📚 AS Cambridge Statistics: Core Knowledge Review | AS剑桥统计：核心知识点梳理
In AS Level Cambridge Statistics (Paper 5: Probability & Statistics 1), students must develop a solid understanding of data handling, probability, and basic distributions. This article covers the essential topics in a clear, structured manner to help you revise effectively for the exam.
在AS剑桥统计（试卷5：概率与统计1）中，学生必须牢固掌握数据处理、概率和基本分布。本文以清晰的结构梳理核心知识点，帮助你高效复习备考。
1. Types of Data and Sampling | 数据类型与抽样
Data can be classified as qualitative (descriptive) or quantitative (numerical). Quantitative data is further split into discrete data, which take only specific values (e.g., number of students), and continuous data, which can take any value within a range (e.g., height).
数据可分为定性数据（描述性）和定量数据（数值型）。定量数据又分为离散数据（只能取特定值，如学生人数）和连续数据（可在某一范围内取任意值，如身高）。
In sampling, a simple random sample gives every member of the population an equal chance of being chosen. Other methods like stratified sampling ensure subgroups are proportionally represented, reducing bias.
在抽样中，简单随机抽样使总体中每个个体有相等的机会被选中。其他方法如分层抽样能保证各子群体按比例被抽中，从而减少偏差。
A sampling frame is a list of all individuals from which the sample is drawn. If the frame is incomplete, the sample may be biased.
抽样框是用于抽取样本的全体个体名单。如果抽样框不完整，样本就可能存在偏差。
2. Measures of Central Tendency | 集中趋势的度量
The mean, median, and mode are the three main measures of central tendency. The mean x̄ = Σx/n gives the arithmetic average and uses all data values, making it sensitive to outliers.
均值、中位数和众数是集中趋势的三个主要度量。均值 x̄ = Σx/n 给出算术平均数，它使用了所有数据值，因此受异常值影响较大。
The median is the middle value when the data are ordered and is robust to extreme values. For n observations, the median position is (n+1)/2.
中位数是数据排序后位于中间的值，它对极端值不敏感。对于 n 个观测值，中位数的位置为 (n+1)/2。
The mode is the most frequently occurring value and is especially useful for qualitative data. A data set may have no mode, one mode, or multiple modes.
众数是出现频率最高的值，对定性数据尤其有用。一个数据集可能没有众数、有一个众数或有多个众数。
For grouped data, the mean is estimated using midpoints: x̄ ≈ Σ(f × mid-value) / Σf, where f is the frequency of each class.
对于分组数据，使用组中值来估计均值：x̄ ≈ Σ（频数 × 组中值）/ Σf，其中 f 是各组的频数。
3. Measures of Dispersion | 离散程度的度量
Dispersion describes how spread out the data are. The range is the simplest measure: maximum minus minimum, but it is heavily affected by outliers.
离散程度描述数据的分散情况。极差是最简单的度量：最大值减最小值，但它极易受异常值影响。
The interquartile range (IQR = Q3 – Q1) measures the spread of the middle 50% of the data and is robust to extreme values. It is the preferred measure to accompany the median.
四分位距（IQR = Q3 – Q1）度量了数据中间50%的范围，对极端值稳健。它是与中位数配合使用的首选离散度量。
Variance and standard deviation measure the average squared deviation from the mean. For a population, variance σ² = Σ(x-μ)²/N; for a sample, s² = Σ(x-x̄)²/(n-1). In AS exams, the formula with denominator n is often used for a set of data treated as the population.
方差和标准差度量了各数据值与均值的平均平方偏差。对于总体，方差 σ² = Σ(x-μ)²/N；对于样本，s² = Σ(x-x̄)²/(n-1)。在 AS 考试中，经常把所给数据集视为总体并使用分母为 n 的公式。
The standard deviation is the square root of the variance. It has the same units as the original data, making it easier to interpret.
