📚 Core Topics in A-Level Cambridge Statistics | A-Level剑桥统计核心知识点梳理
The Cambridge International A-Level Mathematics syllabus includes two statistics modules, Statistics 1 (S1) and Statistics 2 (S2), which build a solid foundation in data analysis, probability theory, and statistical inference. Mastering these core topics is essential for success in both the examinations and further studies in data-related fields. This article provides a structured review of the key concepts, formulas, and methods commonly examined.
剑桥国际A-Level数学教学大纲包含两个统计模块——统计1 (S1) 和统计2 (S2)，为数据分析、概率论和统计推断打下坚实的基础。掌握这些核心知识点对于考试成功以及未来在数据相关领域的深造至关重要。本文系统梳理了常考的核心概念、公式和方法。
1. Representation of Data | 数据表示
Stem-and-leaf diagrams order and display all data values, making it easy to see the shape of the distribution while retaining the original numbers. A key must be provided, and back-to-back stem-and-leaf diagrams can compare two data sets side by side.
茎叶图对所有数据值进行排序和展示，便于观察分布形态并保留原始数字。必须提供图例，背靠背茎叶图可以并排比较两组数据。
Box-and-whisker plots show the median, quartiles and extreme values. Outliers are typically identified as points lying more than 1.5 times the interquartile range (IQR) below Q1 or above Q3. These plots are ideal for comparing skewness and spread across several samples.
箱线图显示中位数、四分位数和极值。异常值通常被定义为低于 Q1-1.5×IQR 或高于 Q3+1.5×IQR 的点。这些图形非常适合比较多组样本的偏态和离散程度。
Histograms use area to represent frequency. For unequal class widths, frequency density must be calculated:
直方图用面积表示频率。当组距不相等时，必须计算频率密度：
Frequency density = frequency / class width
Cumulative frequency curves are constructed by plotting cumulative frequency against the upper class boundary. They are used to estimate the median, quartiles and percentiles by reading off the horizontal axis at the corresponding cumulative frequency.
累积频率曲线通过将累积频率对组上限描点而构建。它可用于通过读取相应累积频率在水平轴上的值来估计中位数、四分位数和百分位数。
2. Measures of Central Tendency and Variation | 集中趋势与变异度量
ean, median and mode describe the centre of a data set. The sample mean is calculated as x̄ ations. The median is the middle value when data are ordered, and the mode is the most frequent value.
均值、中位数和众数描述数据集的中心。样本均值计算公式为 x̄ = Σx / n，其中 n 为观测值个数。中位数是排序后数据的中间值，众数是出现频率最高的值。
Variance and standard deviation quantify spread. The sample variance s² is given by:
方差和标准差用于度量离散程度。样本方差 s² 的公式为：
s² = Σ (x − x̄)² / (n − 1)
e (IQR = Q3 − Q1) is a resistant measure of spread that is not affected by extreme values.
标准差 s 是方差的正平方根。四分位距 (IQR = Q3 − Q1) 是一种不受极端值影响的稳健离散度量。
implifies calculations, with mean(x) = a + b × mean(y) and variance(x) = b² × variance(y).
对于分组数据，使用组中值近似计算均值和方差。编码公式 y = (x − a)/b 能够简化计算，此时 mean(x) = a + b × mean(y)，variance(x) = b² × variance(y)。
3. Probability | 概率
. The sample space S contains all possible outcomes. For any event A, P(not A) = 1 − P(A).
概率度量事件发生的可能性，满足 0 ≤ P(A) ≤ 1。样本空间 S 包含所有可能的结果。对于任意事件 A，P(非 A) = 1 − P(A)。
The addition rule for mutually exclusive events states P(A ∪ B) = P(A) + P(B). If events are not mutually exclusive, the general addition rule is:
互斥事件的加法法则为 P(A ∪ B) = P(A) + P(B)。若事件不互斥，则一般加法法则为：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Independent events satisfy P(A ∩ B) = P(A) × P(B). Conditional probability P(A | B) is the probability of A given that B has occurred, defined as:
独立事件满足 P(A ∩ B) = P(A) × P(B)。条件概率 P(A | B) 表示在 B 已经发生的条件下 A 发生的概率，其定义为：
P(A | B) = P(A ∩ B) / P(B) , provided P(B) > 0
Tree diagrams and Venn diagrams are effective tools for organising multi-stage probability problems and for visualising intersections and unions.
