📚 Year 9 AQA Statistics: Formula & Theorem Quick Reference Handbook | Year 9 AQA 统计：公式定理速查手册
This quick reference handbook covers the essential formulas, rules and concepts required for the Year 9 AQA Statistics topic. It is designed to help you revise data summaries, probability theory and graphical analysis efficiently.
本速查手册涵盖了九年级 AQA 统计主题所需的必备公式、规则和概念，旨在帮助你高效复习数据汇总、概率理论和图形分析。
1. Measures of Central Tendency | 集中趋势的度量
The mean is the most commonly used average. It is found by summing all data values and dividing by the total number of data points.
平均值是最常用的集中量数，通过将所有数据值相加再除以数据个数得到。
Mean = Σx / n
The median is the middle value when data are ordered. If there is an even number of observations, the median is the average of the two central numbers. The position of the median can be located with the formula below.
中位数是排序后数据中间的值。若数据个数为偶数，中位数为中间两个数的平均值。中位数的位置可使用下方公式确定。
Median position = (n + 1) / 2
The mode is the value that appears most frequently. A data set can have one mode, more than one mode (bimodal or multimodal) or no mode at all if all values occur with equal frequency.
众数是出现频率最高的值。一个数据集可以有一个众数、多个众数（双峰或多峰），如果所有值出现频率相同则没有众数。
2. Measures of Spread | 离散程度的度量
The range measures the simplest form of spread. It is the difference between the maximum and minimum values, but it is strongly affected by extreme outliers.
极差是最简单的离散量数，它是最大值与最小值之差，但极易受极端异常值影响。
Range = Max – Min
Quartiles split ordered data into four equal parts. The lower quartile (Q₁) is the median of the lower half of the data; the upper quartile (Q₃) is the median of the upper half. When the total number of data points is odd, the overall median is excluded from both halves.
四分位数将有序数据分成四等份。下四分位数 (Q₁) 是数据下半部分的中位数；上四分位数 (Q₃) 是数据上半部分的中位数。当数据总数为奇数时，整体中位数不纳入任何一半的计算。
Q₁ and Q₃ are found by splitting the ordered data
Interquartile range (IQR) = Q₃ – Q₁
The interquartile range represents the spread of the middle 50% of the data, making it robust against outliers.
四分位距表示中间 50% 数据的分散程度，能较好地抵抗异常值的影响。
3. Frequency Tables and Grouped Data | 频数表和分组数据
When raw data is organised into frequency tables, the mean can be estimated by multiplying each class midpoint (x) by its frequency (f), summing these products and dividing by the total frequency.
当原始数据整理为频数表时，可用每个组距的中点 (x) 乘以其频数 (f)，再将乘积求和后除以总频数来估计平均值。
Estimated Mean = Σ(f × x) / Σf
The modal class is the group interval with the highest frequency; it shows where data are most concentrated in the range.
模态组是频数最高的组别，表明数据在该范围内最为集中。
To locate the median class, calculate cumulative frequencies as you progress through the groups. The median class is the first interval where the cumulative frequency reaches or exceeds half the total frequency (n/2).
要确定中位数组，需依次计算累加频数。第一个累加频数达到或超过总频数一半 (n/2) 的组即为中位数组。
4. Probability Basics | 概率基础
Probability is a measure on a scale from 0 to 1 that quantifies the likelihood of an event. A probability of 0 means the event is impossible, while a probability of 1 means it is certain.
概率是度量事件可能性的数值，范围在 0 到 1 之间。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。
P(Event) = Number of favourable outcomes / Total number of equally likely outcomes
Experimental probability, also called relative frequency, is based on actual trials. It tends to get closer to the theoretical probability as more trials are performed.
实验概率也称相对频率，基于真实试验数据。随着试验次数增加，它会趋近于理论概率。
Relative Frequency = Number of successful trials / Total number of trials
The probability of an event not occurring is the complement of the event. This rule is extremely useful for simplifying calculations.
事件不发生的概率是该事件的补集，这个规则对于简化计算非常有用。
P(not A) = 1 – P(A)
5. Combined Events: AND/OR Rules | 组合事件：与/或规则
For two independent events, the occurrence of one does not influence the probability of the other. The probability of both happening is the product of their individual probabilities.
对于两个独立事件，一个事件的发生不影响另一个的概率。两者同时发生的概率是它们各自概率的乘积。
P(A and B) = P(A) × P(B)
If two events are mutually exclusive, they cannot occur simultaneously. The probability that either one happens is the sum of their separate probabilities.
如果两个事件互斥，它们不能同时发生。任一事件发生的概率是各自概率的和。
P(A or B) = P(A) + P(B)
When events are not mutually exclusive, you must subtract the probability that both occur to avoid counting the overlap twice.
当事件非互斥时，必须减去两者同时发生的概率，以避免重复计算重叠部分。
P(A or B) = P(A) + P(B) – P(A and B)
6. Sample Space Diagrams | 样本空间图
A sample space diagram systematically lists all possible outcomes of a probability experiment. A two-way table is a common form, making it straightforward to identify and count favourable outcomes.
样本空间图系统地列出概率实验的所有可能结果。双向表是一种常见形式，便于识别和计数有利结果。
For example, when rolling two fair dice, a 6 by 6 grid shows 36 equally likely ordered pairs. You can then directly read off probabilities for sums, differences or specific combinations.
例如，掷两个公平骰子时，一个 6×6 的网格显示 36 个等可能的有序数对。你可以由此直接读取各种和、差或特定组合的概率。
The probability of any event is simply the count of successful cells divided by the total number of cells in the diagram.
任何事件的概率只是成功格子的数量除以图中格子的总数。
7. Tree Diagrams | 树状图
Tree diagrams branch out to show sequences of events and their associated probabilities. Each branch is labelled with the probability of that outcome.
树状图通过分支来展示事件序列及其对应的概率，每条分支都标有该结果的概率。
To find the probability of a specific path, multiply the probabilities along the branches. If more than one path fulfils the desired outcome, add the probabilities of those paths together.
要计算某个特定路径的概率，将沿途各分支的概率相乘。如果不止一条路径能达到要求的结果，则将这些路径的概率相加。
When items are drawn with replacement, the probabilities stay the same at each stage because the sample space is unchanged. Without replacement, the probabilities change because the number of items decreases.
当抽取物品后放回时，概率在每一阶段保持不变，因为样本空间未变。无放回抽取时，由于物品数量减少，概率会发生改变。
8. Venn Diagrams and Two-Way Tables | 维恩图和双向表
Venn diagrams use overlapping circles to represent sets and the outcomes that belong to them. The overlap (intersection) shows outcomes common to two or more sets.
维恩图用重叠的圆表示集合及其所属元素。重叠部分（交集
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