Year 9 CIE Statistics: Formula & Theorem Quick Reference Handbook

📚 Year 9 CIE Statistics: Formula & Theorem Quick Reference Handbook | 公式定理速查手册

This quick reference handbook covers the essential formulas and theorems you need for Year 9 CIE Statistics. It is designed to help you revise key concepts in data handling, probability, and statistical measures, allowing you to tackle problems with confidence. Keep this guide handy for homework, class tests, and end-of-year examinations.

本速查手册涵盖了九年级CIE统计课程所需的核心公式和定理，旨在帮你复习数据处理、概率和统计度量等关键概念，让你能自信地解决各类问题。无论是课后作业、课堂测验还是年终考试，这本手册都是你随时查阅的好帮手。

1. Mean, Median, and Mode | 平均数、中位数与众数

The three most common measures of central tendency are the mean, median, and mode. They each summarise a data set with a single representative value, but are calculated differently and are affected by extreme values in distinct ways.

最常用的三种集中趋势量数是平均数、中位数和众数。它们各用一个代表性数值概括一组数据，但计算方法不同，受极端值影响的程度也各不相同。

The mean (arithmetic average) is found by adding all the data values and dividing by the number of values. The formula is:

平均数（算术平均值）是将所有数据值相加再除以数据个数得到的，公式为：

x̅ = Σx / n

where x represents each individual data value, Σx is the sum of all values, and n is the total number of observations. The mean is sensitive to outliers – a single very large or very small value can pull the mean away from the centre of the rest of the data.

其中 x 代表每一个数据值，Σx 为所有数据值之和，n 为观测值的总数。平均数对异常值敏感——一个特别大或特别小的数值就可能把平均数拉向偏离其他数据中心的方向。

The median is the middle value when the data are arranged in ascending order. If n is odd, the median is the value at position (n+1)/2. If n is even, it is the average of the values at positions n/2 and (n/2)+1. Because the median depends only on the order of values, it is not affected by outliers.

中位数是将数据按升序排列后位于中间位置的数值。若 n 为奇数，中位数即第 (n+1)/2 个数据；若 n 为偶数，则为第 n/2 个与第 (n/2)+1 个数据的平均值。因为中位数只依赖于数据的排列顺序，所以不受异常值的影响。

The mode is the value that appears most frequently in a data set. A set can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all if all values occur with the same frequency. The mode is particularly useful for categorical data.

众数是数据集中出现频率最高的值。一组数据可以有一个众数（单峰）、多个众数（双峰或多峰），或者如果所有值的频率都相同，则没有众数。众数特别适用于分类数据。

2. Range and Interquartile Range | 极差与四分位距

Measures of dispersion describe how spread out the data values are. The simplest measure is the range, and a more robust measure is the interquartile range (IQR).

离散量数用于描述数据值的分散程度。最简单的离散量数是极差，而更为稳健的量数是四分位距（IQR）。

The range is the difference between the largest and smallest values: Range = Maximum – Minimum. While easy to compute, the range can be heavily influenced by a single outlier.

极差是最大值与最小值之差：极差 = 最大值 – 最小值。极差虽然计算简单，但极易受单个异常值的影响。

The interquartile range focuses on the middle 50% of the data. It is the difference between the upper quartile (Q₃) and the lower quartile (Q₁): IQR = Q₃ – Q₁. The lower quartile is the median of the lower half of the data, and the upper quartile is the median of the upper half. The IQR is resistant to outliers, making it a better indicator of spread for skewed distributions.

四分位距关注的是数据中间50%的部分，它是上四分位数（Q₃）与下四分位数（Q₁）之差：IQR = Q₃ – Q₁。下四分位数是数据下半部分的中位数，上四分位数是数据上半部分的中位数。四分位距不受异常值干扰，因此在描述偏态分布时是更好的离散程度指标。

To find quartiles from a list, first order the data, locate the median, and then find the medians of the two halves. When n is even, the halves split naturally; when n is odd, the median value is usually omitted from both halves before finding Q₁ and Q₃.

