📚 Year 9 Cambridge Statistics: Formulas & Theorems Quick-Reference Handbook | 九年级剑桥统计：公式定理速查手册
This handbook provides a concise summary of the essential formulas, theorems, and statistical concepts required for the Year 9 Cambridge Statistics course. It is designed as a rapid revision aid, covering data handling, measures of central tendency and spread, probability fundamentals, and common diagrammatic representations.
本手册简要总结了九年级剑桥统计课程所需的重要公式、定理和统计概念。它旨在作为快速复习工具，涵盖数据处理、集中趋势与离散程度的量数、概率基础以及常见的图示表示方法。
1. Measures of Central Tendency | 集中趋势量数
The mean, often called the average, is calculated by adding all data values together and then dividing by the number of values. It is the most commonly used measure of central tendency.
均值，通常称为平均数，通过将所有数据值相加然后除以数据个数来计算。它是最常用的集中趋势量数。
Mean = x̄ = (Σx) / n
The median is the middle value when the data is arranged in ascending order. If there is an even number of values, the median is the mean of the two middle numbers.
中位数是将数据按升序排列后位于中间的值。如果数据个数为偶数，中位数则是中间两个数的均值。
Median position = (n + 1) / 2
The mode is the value that appears most frequently in a data set. A data set can have one mode, more than one mode, or no mode at all.
众数是数据集中出现频率最高的值。一个数据集可能有一个众数、多个众数或没有众数。
2. Finding the Mean from a Frequency Table | 从频率表求均值
When data is organised in a frequency table, the mean is estimated by multiplying each value by its frequency, summing these products, and then dividing by the total frequency. This method works for both ungrouped and grouped frequency tables.
当数据整理成频率表时，通过将每个值与其频数相乘、对这些乘积求和，再除以总频数来估算均值。该方法适用于未分组和分组频率表。
x̄ = Σ(f × x) / Σf
| Value (x) | Frequency (f) | f × x |
|---|---|---|
| 2 | 5 | 10 |
| 3 | 8 | 24 |
| 4 | 2 | 8 |
| Σf = 15 | Σ(f×x) = 42 | |
Always check that the total of the frequency column is used correctly in the denominator.
在此例中，估算均值为 42 / 15 = 2.8。务必确保分母中使用的是频数列的总和。
3. Median, Quartiles, and Interquartile Range | 中位数、四分位数与四分位距
Quartiles divide an ordered data set into four equal parts. The lower quartile (Q₁) is the value one-quarter of the way through the data, and the upper quartile (Q₃) is the value three-quarters of the way through.
四分位数将有序数据集分成四个相等部分。下四分位数（Q₁）是位于排序数据四分之一处的值，上四分位数（Q₃）是位于四分之三处的值。
Q₁ position = (n + 1) / 4
Q₃ position = 3(n + 1) / 4
The interquartile range (IQR) measures the spread of the middle 50% of the data. It is less affected by extreme values than the range.
四分位距（IQR）衡量中间 50% 数据的离散程度。与极差相比，它受极端值的影响较小。
IQR = Q₃ − Q₁
e positions (1/2) × total frequency, (1/4) × total frequency, and (3/4) × total frequency.
要从累积频率曲线找出中位数和四分位数，读取对应总频数二分之一、四分之一和四分之三位置处的值。
4. Range and Measures of Spread | 极差与离散程度
The range is the simplest measure of spread, found by subtracting the smallest value from the largest value in the data set. It gives a quick indication of how spread out the data is.
极差是最简单的离散程度量数，即数据集中最大值减去最小值。它能快速体现数据的分散程度。
Range = max − min
Although easy to calculate, the range can be heavily influenced by outliers. For a more robust picture, combine it with the interquartile range.
尽管易于计算，但极差容易受到异常值的严重影响。要获得更稳健的数据描述，应结合四分位距使用。
5. Cumulative Frequency and Curves | 累积频数与累积频率曲线
A cumulative frequency table extends a normal frequency table by adding a running total of the frequencies. The final cumulative frequency must equal the total frequency.
累积频数表在普通频率表的基础上增加频数的累计总和。最终的累积频数必须等于总频数。
Cumulative frequency curves (or ogives) are drawn by plotting cumulative frequency against the upper class boundaries for grouped data, or against the data values for discrete data, and joining the points with a smooth curve.
累积频率曲线（或拱形图）的绘制方法是：对于分组数据，将累积频数对应上组界描点；对于离散数据，对应数据值描点，然后用平滑曲线连接各点。
From the curve, you can estimate the median at the 50th percentile, Q₁ at the 25th percentile, and Q₃ at the 75th percentile. These estimates are widely used in box plots.
