📚 Year 9 Cambridge Statistics: Formula & Theorem Quick Reference | 九年级剑桥统计：公式定理速查手册
This quick reference handbook compiles all essential formulas, definitions and statistical theorems covered in the Year 9 Cambridge Mathematics Statistics syllabus. It is designed to help you revise data handling, probability and diagram interpretation efficiently before tests and exams.
本速查手册汇集了九年级剑桥数学统计大纲中所有的核心公式、定义和统计定理，旨在帮助你在测试和考试前高效复习数据处理、概率和图表解读。
1. Types of Data | 数据类型
Data can be classified as qualitative (categorical) or quantitative (numerical). Qualitative data describe attributes or categories and cannot be measured numerically. Quantitative data are numbers and can be either discrete or continuous.
数据可分为定性（分类）数据和定量（数值）数据。定性数据描述属性或类别，无法用数字测量。定量数据是数字，可以分为离散型或连续型。
Qualitative (Categorical): e.g., favourite colour, type of pet, brand of phone.
定性（分类）数据：例如，最喜欢的颜色、宠物类型、手机品牌。
Quantitative Discrete: data that can only take specific, separate values, usually counts. e.g., number of students in a class, goals scored in a match.
定量离散数据：只能取特定、分离的值，通常为计数。例如，班级学生人数、比赛进球数。
Quantitative Continuous: data that can take any value within a range, usually measurements. e.g., height, mass, time taken to run 100m.
定量连续数据：可以在一个范围内取任意值，通常为测量值。例如，身高、质量、跑100米所用时间。
2. Measures of Central Tendency | 集中趋势度量
Mean: The mean is calculated by summing all the values and dividing by the number of values. It is often referred to as the average and is affected by extreme values.
平均数：平均数是将所有数值相加后除以数值的个数。它常被称为均值，受极端值影响。
Mean = (∑x) / n
平均数 = (∑x) / n
where ∑x is the sum of all data values, and n is the total number of values.
其中 ∑x 是所有数据值的和，n 是数据的总个数。
Median: The median is the middle value when the data are arranged in order. For an odd number of values, it is the central one. For an even number of values, it is the mean of the two central values.
中位数：中位数是将数据排序后位于中间的值。当数据个数为奇数时，取正中间的那个值；当个数为偶数时，取中间两个数的平均数。
Mode: The mode is the value that occurs most frequently. A data set can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all if no value repeats.
众数：众数是出现次数最多的值。一组数据可以有一个众数（单峰），多个众数（双峰或多峰），或者如果没有值重复则没有众数。
Choosing the best measure: The mean uses all data but is sensitive to outliers. The median is robust to outliers and skewed data. The mode is useful for categorical data.
选用最佳度量：平均数使用了所有数据但对异常值敏感；中位数对异常值和偏态分布稳健；众数适用于分类数据。
3. Measures of Spread | 离散程度度量
Range: The range is the difference between the largest and smallest values. It shows the total spread of the data but can be distorted by extreme values.
极差：极差是最大值与最小值的差，表示数据总体的分散范围，但容易被极端值扭曲。
Range = Maximum – Minimum
极差 = 最大值 – 最小值
Interquartile Range (IQR): The IQR is the range of the middle 50% of the data. It is found by first locating the lower quartile (Q1) and upper quartile (Q3), then calculating Q3 – Q1.
四分位距 (IQR)：四分位距是中间50%数据的范围。先找到下四分位数（Q1）和上四分位数（Q3），然后计算 Q3 – Q1。
Calculating Quartiles for small data sets:
小数据集求四分位数：
1. Arrange data in ascending order. Find the median (Q2).
1. 将数据从小到大排序，找出中位数（Q2）。
2. Q1 is the median of the lower half (values below Q2). Q3 is the median of the upper half (values above Q2).
2. Q1 是下半部分（中位数以下的数据）的中位数，Q3 是上半部分（中位数以上的数据）的中位数。
3. Calculate IQR = Q3 – Q1.
3. 计算 IQR = Q3 – Q1。
The IQR is a more reliable measure of spread than the range because it ignores extreme values.
四分位距比极差更可靠，因为它忽略了极端值。
4. Frequency Tables and Grouped Data | 频数表与分组数据
When data are presented in a frequency table, the mean can be estimated using midpoints for grouped data. You multiply each value (or midpoint) by its frequency, sum these products, and divide by the total frequency.
当数据以频数表呈现时，对于分组数据可以使用组中值来估算平均数。将每一个值（或组中值）乘以其频数，将这些乘积求和，再除以总频数。
Mean from frequency table = (∑f x) / ∑f
频数表求平均数 = (∑f x) / ∑f
For grouped data: Estimated Mean = (∑f × midpoint) / ∑f
分组数据：估算平均数 = (∑f × 组中值) / ∑f
Modal class (grouped data): The class interval with the highest frequency.
众数组（分组数据）：频数最高的组区间。
Median from grouped frequency table: Use cumulative frequencies to find the class containing the median position (n/2). A more accurate estimate can be made using a cumulative frequency graph (see next section).
