📚 Year 12 AQA Statistics: Core Knowledge Summary | AQA统计核心知识点梳理
Welcome to the core knowledge summary for Year 12 AQA Statistics. This article distils the essential topics you need to master for the AS-level examination, from collecting and representing data through to probability distributions and hypothesis testing. Each section pairs concise explanations with practical examples, ensuring you build a solid foundation in statistical thinking.
欢迎阅读Year 12 AQA统计核心知识点梳理。本文凝练了AS阶段考试必须掌握的重要主题，从数据收集与呈现到概率分布和假设检验。每个小节将精准解释与实例相结合，帮助你建立扎实的统计思维基础。
1. Statistical Sampling | 统计抽样
A population is the entire set of individuals or items that we want to study. A sample is a subset selected to represent the population. A sampling frame is a list of all members of the population from which the sample is drawn.
总体是我们希望研究的全部个体或对象的集合。样本是从中选出的代表总体的子集。抽样框是列出总体所有成员的清单，用于抽取样本。
Simple random sampling gives every member an equal chance of selection, often using random number generators or lottery methods. It is unbiased but requires a complete sampling frame.
简单随机抽样使每个成员被选中的机会均等，常用随机数生成器或抽签法。它无偏，但需要完整的抽样框。
Systematic sampling selects every k-th member from the sampling frame after a random start. It is quick and easy, but can introduce periodicity bias.
等距抽样从抽样框中以随机起点每隔 k 个选取一个成员。操作快捷简便，但可能引入周期性偏差。
Stratified sampling divides the population into strata based on characteristics, then takes a random sample from each stratum proportionally. This improves representativeness for known groups.
分层抽样按特征将总体划分为层，然后在每层中按比例随机抽样。这能提高已知群体的代表性。
Quota sampling is a non-random method where interviewers select a predetermined number of individuals with certain characteristics. It is cheap and does not require a frame, but may be biased.
配额抽样是一种非随机方法，调查员按规定特征选取预设数量的个体。成本低且无需抽样框，但可能存在偏差。
Opportunity (convenience) sampling involves selecting the most easily available individuals. It is very easy, but unlikely to be representative.
便利抽样选取最容易获得的个体。非常简便，但难以具有代表性。
Larger samples generally reduce sampling variability and give more precise estimates, but cost and practicality must be considered.
较大的样本通常会降低抽样变异性并给出更精确的估计，但必须考虑成本与可行性。
2. Data Representation | 数据表示
Data can be visualised using histograms, cumulative frequency curves, box plots and scatter diagrams. Each type highlights different features: histograms show distribution shape, cumulative frequency graphs help find medians and percentiles, and box plots summarise spread and outliers.
数据可用直方图、累积频率曲线、箱线图和散点图进行可视化。每种图突出不同的特征：直方图展示分布形状，累积频率图有助于求中位数和百分位数，箱线图概括离散程度和异常值。
For a histogram, frequency density = frequency ÷ class width. The area of each bar is proportional to the frequency. Unequal class widths require the frequency density to be plotted on the vertical axis.
在直方图中，频数密度 = 频数 ÷ 组距。每个条形的面积与频数成正比。组距不相等时，需将频数密度绘于纵轴。
A cumulative frequency graph plots the running total of frequencies against the upper class boundary. The median, quartiles and percentiles can be read directly. Plot points at the upper boundary of each class.
累积频率图将频数的累计值相对于组上限描点。中位数、四分位数和百分位数可直接从图上读取。描点应位于各组的组上限。
Box plots use the five‑number summary: minimum, lower quartile Q₁, median Q₂, upper quartile Q₃ and maximum. Outliers are often marked individually. A box plot clearly shows the central 50% of the data and any skewness.
箱线图使用五数概括：最小值、下四分位数 Q₁、中位数 Q₂、上四分位数 Q₃ 和最大值。异常值通常单独标出。箱线图清晰地显示中间 50% 数据的分布和偏态。
Outliers can be identified using the 1.5 × IQR rule: a value is an outlier if it is less than Q₁ – 1.5 × IQR or greater than Q₃ + 1.5 × IQR.
