📚 AS WJEC Statistics: Core Concepts Summary | AS WJEC统计：核心知识点梳理
This comprehensive revision guide covers the essential topics in the AS WJEC Statistics specification. From the foundations of data collection and sampling to probability distributions and hypothesis testing, the following sections are designed to help you master key concepts and feel confident in your exam. Each point is presented in both English and Chinese to support bilingual learners.
这份全面的复习指南涵盖了AS WJEC统计大纲中的核心主题。从数据收集和抽样的基础到概率分布及假设检验，以下内容旨在帮助你掌握关键概念并在考试中充满信心。每个要点都以英中双语呈现，以支持双语学习者。
1. Statistical Modelling and Data Collection | 统计模型与数据收集
A statistical model is a simplified mathematical description of a real-world process. It allows us to make predictions, understand variation, and estimate probabilities. Models are not perfect copies of reality, but they provide useful approximations when assumptions are reasonable.
统计模型是对现实过程的简化数学描述。它使我们能够进行预测、理解变异并估计概率。模型并非现实的完美复制，但当假设合理时，它们能提供有用的近似。
Data can be classified as quantitative (numerical, such as height or test scores) or qualitative (categorical, such as eye colour or type of car). Quantitative data can be further split into discrete (countable) and continuous (measurable) types. Understanding the data type is crucial for choosing appropriate diagrams and analysis methods.
数据可分为定量（数值型，如身高或考试成绩）和定性（类别型，如眼睛颜色或汽车类型）。定量数据还能细分为离散型（可数）和连续型（可测）。了解数据类型对于选择合适的图表和分析方法至关重要。
2. Sampling Methods | 抽样方法
Sampling is the process of selecting a subset of individuals from a population to estimate characteristics of the whole group. A sample should be representative to avoid bias. Key sampling techniques include simple random sampling, stratified sampling, systematic sampling, and quota sampling.
抽样是从总体中选取一部分个体以估计整个群体特征的过程。样本应具有代表性以避免偏差。主要的抽样技术包括简单随机抽样、分层抽样、系统抽样和配额抽样。
| Method (方法) | Description (描述) |
|---|---|
| Simple random sampling | Every member has an equal chance of being chosen; often uses random number generators. |
| Stratified sampling | Population divided into distinct groups (strata); a random sample is taken from each stratum in proportion to its size. |
| Systematic sampling | Elements are chosen at regular intervals from an ordered list. |
| Quota sampling | Interviewers select a fixed number of people from different categories; non-random and can introduce bias. |
Each method has advantages and disadvantages. For example, simple random sampling is free from bias but can be impractical for large populations, while stratified sampling improves representativeness but requires detailed population information.
每种方法都有优缺点。例如，简单随机抽样无偏差，但对大规模总体可能不切实际；而分层抽样提高了代表性，但需要详细的总体信息。
3. Data Representation | 数据表示
Selecting the right diagram helps reveal patterns in data. Common representations include bar charts, histograms, cumulative frequency curves, box plots (box-and-whisker plots), and scatter diagrams. For a single continuous variable, a histogram with equal (or unequal) class widths is standard; for comparing distributions, box plots are excellent because they show median, quartiles, and outliers.
选择合适的图表有助于揭示数据中的模式。常见的表示方式包括条形图、直方图、累积频率曲线、箱线图（箱须图）和散点图。对于单个连续变量，等宽（或不等宽）的直方图是标准方法；对于比较分布，箱线图非常出色，因为它们能显示中位数、四分位数和异常值。
When drawing a histogram, frequency density is used for unequal class widths: frequency density = frequency / class width. A cumulative frequency graph can be used to estimate medians, quartiles, and percentiles. Box plots are constructed from the five-number summary: minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum.
绘制直方图时，若组距不等则用频率密度：频率密度 = 频率 / 组距。累积频率图可用于估计中位数、四分位数和百分位数。箱线图由五数概括构建：最小值、下四分位数（Q1）、中位数（Q2）、上四分位数（Q3）和最大值。
4. Measures of Central Tendency | 集中趋势的测量
A measure of central tendency gives a typical value for a data set. The three main measures are the mean, the median, and the mode. The mean (often denoted by x̅ for a sample) is the arithmetic average; the median is the middle value when data are ordered; the mode is the most frequent value.
