
The mean uses all data but is affected by outliers. The median is robust to outliers. The mode is the only measure suitable for non-numeric data.
平均数使用了所有数据，但易受异常值影响。中位数对异常值稳健。众数是唯一适用于非数值数据的度量。
Use range alongside the median for skewed data; use range with the mean for symmetric data. In KS3, you will often be asked to justify your choice.
分析偏态数据时常将极差与中位数搭配使用；分析对称数据时常用极差与平均数。在 KS3 中，经常会被要求解释这样选择的理由。
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