📚 IB & WJEC Maths: Statistics Revision Guide | IB 与 WJEC 数学：统计考点精讲
Statistics forms a vital part of both the IB and WJEC mathematics curricula, demanding a blend of conceptual understanding and practical data-handling skills. This guide walks you through the core topics, from sampling techniques to hypothesis testing, with clear examples and bilingual explanations to reinforce your learning and exam readiness.
统计是 IB 和 WJEC 数学课程的重要组成部分，要求你既能理解概念，又能掌握实际处理数据的能力。本文带你梳理核心考点，从抽样方法到假设检验，搭配清晰的例子和中英双语讲解，帮助你巩固知识、备战考试。
1. Types of Data and Sampling Methods | 数据类型与抽样方法
Data can be classified as qualitative (categorical) or quantitative (numerical), and quantitative data can be further split into discrete and continuous variables. Understanding this classification is the first step in choosing appropriate statistical techniques for description and inference.
数据可分为定性（分类）数据和定量（数值）数据，其中定量数据又可细分为离散变量和连续变量。理解这种分类是选择合适的描述和推断统计方法的第一步。
Common sampling methods include simple random sampling, stratified sampling, systematic sampling, and convenience sampling. Each has advantages and limitations regarding bias, representativeness, and ease of implementation; exam questions often require you to identify the method used in a scenario or suggest an improvement to reduce bias.
常见的抽样方法包括简单随机抽样、分层抽样、系统抽样和便利抽样。每种方法在偏差、代表性和实施便利性方面各有优缺点；考试题目经常要求你识别情境中使用的方法，或提出改进建议以减少偏差。
2. Measures of Central Tendency and Spread | 集中趋势与离散程度的度量
Mean, median, and mode summarise the centre of a data set. The mean is sensitive to outliers, while the median provides a robust alternative for skewed distributions. The mode is most useful for categorical data or discrete data with repetition.
平均数、中位数和众数概括了数据集的中心位置。平均数对异常值敏感，而中位数在偏态分布中更为稳健。众数最适用于分类数据或有重复的离散数据。
Spread is captured by range, interquartile range (IQR), variance, and standard deviation. The IQR is the difference between the upper and lower quartiles and is resistant to outliers, making it particularly valuable in box‑and‑whisker plots. Standard deviation measures the average distance from the mean and is used extensively in normal distribution contexts.
数据的离散程度通过极差、四分位距 (IQR)、方差和标准差来衡量。IQR 是上四分位数与下四分位数之差，不受异常值影响，因此在箱线图中尤为有用。标准差衡量数据与平均值的平均距离，广泛用于正态分布相关的情境。
3. Representing Data Visually | 数据可视化表示
Bar charts, pie charts, histograms, cumulative frequency graphs, and box plots each serve a different purpose. Histograms are used for continuous data grouped into classes of equal or unequal width; the area of each bar is proportional to frequency, so frequency density must be calculated when class widths differ.
条形图、饼图、直方图、累积频率图和箱线图各有不同的用途。直方图用于按相等或不等宽度分组表示的连续数据；每个长方形的面积与频数成正比，因此在组距不相等时必须计算频率密度。
Box‑and‑whisker plots display the five‑number summary (minimum, Q1, median, Q3, maximum) and are excellent for comparing distributions and identifying outliers. In WJEC and IB exams, you may be asked to construct a box plot from raw data or to interpret it in context alongside other diagrams.
箱线图展示了五数概括（最小值、下四分位数、中位数、上四分位数、最大值），非常适合比较分布和识别异常值。在 WJEC 和 IB 考试中，你可能会被要求根据原始数据绘制箱线图，或结合上下文与其他图形一起进行解读。
4. Probability Foundations | 概率基础
Probability is the measure of the likelihood of an event, ranging from 0 to 1. The complement rule states that P(A’) = 1 − P(A). For mutually exclusive events, P(A ∪ B) = P(A) + P(B), while for non‑mutually exclusive events the addition rule subtracts the intersection: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
概率衡量事件发生的可能性，取值范围从 0 到 1。补集规则是 P(A’) = 1 − P(A)。对于互斥事件，P(A ∪ B) = P(A) + P(B)；而对于非互斥事件，加法规则需要减去交集：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
Conditional probability, expressed as P(A|B) = P(A ∩ B) / P(B), plays a central role in tree diagrams and two‑way tables. Independent events satisfy P(A ∩ B) = P(A) × P(B), and this independence is often tested by verifying whether the product of individual probabilities equals the joint probability.