集中趋势的测量给出了数据集的典型值。三种主要的测量是平均数、中位数和众数。平均数（样本通常记为x̅）是算术平均值；中位数是排序后位于中间的值；众数是出现频率最高的值。
Mean = Σx / n, Median = middle value, Mode = most frequent
For symmetric distributions the mean and median are close; in skewed distributions the mean is pulled towards the tail. The choice of measure depends on the data and the presence of outliers: the median is resistant to outliers, while the mean is sensitive but uses all data values.
对于对称分布，平均数和中位数接近；在偏斜分布中，平均数会被拉向尾部。测量方法的选择取决于数据和异常值的存在：中位数对异常值不敏感，而平均数敏感但使用了所有数据值。
5. Measures of Dispersion | 离散程度的测量
Dispersion measures the spread of data. Key measures include the range, interquartile range (IQR), variance, and standard deviation. The range is simply the maximum minus the minimum, but it is heavily affected by outliers. The IQR (Q3 – Q1) is more robust and measures the spread of the middle 50% of the data.
离散程度衡量数据的散布情况。关键的测量包括极差、四分位距（IQR）、方差和标准差。极差只是最大值减去最小值，但极易受异常值影响。IQR（Q3 – Q1）则更稳健，衡量中间50%数据的离散程度。
Variance is the average of the squared deviations from the mean: for a population,
σ² = Σ(xi − μ)² / N
The standard deviation is the square root of the variance, σ = √σ², and has the same units as the original data. When calculating from a sample, the divisor n−1 is used to obtain an unbiased estimate of the population variance.
方差是各数据值与均值离差平方的平均值：对于总体，σ² = Σ(xi − μ)² / N。标准差是方差的平方根，σ = √σ²，与原数据单位相同。当用样本计算时，使用除数n−1可以得到总体方差的无偏估计。
6. Basic Probability | 概率基础
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For any event A, 0 ≤ P(A) ≤ 1. The sample space is the set of all possible outcomes. The complement rule states P(not A) = 1 − P(A).
概率衡量事件发生的可能性，范围从0（不可能）到1（必然）。对任何事件A，0 ≤ P(A) ≤ 1。样本空间是所有可能结果的集合。互补规则为P(非A) = 1 − P(A)。
Two events are mutually exclusive if they cannot occur together, so P(A ∩ B) = 0. The addition rule for mutually exclusive events: P(A ∪ B) = P(A) + P(B). If events are not mutually exclusive, use the general addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
如果两个事件不能同时发生，则它们是互斥的，因此P(A ∩ B) = 0。互斥事件的加法规则：P(A ∪ B) = P(A) + P(B)。如果事件不互斥，则使用一般加法规则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
Independent events are those where the occurrence of one does not affect the probability of the other. For independent events, P(A ∩ B) = P(A) × P(B). Conditional probability P(A|B) = P(A ∩ B) / P(B). Tree diagrams are very useful for solving multi-stage probability problems.
独立事件是指一个事件的发生不影响另一个事件概率的事件。对于独立事件，P(A ∩ B) = P(A) × P(B)。条件概率P(A|B) = P(A ∩ B) / P(B)。树状图对解决多阶段概率问题非常有用。
7. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of possible values, each with an associated probability. The probability function P(X = x) must satisfy two conditions: 0 ≤ P(X = x) ≤ 1 for all x, and Σ P(X = x) = 1 over all possible values.
离散随机变量X取可数个可能值，每个值都有相应概率。概率函数P(X = x)必须满足两个条件：对所有x，0 ≤ P(X = x) ≤ 1，且所有可能值的Σ P(X = x) = 1。
The expected value (mean) of X is a measure of central tendency: E(X) = Σ x · P(X = x). The variance is given by Var(X) = E(X²) − [E(X)]², where E(X²) = Σ x² · P(X = x). The standard deviation is √Var(X).
期望值（均值）E(X)衡量集中趋势：E(X) = Σ x · P(X = x)。方差公式为Var(X) = E(X²) − [E(X)]²，其中E(X²) = Σ x² · P(X = x)。标准差为√Var(X)。
For simple problems, these calculations are best organised in a probability distribution table, listing x, P(X=x), x·P, and x²·P.