树状图和维恩图是组织多阶段概率问题以及直观展示交集和并集的有效工具。
4. Discrete Random Variables | 离散随机变量
bution lists each value x and the corresponding probability P(X = x), with Σ P(X = x) = 1.
离散随机变量 X 取可数个值。其概率分布列出了每个值 x 及相应的概率 P(X = x)，且 Σ P(X = x) = 1。
ean) of X is E(X) = Σ x P(X = x). The variance can be computed using two equivalent forms:
X 的期望值（均值）为 E(X) = Σ x P(X = x)。方差可通过两种等价形式计算：
Var(X) = Σ (x − μ)² P(X = x) = E(X²) − [E(X)]²
r(aX + b) = a² Var(X). These rules are particularly useful for coding and standardisation.
线性变换具有简单的性质：E(aX + b) = aE(X) + b 且 Var(aX + b) = a² Var(X)。这些规则在编码和标准化时非常有用。
5. 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 denoted by X ~ B(n, p).
二项分布模型描述在固定次数的 n 次独立试验中成功的次数，每次试验的成功概率相同，记为 p。记作 X ~ B(n, p)。
The probability of exactly r successes is given by the binomial probability formula:
恰好成功 r 次的概率由二项概率公式给出：
P(X = r) = nCr pr (1 − p)n−r
where nCr = n! / [r!(n − r)!].
其中 nCr = n! / [r!(n − r)!]。
. Cumulative binomial probabilities can be found using statistical tables or a calculator.
二项随机变量的均值和方差分别为 E(X) = np 和 Var(X) = np(1 − p)。二项累积概率可通过统计表或计算器查找。
6. Normal Distribution | 正态分布
bution with a bell-shaped curve, fully defined by its mean μ and variance σ², written as X ~ N(μ, σ²). About 68% of values fall within μ ± σ, and about 95% within μ ± 2σ.
正态分布是一种钟形曲线的连续分布，完全由其均值 μ 和方差 σ² 决定，记作 X ~ N(μ, σ²)。大约 68% 的值落在 μ ± σ 内，约 95% 落在 μ ± 2σ 内。
To calculate probabilities, we standardise the variable to the standard normal distribution N(0, 1) using:
为计算概率，我们通过标准化将变量转换为标准正态分布 N(0, 1)：
Z = (X − μ) / σ
g with a normal approximation to the binomial, a continuity correction of ±0.5 is applied.
标准正态表给出 Φ(z) = P(Z < z)。已知概率可反查 z，然后通过 X = μ + zσ 复原 X。当使用正态分布近似二项分布时，需要应用 ±0.5 的连续性修正。
7. Poisson Distribution | 泊松分布
The Poisson distribution models the number of random events occurring independently in a fixed interval of time or space. It is characterised by the mean rate λ and denoted by X ~ Po(λ).
泊松分布模型描述在固定时间或空间间隔内独立发生的随机事件数量。它由平均发生率 λ 表征，记作 X ~ Po(λ)。
The probability of observing exactly r events is:
观察到恰好 r 个事件的概率为：
P(X = r) = e−λ λr / r!
the mean and variance equal λ: E(X) = Var(X) = λ. When two independent Poisson variables X ~ Po(λ₁) and Y ~ Po(λ₂) are added, the sum follows X + Y ~ Po(λ₁ + λ₂).
对于泊松变量，均值和方差都等于 λ，即 E(X) = Var(X) = λ。若两个独立的泊松变量 X ~ Po(λ₁) 和 Y ~ Po(λ₂) 相加，其和服从 X + Y ~ Po(λ₁ + λ₂)。
approximate a binomial distribution B(n, p) when n is large and p is small, using λ = np.
作为经验法则，当 n 很大而 p 很小时，可用 λ = np 的泊松分布近似二项分布 B(n, p)。
8. Continuous Random Variables | 连续随机变量
nsity function (pdf) f(x) that satisfies f(x) ≥ 0 and the total area under the curve is 1:
连续随机变量 X 具有概率密度函数 (pdf) f(x)，满足 f(x) ≥ 0 且曲线下的总面积为 1：
∫−∞<
Published by TutorHao | A-Level 统计 Revision Series | aleveler.com 更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图