异常值可用 1.5 × IQR 法则识别：若某值小于 Q₁ – 1.5 × IQR 或大于 Q₃ + 1.5 × IQR，即为异常值。
3. Measures of Central Tendency | 集中趋势量数
The mean (x̄) is the arithmetic average: sum all values and divide by the number of values. It uses every data point but is sensitive to outliers.
均值（x̄）是算术平均数：将所有数值求和后除以数值的个数。它使用了所有数据点，但对异常值敏感。
The median is the middle value when data are ordered. It is not affected by extreme values and is often used for skewed distributions.
中位数是数据排序后居于中间位置的值。它不受极端值影响，常用于偏态分布。
The mode is the most frequent value. A data set can have more than one mode or no mode at all. It is the only measure suitable for categorical data.
众数是出现频率最高的值。数据集可以有多个众数或没有众数。它是唯一适用于分类数据的度量。
For grouped data, the exact mean cannot be calculated; we estimate it using the midpoint of each class interval multiplied by its frequency, then divide by the total frequency. The median group is the class containing the (n/2)-th observation.
对于分组数据，无法计算精确均值；我们使用各组的组中值乘以频数求和，再除以总频数来估算。中位数所在组是包含第 (n/2) 个观测值的组。
4. Measures of Dispersion | 离散程度量数
The range is the difference between the maximum and minimum values. It is easy to compute but ignores the distribution of the middle values.
极差是最大值与最小值之差。计算简单，但忽略了中间值的分布。
The interquartile range (IQR) = Q₃ – Q₁. It measures the spread of the middle 50% and is resistant to outliers.
四分位距（IQR）= Q₃ – Q₁。它测量中间 50% 数据的离散程度，且耐抗异常值。
Variance measures the average squared deviation from the mean. For a population, σ² = Σ(x – μ)² / N. For a sample, s² = Σ(x – x̄)² / (n – 1), using n – 1 to give an unbiased estimate of the population variance.
方差衡量数据与均值偏差的平方平均值。总体方差 σ² = Σ(x – μ)² / N。样本方差 s² = Σ(x – x̄)² / (n – 1)，使用 n – 1 以得到总体方差的无偏估计。
Standard deviation is the positive square root of the variance: s = √[Σ(x – x̄)² / (n – 1)]. It has the same units as the original data and is the most widely used measure of spread.
标准差是方差的正平方根：s = √[Σ(x – x̄)² / (n – 1)]。它与原始数据单位相同，是最常用的离散程度度量。
When data are coded using a linear transformation y = (x – a)/b, the mean and standard deviation transform accordingly: ȳ = (x̄ – a)/b and sy = sx/|b|. Coding simplifies calculations.
当数据通过线性变换 y = (x – a)/b 进行编码时，均值和标准差随之变换：ȳ = (x̄ – a)/b，sy = sx/|b|。编码可简化计算。
5. Probability Basics | 概率基础
The probability of an event A, denoted P(A), is a measure of the likelihood that A occurs, satisfying 0 ≤ P(A) ≤ 1. The sample space S contains all possible outcomes.
事件 A 的概率 P(A) 是对 A 发生可能性的度量，满足 0 ≤ P(A) ≤ 1。样本空间 S 包含所有可能的结果。
For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B). They cannot occur simultaneously. For non‑mutually exclusive events, P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
对于互斥事件 A 和 B，P(A ∪ B) = P(A) + P(B)。它们不能同时发生。对于非互斥事件，P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。
Complement rule: P(A’) = 1 – P(A), where A’ is the event ‘not A’.
互补规则：P(A’) = 1 – P(A)，其中 A’ 表示“非 A”事件。
Two events A and B are independent if P(A ∩ B) = P(A) × P(B). For dependent events, P(A ∩ B) = P(A) × P(B|A). Conditional probability P(B|A) = P(A ∩ B) / P(A), provided P(A) > 0.