从列表中找出四分位数时，需先将数据排序、找到中位数，再分别找出两半部分的中位数。当 n 为偶数时，两半部分自然划分；当 n 为奇数时，通常会在求 Q₁ 和 Q₃ 之前将整体中位数从两半数据中排除。

3. Frequency Tables and the Mean | 频率表与均值计算

When data are presented in a frequency table, we can calculate the mean using the frequency distribution. If the variable x takes values x₁, x₂, …, with corresponding frequencies f₁, f₂, …, then the mean is given by:

当数据以频率表的形式呈现时，我们可以利用频率分布来计算平均数。如果变量 x 的取值为 x₁, x₂, …，对应的频数为 f₁, f₂, …，则平均数公式为：

x̅ = Σ(fx) / Σf

Here Σ(fx) means sum of (frequency × data value) for all rows, and Σf is the total frequency. This is essentially the same as the ordinary mean but accounts for repeated values.

这里 Σ(fx) 表示所有行的（频数 × 数据值）之和，Σf 为总频数。这本质上与普通平均数相同，只是考虑了重复出现的值。

For grouped data, individual values are unknown, so we use the midpoint of each class interval as an estimate of x. The formula becomes Mean ≈ Σ(f × midpoint) / Σf. This gives an estimated mean, and its accuracy depends on how evenly the data are spread within each interval.

对于分组数据，由于不知道每个具体数值，我们使用每个组区间的中点作为 x 的估计值，公式变为：平均数 ≈ Σ（频数 × 组中点）/ Σf。这样算出的是估算平均数，其准确性取决于各组内部数据分布的均匀程度。

Remember that when dealing with class intervals such as 10 ≤ x < 20, the midpoint is 15. Be careful to identify whether boundaries are inclusive or exclusive, as this affects the midpoint.

注意，当处理如 10 ≤ x < 20 这样的组区间时，组中点为 15。务必分清区间边界是包含还是不包含，因为这会影响中点的取值。

4. Cumulative Frequency and Quartiles | 累积频率与四分位数

A cumulative frequency table adds a running total of frequencies. It can be used to construct a cumulative frequency curve (ogive), which allows us to estimate the median, quartiles, and percentiles directly from the graph.

累积频率表就是将频数逐行累加而成的表格。利用它可以绘制累积频率曲线（卵形线），借助该曲线我们能直接从图中估算中位数、四分位数和百分位数。

To draw an ogive, plot the upper class boundary of each interval against the cumulative frequency, then join the points with a smooth curve. The median corresponds to the 50th percentile, so on the cumulative frequency axis, locate half the total frequency and read off the corresponding value on the data axis. Similarly, the lower quartile is at 25% of total frequency and the upper quartile at 75%.

绘制累积频率曲线时，将每个区间的上区间边界与累积频数对应描点，再用光滑曲线连接各点。中位数对应第50百分位数，因此在累积频率轴上找到总频数一半的位置，横向读取对应的数据轴数值。同理，下四分位数位于总频数的25%处，上四分位数位于75%处。

The interquartile range can then be found by subtracting the lower quartile value from the upper quartile value obtained from the curve. The ogive is also extremely useful for finding the number of observations below a certain value or for estimating the proportion of data within an interval.

然后，用从曲线上读取的上四分位数减去下四分位数即可得到四分位距。累积频率曲线在确定低于某数值的观测个数、或估计某区间内数据占比时也非常有用。

5. Probability Basics | 概率基础

Probability measures the chance that an event will occur. It is always a number between 0 and 1, where 0 means impossible and 1 means certain. The basic formula is:

概率用来衡量某一事件发生的可能性大小，其值总是介于0到1之间，0表示不可能发生，1表示必然发生。基础公式为：

P(A) = Number of favourable outcomes / Total number of possible outcomes

This formula assumes that all outcomes are equally likely. The sum of the probabilities of all possible outcomes in a sample space is equal to 1.

该公式假设所有可能结果的出现是等可能的。样本空间中所有可能结果的概率之和等于1。

The complement of an event A, written as A’, is the event that A does not happen. Its probability is P(A’) = 1 – P(A). This rule is often helpful when it is easier to calculate the chance that something does not happen.

事件 A 的补事件，记为 A’，表示 A 不发生的事件。其概率为 P(A’) = 1 – P(A)。当计算某事件不发生的概率更为简单时，这个规则尤为有用。

Probabilities can be expressed as fractions, decimals, or percentages. In Year 9 CIE Statistics, you should be comfortable converting between these forms and using them in tree diagrams and two-way tables.