从曲线上，你可以在第 50 百分位处估计中位数，在第 25 百分位处估计 Q₁，在第 75 百分位处估计 Q₃。这些估计值广泛用于箱线图中。
6. Box-and-Whisker Plots | 箱线图
A box-and-whisker plot provides a visual summary of the data using five key values: the minimum, lower quartile (Q₁), median, upper quartile (Q₃), and maximum.
箱线图利用五个关键值提供数据的可视化摘要：最小值、下四分位数 (Q₁)、中位数、上四分位数 (Q₃) 和最大值。
The box spans from Q₁ to Q₃ with a line inside marking the median.
箱体从 Q₁ 延伸到 Q₃，内部有一条线标示中位数。
The whiskers extend from the box to the minimum and maximum values, unless outliers are present.
须线从箱体延伸至最小值和最大值（除非存在异常值）。
The length of the box represents the interquartile range (IQR).
箱体长度代表四分位距 (IQR)。
Box plots are especially useful for comparing distributions of two or more data sets side by side.
箱线图特别适用于并排比较两个或多个数据集的分布。
7. Scatter Diagrams and Correlation | 散点图与相关性
A scatter diagram plots paired numerical data on a graph to investigate the relationship between two variables. The pattern of points suggests whether a correlation exists.
散点图将成对的数值数据绘制在图上，以研究两个变量之间的关系。点的分布模式可指示是否存在相关性。
Positive correlation: as one variable increases, the other tends to increase.
正相关：一个变量增加时，另一个也趋于增加。
Negative correlation: as one variable increases, the other tends to decrease.
负相关：一个变量增加时，另一个趋于减少。
No correlation: there is no clear pattern.
无相关：没有明显的模式。
Correlation does not imply causation. Even a strong correlation does not necessarily mean one variable causes the other to change.
相关关系并不意味着因果关系。即使存在强相关，也不一定意味一个变量导致另一个变化。
8. Basic Probability Concepts | 基本概率概念
Probability is a measure of how likely an event is to happen. The probability of an event A is calculated by dividing the number of favourable outcomes by the total number of possible outcomes, assuming all outcomes are equally likely.
概率是对事件发生可能性的度量。在假设所有结果等可能的情况下，事件 A 的概率由有利结果数除以可能结果总数来计算。
P(A) = n(A) / n(S)
The sample space (S) is the set of all possible outcomes of an experiment. An event is any subset of the sample space. The probability of an impossible event is 0, and the probability of a certain event is 1.
样本空间 (S) 是试验所有可能结果的集合。事件是样本空间的任意子集。不可能事件的概率为 0，必然事件的概率为 1。
0 ≤ P(A) ≤ 1
9. Complementary Events and Relative Frequency | 互补事件与相对频数
The complement of event A, written as 'not A' or A', consists of all outcomes not in A. The sum of the probabilities of an event and its complement is always 1.
事件 A 的互补事件，记作“非 A”或 A'，包含所有不在 A 中的结果。一个事件与其互补事件的概率之和总是 1。
P(A') = 1 − P(A)
When actual trials are performed, the relative frequency of an event can be used as an estimate of its probability. Relative frequency stabilises around the true probability as the number of trials increases.
当实际进行试验时，事件的相对频数可作为其概率的估计值。随着试验次数增加，相对频数会稳定在真实概率附近。
Relative frequency = number of times the event occurs / total number of trials
10. Expected Frequency | 期望频数
Expected frequency predicts how many times an event is likely to occur in a number of trials. It is found by multiplying the probability of the event by the number of trials.
期望频数预测在若干次试验中事件可能发生的次数。它由事件的概率乘以试验次数得出。
Expected frequency = P(event) × number of trials
This is often used in survey predictions, quality control, and games of chance to determine whether observed results are consistent with theoretical probability.
这通常用于调查预测、质量控制和机会游戏中，以判断观察结果是否与理论概率一致。
11. Tree Diagrams | 树形图
Tree diagrams are used to display all possible outcomes of two or more events in a clear, branching structure. They are especially helpful for independent events where the outcome of one does not affect the other.
树形图以清晰的分支结构展示两个或多个事件的所有可能结果。它们对于独立事件特别有用，即一个事件的结果不影响另一个。
To find the probability of a combined path of events, multiply the probabilities along the branches. To find the total probability of two or more paths leading to the same final outcome, add their probabilities.
要找到事件组合路径的概率，将各分支上的概率相乘。要找到通向同一最终结果的两条或多条路径的总概率，将这些概率相加。
P(A and B) = P(A) × P(B) (for independent events)
Always check that the sum of probabilities on branches from a single point equals 1. Tree diagrams are powerful tools for solving multi-stage probability problems.
始终检查从一个点出发的所有分支概率之和是否等于 1。树形图是解决多阶段概率问题的有力工具。
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