若 P(A ∩ B) = P(A) × P(B)，则事件 A 与 B 独立。对于相依事件，P(A ∩ B) = P(A) × P(B|A)。条件概率 P(B|A) = P(A ∩ B) / P(A)，前提是 P(A) > 0。
Tree diagrams help model sequential events: multiply along branches for intersections and add appropriate branch probabilities for unions.
树状图有助于模拟序贯事件：沿分支相乘求交集，将相应分支概率相加求并集。
6. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of values with assigned probabilities. The probability distribution must satisfy Σ P(X = x) = 1.
离散随机变量 X 取可数个值，每个值有指定概率。概率分布必须满足 Σ P(X = x) = 1。
The expected value E(X) = Σ x·P(X = x), giving the theoretical mean or long‑run average. For a linear function, E(aX + b) = a E(X) + b.
期望值 E(X) = Σ x·P(X = x)，给出理论均值或长期平均值。对于线性函数，E(aX + b) = a E(X) + b。
Variance Var(X) = E[(X – μ)²] = Σ (x – μ)² P(X = x) = E(X²) – [E(X)]². The standard deviation is √Var(X). For a linear transformation, Var(aX + b) = a² Var(X).
方差 Var(X) = E[(X – μ)²] = Σ (x – μ)² P(X = x) = E(X²) – [E(X)]²。标准差为 √Var(X)。线性变换下方差满足 Var(aX + b) = a² Var(X)。
These properties hold for any discrete random variable and are used extensively in binomial and other distributions.
这些性质适用于任意离散随机变量，并广泛用于二项分布及其他分布。
7. Binomial Distribution | 二项分布
A binomial random variable X counts the number of successes in a fixed number n of independent trials, where each trial has the same probability of success p. It is written X ~ B(n, p).
二项随机变量 X 计数在固定次数 n 次独立试验中成功的次数，每次试验成功的概率恒为 p。记作 X ~ B(n, p)。
The probability of exactly r successes is given by the formula:
恰好获得 r 次成功的概率由下式给出：
P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ
where ⁿCᵣ = n! / [r!(n – r)!] is the binomial coefficient. Cumulative probabilities can be found using tables or calculators.
其中 ⁿCᵣ = n! / [r!(n – r)!] 为二项系数。累积概率可使用表格或计算器求得。
The mean and variance of a binomial random variable are E(X) = np and Var(X) = np(1 – p). These are valid only when the binomial conditions hold.
二项随机变量的均值与方差分别为 E(X) = np 和 Var(X) = np(1 – p)。仅当二项条件成立时有效。
Conditions for a binomial model: fixed number of trials n; trials independent; exactly two outcomes (success/failure); probability p constant for all trials.
二项模型的条件：试验次数 n 固定；各试验独立；仅有两种结果（成功/失败）；所有试验的成功概率 p 恒定。
8. Hypothesis Testing for Binomial Distribution | 二项分布的假设检验
A hypothesis test uses sample data to assess a claim about a population parameter, typically the probability p in a binomial setting. The null hypothesis H₀ states the assumed value of p, while the alternative hypothesis H₁ specifies the direction of the test (one‑tailed or two‑tailed).
假设检验利用样本数据评估关于总体参数的断言，在二项情境下通常是针对概率 p。原假设 H₀ 给出 p 的假定值，备择假设 H₁ 指明检验的方向（单尾或双尾）。
The significance level α (often 0.05) is the probability of rejecting H₀ when it is true. The critical region is the set of values of the test statistic for which H₀ is rejected. The acceptance region contains the remaining values.
显著性水平 α（常取 0.05）是 H₀ 为真时拒绝 H₀ 的概率。拒绝域是检验统计量取值导致拒绝 H₀ 的区域。接受域包含其余取值。
Under the assumption that H₀ is true, we calculate the probability of obtaining a result at least as extreme as the observed test statistic. If this p‑value ≤ α, we reject H₀; otherwise, there is insufficient evidence to reject H₀.