概率可以用分数、小数或百分数表示。在九年级CIE统计课程中，你需要熟练掌握这些形式之间的转换，并在树状图和双向表中灵活运用。

6. Mutually Exclusive Events | 互斥事件

Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling a die, the events ‘roll a 2’ and ‘roll a 5’ are mutually exclusive because a single roll cannot result in both outcomes.

如果两个事件不能同时发生，则称它们为互斥事件。例如，掷一个骰子时，“掷出2点”和“掷出5点”是互斥的，因为单次掷骰不可能同时出现两个结果。

For mutually exclusive events A and B, the probability that either A or B occurs is the sum of their individual probabilities:

对于互斥事件 A 和 B，事件 A 或 B 发生的概率等于各自概率之和：

P(A or B) = P(A) + P(B)

This is called the addition rule for mutually exclusive events. When events are not mutually exclusive, we must subtract the probability of the intersection to avoid double counting:

这被称为互斥事件的加法法则。当事件并非互斥时，必须减去二者交集的概率以避免重复计算：

P(A or B) = P(A) + P(B) – P(A and B)

You can often identify mutually exclusive situations from a clear context, like selecting a single card that cannot be both a King and a Queen at the same time. When working with two-way tables, these rules help you correctly combine probabilities.

通常可以从明确的情境中辨别互斥情况，比如抽取一张牌，它不可能同时是 K 和 Q。当使用双向表处理概率时，这些规则有助于正确合并概率。

7. Independent Events and Tree Diagrams | 独立事件与树状图

Two events are independent if the outcome of one does not affect the probability of the other. For independent events A and B, the probability that both occur is the product of their individual probabilities:

如果一事件的发生不影响另一事件发生的概率，则这两个事件是独立的。对于独立事件 A 和 B，两者同时发生的概率等于各自概率的乘积：

P(A and B) = P(A) × P(B)

Tree diagrams are ideal for displaying sequences of independent events, such as flipping a coin and then rolling a die. Each branch represents an outcome with its probability written on the branch. To find the probability of a specific combination of outcomes, multiply the probabilities along the relevant branches.

树状图非常适合展示一系列独立事件，例如先抛硬币再掷骰子。每条分支代表一个结果，分支上标注其概率。要计算某种特定结果组合的概率，只需将相应路径上的概率相乘即可。

When the tree shows all possible outcomes, the sum of the probabilities at the final ends of all branches equals 1. If an event is repeated (e.g., two coin tosses), the probabilities remain the same on each set of branches because the events are independent.

若树状图展示了所有可能结果，所有末梢概率之和等于1。若某事件重复进行（如两次抛硬币），由于事件独立，每组分支上的概率保持不变。

Be careful: if events are not independent, you cannot simply multiply the original probabilities. Instead, probabilities on the second set of branches must be conditioned on the first outcome. This leads to conditional probability, discussed in the next section.

注意：如果事件不独立，就不能简单地将原始概率相乘。此时，第二组分支上的概率必须基于第一个结果的条件来调整。这就引出了下一节将要讨论的条件概率。

8. Conditional Probability | 条件概率

Conditional probability is the probability of an event occurring given that another event has already occurred. The notation P(A|B) reads ‘the probability of A given B’. The formula is:

条件概率是指在另一事件已经发生的情况下，某事件发生的概率。记号 P(A|B) 读作“在 B 发生的条件下 A 发生的概率”。其公式为：

P(A|B) = P(A and B) / P(B)

provided P(B) > 0. This formula is fundamental when dealing with successive events that are not independent, such as drawing two cards from a deck without replacement.

前提是 P(B) > 0。在处理非独立连续事件时（例如从一副牌中不放回地抽取两张牌），这个公式至关重要。

You can rearrange the formula to find the joint probability: P(A and B) = P(B) × P(A|B). In a tree diagram for dependent events, the probabilities on the second set of branches are conditional probabilities. They are written differently because the sample space has changed after the first event.

你可以将公式变形来求联合概率：P(A and B) = P(B) × P(A|B)。在相依事件的树状图中，第二组分支上的概率就是条件概率。这些概率的写法会有所不同，因为第一次事件发生后样本空间已经改变。

Common textbook problems involve coloured balls in a bag where selections are made without replacement. Always adjust the denominator and the numerator to reflect the items remaining after each draw.