条件概率表示为 P(A|B) = P(A ∩ B) / P(B)，在树状图和双向表中起着核心作用。独立事件满足 P(A ∩ B) = P(A) × P(B)，通过检验个体概率的乘积是否等于联合概率，经常用于验证独立性。
5. Discrete Random Variables | 离散随机变量
A discrete random variable takes distinct, countable values, and its probability distribution is given by a table or function. The sum of all probabilities must equal 1. The expected value E(X) = Σ [x · P(X = x)] represents the theoretical long‑run average, while the variance Var(X) = E(X²) − [E(X)]² quantifies spread.
离散随机变量取有限个或可数个值，其概率分布由表格或函数给出。所有概率之和必须等于 1。期望值 E(X) = Σ [x · P(X = x)] 代表理论上的长期平均值，而方差 Var(X) = E(X²) − [E(X)]² 则量化了离散程度。
Applications of expected value extend to fair games and decision‑making: if the expected gain is zero, the game is considered fair. This concept is regularly examined in IB in the context of probability distributions and in WJEC as part of risk analysis.
期望值的应用延伸到公平游戏和决策：如果期望收益为零，则游戏被认为是公平的。这一概念在 IB 的概率分布和 WJEC 的风险分析中经常出现。
6. Binomial Distribution | 二项分布
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success, p. If X ~ B(n, p), then P(X = r) = (nCr) p r (1 − p) n−r, and the mean is np with variance np(1 − p).
二项分布适用于固定次数的独立试验中成功次数的建模，每次试验的成功概率均为 p。若 X ~ B(n, p)，则 P(X = r) = (nCr) p r (1 − p) n−r，均值为 np，方差为 np(1 − p)。
Candidates must be able to calculate probabilities using the formula, tables, or a calculator’s binomial functions, and to recognise the conditions for a binomial model: fixed number of trials, two possible outcomes per trial, constant probability, and independence.
考生需能够使用公式、表格或计算器的二项功能计算概率，并能识别二项模型的条件：试验次数固定、每次试验只有两种可能结果、概率恒定以及独立性。
7. Normal Distribution | 正态分布
The normal distribution is a continuous symmetric bell‑shaped curve defined by its mean, μ, and standard deviation, σ. Approximately 68% of data lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ. The standard normal variable Z = (X − μ) / σ allows the use of Z‑tables for probability calculations.
正态分布是一条连续对称的钟形曲线，由其均值 μ 和标准差 σ 确定。大约 68% 的数据落在 μ ± σ 范围内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。标准正态变量 Z = (X − μ) / σ 使我们能够使用 Z 值表进行概率计算。
Working backwards from a probability to find an unknown mean, standard deviation, or quantile requires inverse normal techniques. IB and WJEC problems frequently involve real‑world contexts such as packaging weights, examination scores, and measurement errors, where assumptions of normality must be justified.
从概率反推求未知的均值、标准差或分位数，需要运用逆正态方法。IB 和 WJEC 的题目经常涉及实际情境，如包装重量、考试成绩和测量误差，此时必须说明正态性假设的合理性。
8. Correlation and Linear Regression | 相关性与线性回归
Scatter diagrams help visualise the relationship between two quantitative variables. Pearson’s product‑moment correlation coefficient, r, measures the strength and direction of a linear relationship, taking values between −1 and 1. A value close to 0 suggests no linear correlation, but there may still be a non‑linear relationship.
散点图有助于直观地展示两个定量变量之间的关系。皮尔逊积矩相关系数 r 衡量线性关系的强度和方向，取值介于 −1 和 1 之间。接近 0 的值表明没有线性相关性，但仍可能存在非线性关系。
The least‑squares regression line has the form y = a + bx, where b = Sxy / Sxx and a = ȳ − b x̄. The line can be used for interpolation within the data range, but extrapolation beyond the range is unreliable. Residual plots can assess the goodness of fit and identify patterns that violate linear assumptions.
最小二乘回归直线形式为 y = a + bx，其中 b = Sxy / Sxx，a = ȳ − b x̄。该直线可用于数据范围内的内插，但超出范围的外推不可靠。残差图可以评估拟合优度，并能识别出违反线性假设的模式。
9. Probability Distributions for Continuous Random Variables | 连续随机变量的概率分布
For a continuous random variable, probabilities are found by calculating areas under a probability density function (pdf). The total area under the curve is 1, and the probability of the variable falling within an interval [a, b] is the definite integral of the pdf from a to b. This concept underlies the normal distribution and other continuous models.