标准差是方差的平方根，它的单位与原数据一致，因此更容易解释。
4. Data Representation | 数据表示
Stem-and-leaf diagrams organise data while preserving original values. A key must be given, e.g., ‘3 | 4 means 34’. Back-to-back stems can compare two data sets.
茎叶图既能整理数据又能保留原始数值。必须给出图例，如“3|4 表示 34”。背靠背茎叶图可以用来比较两组数据。
Box-and-whisker plots display the minimum, Q1, median, Q3, and maximum. Outliers are usually defined as values more than 1.5 × IQR below Q1 or above Q3 and are plotted as individual points.
箱线图展示最小值、Q1、中位数、Q3 和最大值。异常值通常定义为低于 Q1 – 1.5 × IQR 或高于 Q3 + 1.5 × IQR 的数值，它们在图上以单独的点标出。
Histograms use area to represent frequency. With unequal class widths, frequency density = frequency / class width must be plotted on the vertical axis.
直方图用面积表示频数。当组距不相等时，必须在纵轴上标绘频数密度 = 频数 / 组距。
Cumulative frequency curves (ogives) are used to estimate medians, quartiles, and percentiles. They always start at the lower class boundary with cumulative frequency zero.
累积频数曲线（肩形图）用于估计中位数、四分位数和百分位数。曲线总是从最低组的下限开始，累积频数为零。
5. Coding and Its Effect on Summary Statistics | 数据编码及其对汇总统计量的影响
Coding is a linear transformation of data, typically y = (x – a)/b or y = ax + b. It simplifies calculation when original values are large or awkward.
编码是对数据进行的线性变换，常见形式为 y = (x – a)/b 或 y = ax + b。当原始数值较大或不便计算时，编码能使计算简化。
If y = ax + b, then the mean of y is a × mean(x) + b, and the standard deviation of y is |a| × standard deviation of x. Adding a constant shifts the mean but does not affect spread.
若 y = ax + b，则 y 的均值 = a × x 的均值 + b，y 的标准差 = |a| × x 的标准差。加上常数会平移均值，但不改变数据的离散程度。
Variance is affected only by the scaling factor a: Var(y) = a² Var(x). When using coded data to find the original mean and standard deviation, always reverse the coding correctly.
方差只受缩放因子 a 的影响：Var(y) = a² Var(x)。使用编码数据求原均值和标准差时，务必正确进行反向变换。
6. Probability Basics | 概率基础
The probability of an event A, P(A), satisfies 0 ≤ P(A) ≤ 1. P(A’) = 1 – P(A). For any two events, P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
事件 A 的概率 P(A) 满足 0 ≤ P(A) ≤ 1。P(A’) = 1 – P(A)。对于任意两个事件，P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。
Mutually exclusive events cannot happen simultaneously, so P(A ∩ B) = 0. Independent events satisfy P(A ∩ B) = P(A) × P(B) and P(A|B) = P(A).
互斥事件不可能同时发生，因此 P(A ∩ B) = 0。独立事件满足 P(A ∩ B) = P(A) × P(B) 且 P(A|B) = P(A)。
Conditional probability P(A|B) = P(A ∩ B) / P(B) represents the probability of A given that B has occurred. Tree diagrams are extremely useful for multi-stage experiments when events are conditional.
条件概率 P(A|B) = P(A ∩ B) / P(B) 表示在 B 已发生的条件下 A 发生的概率。在处理多阶段试验且存在条件关系时，树状图是非常有用的工具。
7. Permutations and Combinations | 排列与组合
The counting principle states that if one task can be done in m ways and another in n ways, the total number of ways to do both is m × n.
乘法原理指出，若一项任务有 m 种完成方式，另一项有 n 种，则两者先后完成的方式总数为 m × n。
Permutations are arrangements where order matters. The number of ways to arrange n distinct objects is n! (n factorial). The number of permutations of r objects chosen from n is nPr = n! / (n-r)!.