从分组频数表求中位数：利用累积频数找到含有中位数位置（n/2）的组。更精确的估算可以使用累积频数图（见下一节）。
5. Cumulative Frequency and Box Plots | 累积频数与箱线图
A cumulative frequency table shows the running total of frequencies. Plotting the upper class boundaries against cumulative frequency gives a cumulative frequency curve (ogive).
累积频数表显示频数的累加和。将组上限与累积频数作图，得到累积频数曲线（拱形图）。
Finding quartiles from the graph:
从图中找四分位数：
1. Locate median (Q2) at cumulative frequency = n/2 on the vertical axis, draw a horizontal line to the curve, then a vertical line down to read the value.
1. 在纵轴上找到累积频数 = n/2 的位置，画水平线与曲线相交，再向下画垂直线读取数值，即为中位数。
2. Q1 is found at cumulative frequency = n/4; Q3 at cumulative frequency = 3n/4.
2. 下四分位数（Q1）在累积频数 = n/4 处；上四分位数（Q3）在累积频数 = 3n/4 处。
Five-number summary: minimum, Q1, Q2 (median), Q3, maximum. These five values form a box plot.
五数概括：最小值、下四分位数 Q1、中位数 Q2、上四分位数 Q3、最大值。这五个值构成箱线图。
A box plot displays the central box from Q1 to Q3 with a line at the median, and ‘whiskers’ extending to the minimum and maximum (or to 1.5 IQR limits, where outliers are plotted separately).
箱线图展示从 Q1 到 Q3 的箱子，中位线标记在其中，触须延伸到最小值和最大值（或延伸至 1.5倍IQR 的界限，异常值单独标出）。
Outlier boundaries:
异常值界限：
Lower fence = Q1 – 1.5 × IQR
下界 = Q1 – 1.5 × IQR
Upper fence = Q3 + 1.5 × IQR
上界 = Q3 + 1.5 × IQR
Any data point outside these fences is considered an outlier.
落在这些界线之外的数据点被视为异常值。
6. Probability Basics | 概率基础
Probability is a measure of how likely an event is to occur, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage.
概率是衡量事件发生可能性的度量，范围从 0（不可能）到 1（必然）。可以用分数、小数或百分数表示。
Theoretical probability of an event A, when all outcomes are equally likely:
当所有结果等可能时，事件 A 的理论概率：
P(A) = Number of favourable outcomes / Total number of possible outcomes
P(A) = 有利结果数 / 所有可能结果数
Experimental probability (relative frequency) is based on trials or experiments:
实验概率（相对频数）基于试验或实验：
Experimental Probability = Number of times event occurs / Total number of trials
实验概率 = 事件发生次数 / 总试验次数
As the number of trials increases, experimental probability tends to get closer to theoretical probability.
随着试验次数增加，实验概率会趋近于理论概率。
The sum of probabilities of all possible outcomes of an experiment is 1.
一个实验中所有可能结果的概率之和为 1。
The complement of event A (not A) is written as A’. P(A’) = 1 – P(A).
事件 A 的补集（非 A）记为 A’。P(A’) = 1 – P(A)。
7. Probability Rules and Tree Diagrams | 概率规则与树状图
Two events are mutually exclusive if they cannot happen at the same time. For mutually exclusive events A and B:
如果两个事件不能同时发生，则它们互斥。对于互斥事件 A 和 B：
P(A or B) = P(A) + P(B)
P(A 或 B) = P(A) + P(B)
If events are not mutually exclusive, we must subtract the overlap:
如果事件不互斥，则需减去重叠部分：
P(A or B) = P(A) + P(B) – P(A and B)
P(A 或 B) = P(A) + P(B) – P(A 与 B)
Two events are independent if the occurrence of one does not affect the probability of the other. For independent events A and B:
如果一个事件的发生不影响另一个事件的概率，则称这两个事件独立。对于独立事件 A 和 B：
P(A and B) = P(A) × P(B)
P(A 与 B) = P(A) × P(B)
Tree diagrams show all possible outcomes of one or more events. Label branches with probabilities. To find the probability of a combined sequence of outcomes, multiply along the branches. If multiple branches give the same final outcome, add those probabilities.
树状图显示一个或多个事件的所有可能结果。在分枝上标出概率。求一系列组合结果的概率时，沿分枝相乘。若多条分枝得到相同的最终结果，则将这些概率相加。
Remember: the probabilities on all branches from a single point must sum to 1.
记住：从同一个点发出的所有分枝上的概率之和必须为 1。
8. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph (scatter plot) displays the relationship between two quantitative variables. Each point represents a pair of values (x, y).
散点图显示两个定量变量之间的关系。每个点代表一对数值 (x, y)。
Correlation describes the strength and direction of a linear relationship:
相关性描述线性关系的强度和方向：
- Positive correlation: as x increases, y tends to increase.
- 正相关：当 x 增加时，y 也倾向于增加。
- Negative correlation: as x increases, y tends to decrease.
- 负相关：当 x 增加时，y 倾向于减少。
- No correlation: no clear pattern between x and y.
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