对于连续随机变量，概率是通过计算概率密度函数 (pdf) 下的面积来获得的。曲线下的总面积为 1，变量落在区间 [a, b] 内的概率就是 pdf 从 a 到 b 的定积分。这一概念是正态分布和其他连续模型的基础。
Expectation and variance for continuous distributions are defined analogously using integrals: E(X) = ∫ x f(x) dx and Var(X) = ∫ (x − μ)² f(x) dx. While IB frequently tests these concepts in a theoretical context, they provide a deeper understanding of why the normal distribution behaves as it does.
连续分布的期望和方差用积分类似定义：E(X) = ∫ x f(x) dx，Var(X) = ∫ (x − μ)² f(x) dx。虽然 IB 常以理论形式考查这些概念，但它们能让人更深入地理解正态分布为何具有这样的行为。
10. Confidence Intervals and Hypothesis Testing | 置信区间与假设检验
A confidence interval gives a range of plausible values for a population parameter (e.g., mean or proportion). For a population mean with known variance, a 95% confidence interval is x̄ ± 1.96 × σ/√n. The interpretation focuses on the long‑run capture rate: if we repeated the sampling many times, 95% of such intervals would contain the true mean.
置信区间为总体参数（如均值或比例）提供了一个合理的取值范围。对于已知方差的总体均值，95% 置信区间为 x̄ ± 1.96 × σ/√n。其解读侧重于长期捕获率：如果我们多次重复抽样，95% 的此类区间将包含真实均值。
Hypothesis testing involves stating null and alternative hypotheses, calculating a test statistic, and comparing it to a critical value or using a p‑value. The significance level, α, is the probability of rejecting a true null hypothesis (Type I error). The power of a test is the probability of correctly rejecting a false null hypothesis.
假设检验包括陈述原假设和备择假设，计算检验统计量，并将其与临界值比较或使用 p 值。显著性水平 α 是拒绝一个真实原假设（第一类错误）的概率。检验的功效是正确拒绝一个错误原假设的概率。
11. The Chi‑squared Test for Independence | 独立性卡方检验
The chi‑squared test for independence determines whether there is a significant association between two categorical variables. Observed frequencies are compared to expected frequencies calculated under the assumption of independence, using the formula χ² = Σ [(O − E)² / E].
独立性卡方检验用于判断两个分类变量之间是否存在显著关联。观察频数与在独立性假设下计算出的期望频数进行比较，所用公式为 χ² = Σ [(O − E)² / E]。
Degrees of freedom for an r × c contingency table are (r − 1)(c − 1). The test statistic is compared against a critical value from the chi‑squared distribution. A small expected frequency (typically less than 5) in any cell may require combining categories to maintain validity.
r × c 列联表的自由度为 (r − 1)(c − 1)。检验统计量与卡方分布的临界值进行比较。如果任一单元格的期望频数过小（通常小于 5），可能需要合并类别以保证检验的有效性。
12. Exam Tips and Common Pitfalls | 考试技巧与常见错误
Always define variables and state assumptions clearly. When using the normal approximation to a binomial, remember to apply continuity correction. Check that conditions for tests are satisfied: independence, sample size, and distributional shape. In regression, do not confuse the explanatory and response variables, and be careful with units when interpreting slope.
始终清晰地定义变量并陈述假设。在采用正态近似代替二项分布时，记得进行连续性校正。检验前要检查条件是否满足：独立性、样本量和分布形态。在回归分析中，不要混淆解释变量与响应变量，并在解释斜率时注意单位。
Round final answers sensibly and maintain intermediate precision. When reading tables, confirm whether you need a one‑tailed or two‑tailed critical value. Lastly, use diagrams liberally – a well‑drawn sketch can clarify your reasoning and earn method marks even if the subsequent arithmetic goes slightly astray.
合理地对最终结果进行四舍五入，并保持中间计算精度。查表时，确认是需要单尾还是双尾临界值。最后，请善用图表——一幅清晰的草图可以阐明你的推理过程，即使后续计算稍有偏差，也能获得方法分。
Published by TutorHao | Statistics Revision Series | aleveler.com
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