对简单问题，最好在概率分布表中整理计算，列出x、P(X=x)、x·P和x²·P。
8. Binomial Distribution | 二项分布
The binomial distribution models the number of successes in a fixed number n of independent trials, each with the same probability of success p. It is written as X ~ B(n, p). The conditions for using a binomial model are: a fixed number of trials, each trial has two outcomes (success/failure), trials are independent, and the probability of success p remains constant.
二项分布模拟在固定次数n次独立试验中成功次数的分布，每次试验的成功概率p相同。记作X ~ B(n, p)。使用二项模型的条件是：试验次数固定，每次试验有两种结果（成功/失败），试验独立，且成功概率p恒定。
P(X = r) = nCr pr (1-p)n-r
The binomial coefficient nCr (or nCr ) counts the number of ways to choose r successes from n trials. The mean and variance of a binomial distribution are E(X) = np and Var(X) = np(1-p). Cumulative binomial probabilities can be found using tables or calculators.
二项式系数nCr 计算从n次试验中选取r次成功的方法数。二项分布的均值和方差分别为E(X) = np和Var(X) = np(1-p)。累积二项概率可使用表格或计算器查找。
9. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution with a bell-shaped curve, defined by its mean μ and variance σ². It is written as X ~ N(μ, σ²). The total area under the curve is 1, and it is symmetric about the mean. Many natural phenomena are approximately normally distributed.
正态分布是一种具有钟形曲线的连续概率分布，由其均值μ和方差σ²定义。记作X ~ N(μ, σ²)。曲线下的总面积为1，且关于均值对称。许多自然现象近似服从正态分布。
To find probabilities, we standardise to the standard normal distribution Z ~ N(0, 1) using the z-score:
Z = (X – μ) / σ
Standard normal tables provide the cumulative probability Φ(z) = P(Z < z). For any value, we can transform and use symmetry. Be aware of notation: the tables often give the area to the left. For P(X > a) use 1 − Φ((a-μ)/σ). Solve inverse problems by working backwards from a given probability to find an unknown mean or standard deviation.
标准正态表提供累积概率Φ(z) = P(Z < z)。对于任何值，我们可以进行变换并利用对称性。注意符号：表格通常给出左侧面积。对于P(X > a)使用1 − Φ((a-μ)/σ)。求解反问题时，从给定概率反向推算，以求出未知的均值或标准差。
10. Introduction to Hypothesis Testing | 假设检验简介
A hypothesis test is a formal procedure for checking whether a claim about a population parameter is supported by sample evidence. In AS WJEC Statistics, hypothesis tests are typically carried out on the binomial distribution. The null hypothesis H&sub0; states the assumed value of the parameter p; the alternative hypothesis H&sub1; states that p is different (or greater/less than) the null value.
假设检验是正式的程序，用于检验关于总体参数的说法是否得到样本证据的支持。在AS WJEC统计中，假设检验通常针对二项分布进行。原假设H&sub0;陈述参数p的假定值；备择假设H&sub1;陈述p与该原假设值不同（或大于/小于）。
We calculate the probability of obtaining the observed result (or more extreme) under the assumption that H&sub0; is true. This is the p-value. If the p-value is less than the significance level α (commonly 0.05 or 0.01), we reject H&sub0; in favour of H&sub1;. If not, there is insufficient evidence to reject H&sub0;.
我们计算在原假设H&sub0;成立的条件下，获得所观察结果（或更极端结果）的概率。这就是p值。如果p值小于显著性水平α（通常为0.05或0.01），则拒绝H&sub0;而支持H&sub1;。否则，没有足够证据拒绝H&sub0;。
The critical region is the set of values of the test statistic for which H&sub0; is rejected. For a one-tailed test, the critical region is at one end of the distribution; for a two-tailed test, it is split between both tails, and we halve the significance level. A proper conclusion must be written in the context of the problem, not merely ‘reject H&sub0;’.
拒绝域是导致拒绝H&sub0;的检验统计量的取值集合。对于单尾检验，拒绝域位于分布的一端；对于双尾检验，它分配在双尾，显著性水平需对半分。正确的结论必须结合问题情境来书写，而不只是”拒绝H&sub0;”。
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