在 H₀ 为真的假设下，计算得到至少与实际观测值同等极端的结果的概率。若此 p‑值 ≤ α，则拒绝 H₀；否则，没有足够证据拒绝 H₀。
For a one‑tailed test, all the significance level is in one tail. For a two‑tailed test, α is split equally between the two tails. The critical values are found from binomial tables or via calculator inverses.
单尾检验将所有显著性水平置于单一尾部。双尾检验将 α 均分到两个尾部。临界值通过二项分布表或计算器逆函数求得。
Example: A coin is tossed 10 times, giving 2 heads. Test at the 5% level whether the coin is biased towards tails. H₀: p = 0.5, H₁: p < 0.5. Observed X = 2; p‑value = P(X ≤ 2 | p = 0.5) = 0.0547 > 0.05, so do not reject H₀.
示例：抛硬币 10 次，得到 2 次正面。在 5% 显著性水平下检验硬币是否偏向反面。H₀: p = 0.5，H₁: p < 0.5。观测值 X = 2；p‑值 = P(X ≤ 2 | p = 0.5) = 0.0547 > 0.05，故不拒绝 H₀。
9. Correlation and Regression | 相关与回归
Correlation measures the strength and direction of a linear relationship between two variables. A scatter diagram visually suggests whether the correlation is positive, negative or non‑existent.
相关衡量两个变量之间线性关系的强度与方向。散点图可以直观显示相关是正、负或是无相关性。
The product moment correlation coefficient r satisfies -1 ≤ r ≤ 1. r = 1 indicates perfect positive linear correlation, r = -1 perfect negative, and r = 0 no linear correlation. In AQA AS, you calculate r using a calculator rather than the formula.
积矩相关系数 r 满足 -1 ≤ r ≤ 1。r = 1 表示完全正线性相关，r = -1 完全负线性相关，r = 0 无线性相关。在 AQA AS 考试中，用计算器计算 r 而非公式。
Regression analysis models the relationship with an equation of the form y = a + bx, where y is the response variable and x is the explanatory variable. The least squares regression line minimises the sum of squared vertical distances from the points to the line.
回归分析用方程 y = a + bx 对关系建模，其中 y 为响应变量，x 为解释变量。最小二乘回归线使得各点到直线的垂直距离的平方和最小。
The slope b and intercept a are calculated as:
斜率 b 和截距 a 的计算公式为：
b = Sxy / Sxx, a = ȳ – b x̄
where Sxy = Σ(x – x̄)(y – ȳ) = Σxy – (Σx)(Σy)/n and Sxx = Σ(x – x̄)² = Σx² – (Σx)²/n.
其中 Sxy = Σ(x – x̄)(y – ȳ) = Σxy – (Σx)(Σy)/n，Sxx = Σ(x – x̄)² = Σx² – (Σx)²/n。
The regression line can be used to make predictions within the range of the data (interpolation). Extrapolation beyond the data range is unreliable.
回归线可用于在数据范围内进行预测（内插）。超出数据范围的外推不可靠。
10. Data Cleaning and Outliers | 数据清理与异常值
Before analysis, data should be checked for errors, missing values and unusual observations. Cleaning ensures the reliability of your conclusions.
分析之前应检查数据中的错误、缺失值和异常观测。清理数据以确保结论的可靠性。
A common method to identify outliers is 1.5 × IQR beyond the quartiles. On a box plot, outliers appear as individual points beyond the whiskers. In regression, a point with a large residual may be an outlier and can unduly influence the line.
识别异常值的常用方法是 1.5 × IQR 法则。在箱线图中，异常值显示为须线之外的孤立点。在回归中，具有较大残差的点可能是异常值，并可能对回归线产生不当影响。
It is important to investigate the cause of an outlier — it could be a data entry error, a legitimate extreme value, or indicate that the model is unsuitable.
调查异常值的成因很重要——它可能是数据录入错误、合理的极端值，或表明模型不适合。
11. Working with Large Data Sets | 大型数据集处理
The AQA specification includes a pre‑
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