课本中常见的问题涉及袋子里不同颜色的球，且抽取后不放回。每次抽取后，务必要调整分子和分母，以反映剩余物体的数量。

9. Expected Value | 期望值

Expected value is a form of long-run average. If a discrete random variable X can take values x₁, x₂, …, xₖ with corresponding probabilities p₁, p₂, …, pₖ, the expected value E(X) is calculated as:

期望值是一种长期平均值。如果离散随机变量 X 的可能取值为 x₁, x₂, …, xₖ，对应概率为 p₁, p₂, …, pₖ，则期望值 E(X) 的计算公式为：

E(X) = Σ [xᵢ × P(X = xᵢ)]

In words, multiply each possible outcome by its probability and sum all these products. The expected value need not be a possible outcome itself; for a fair six-sided die, E(X) = 3.5, which is not on any face of the die.

也就是说，将每个可能结果与其概率相乘，再将这些乘积全部相加。期望值本身不一定是可能出现的某个结果；以均匀六面骰子为例，E(X) = 3.5，这并不出现在骰子的任何一面上。

In games of chance, expected value helps to determine whether a game is fair or favourable. If a game costs £C to play and the expected winnings are E(W), the expected gain per game is E(W) – C. A positive value suggests a long-term profit, while a negative value indicates a long-term loss.

在概率游戏中，期望值有助于判断游戏是否公平或对玩家有利。如果一局游戏的成本为 £C，预期赢得的金额为 E(W)，则每局的预期收益为 E(W) – C。正值表明长期可获利，负值则表明长期会亏损。

For a frequency distribution, you can also estimate the expected value by using relative frequencies as probabilities, particularly when analysing data from a survey or an experiment.

对于频率分布，你也可以用相对频率作为概率来估算期望值，尤其是在分析调查或实验数据时这种方法十分常见。

10. Scatter Diagrams and Correlation | 散点图与相关

A scatter diagram (or scatter plot) is used to display the relationship between two numerical variables. Each point on the graph represents a pair of values (x, y). By examining the pattern of points, we can describe the type and strength of correlation.

散点图（或称散布图）用于展示两个数值变量之间的关系。图上的每一个点代表一组 (x, y) 对。通过观察点的分布形态，我们可以描述相关的类型和强弱程度。

Positive correlation means that as x increases, y tends to increase. Negative correlation means that as x increases, y tends to decrease. No correlation indicates no apparent linear relationship between the variables. The strength can be described as strong, moderate, or weak depending on how closely the points cluster around a straight line.

正相关表示当 x 增加时，y 倾向于增加。负相关表示当 x 增加时，y 倾向于减小。无相关则表示变量之间没有明显的线性关系。相关强度可根据数据点围绕一条直线的紧密程度，描述为强、中等或弱。

Often we draw a line of best fit by eye so that it passes through the mean point (x̅, ȳ) and has roughly equal numbers of points above and below the line. This line can be used to estimate one variable given a value of the other. Such predictions are reliable only within the range of the data (interpolation); extrapolation beyond the data range should be treated with caution.

我们通常凭目测绘制一条最佳拟合线，使其穿过均值点 (x̅, ȳ)，并使线上方和线下方的点数大致相等。这条线可用于通过一个变量的值来估计另一个变量的值。此类预测仅在数据范围内（内插）可靠；超出数据范围的外推则需谨慎对待。

The correlation does not imply causation. Even a strong correlation between two variables does not prove that a change in one causes a change in the other; there may be other underlying factors.

相关关系并不意味着因果关系。即使两个变量之间存在强相关，也不能证明其中一个变量的变化会引起另一个的变化；可能存在其他潜在的影响因素。

In Year 9 CIE Statistics, you are expected to plot points accurately on graph paper, draw a reasonable line of best fit, and use it to make predictions, while understanding the limitations of your line.

在九年级CIE统计课程中，你需要能够在坐标纸上准确描点、绘制合理的拟合线，并能用它进行预测，同时要理解这条线的局限所在。

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Year 9 CIE Statistics: Formula & Theorem Quick Reference Handbook | 公式定理速查手册