排列是与顺序有关的安排。n 个不同物体的全排列数为 n!（n 的阶乘）。从 n 个物体中选取 r 个的排列数为 nPr = n! / (n-r)!。
Combinations are selections where order does not matter. The number of ways to choose r objects from n is nCr = n! / [r!(n-r)!]. This is essential for binomial probability calculations.
组合是与顺序无关的选择。从 n 个物体中选取 r 个的组合数为 nCr = n! / [r!(n-r)!]。这一公式在二项分布概率计算中至关重要。
When some objects are identical, the number of distinct arrangements of n items with repetitions is n! / (n₁! n₂! …) where n₁, n₂, … are frequencies of each type.
当部分物体相同时，n 个含有重复物体的排列数为 n! / (n₁! n₂! …)，其中 n₁, n₂, … 是各类相同物体的个数。
8. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of values, each with a corresponding probability P(X = x). The sum of all probabilities must equal 1.
离散随机变量 X 取有限个或可数个值，每个值对应一个概率 P(X = x)，所有概率之和必须等于 1。
The expected value E(X) = Σ x·P(X = x) represents the long-run average. It is the centre of the probability distribution.
期望值 E(X) = Σ x·P(X = x) 表示长期平均值，是概率分布的中心。
Variance Var(X) = E(X²) – [E(X)]² = Σ x² P(X = x) – μ². The standard deviation is √Var(X).
方差 Var(X) = E(X²) – [E(X)]² = Σ x² P(X = x) – μ²。标准差为 √Var(X)。
For a linear function aX + b, we have E(aX+b) = aE(X)+b and Var(aX+b) = a² Var(X). This mirrors the coding rules for data.
对于线性函数 aX + b，有 E(aX+b) = aE(X)+b，Var(aX+b) = a² Var(X)，与数据编码的规则一致。
9. 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. Notation: X ~ B(n, p).
二项分布描述在固定次数 n 的独立试验中成功次数的分布，每次试验成功的概率 p 不变。记作 X ~ B(n, p)。
The probability of exactly r successes is given by P(X = r) = nCr pr (1-p)n-r. Remember that q = 1-p is the failure probability.
恰好成功 r 次的概率由 P(X = r) = nCr pr (1-p)n-r 给出。记住 q = 1-p 是失败的概率。
The mean and variance of a binomial distribution are E(X) = np and Var(X) = npq. These are used in hypothesis testing questions at A2, but at AS you may need to verify them using the formulas for discrete random variables.
二项分布的均值和方差分别为 E(X) = np 和 Var(X) = npq。这些在 A2 的假设检验中会用到，但在 AS 阶段你可能需要用离散随机变量公式来验证它们。
Conditions for a binomial model: fixed n, independent trials, constant p for each trial, and only two outcomes (success/failure). If you recognise these conditions in a word problem, you can apply the binomial distribution.
使用二项分布的条件：固定的 n 次试验、试验之间相互独立、每次试验的 p 相同，以及只有两种结果（成功/失败）。如果在应用题中识别出这些条件，就可以应用二项分布。
10. 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. Notation: X ~ N(μ, σ²).
正态分布是一种连续型概率分布，曲线呈钟形，由均值 μ 和方差 σ² 决定。记作 X ~ N(μ, σ²)。
To find probabilities, we standardise the variable to Z ~ N(0, 1) using z = (x – μ) / σ. The standard normal table then gives Φ(z) = P(Z < z).
为求概率，我们通过 z = (x – μ) / σ 将变量转化为标准正态变量 Z ~ N(0, 1)。然后利用标准正态分布表得到 Φ(z) = P(Z < z)。
For probabilities like P(X > a), use P(X > a) = 1 – Φ((a-μ)/σ). For P(a < X < b), subtract two standardised cumulative probabilities.
对于 P(X > a) 这类概率，使用 P(X > a) = 1 – Φ((a-μ)/σ)。对于 P(a < X < b)，只需将两个标准化后的累积概率相减。
You may be asked to find an unknown mean or standard deviation given a probability. Set up the equation Φ(z) = known probability, find the z-value from the table in reverse, and solve for μ or σ.
考试中可能会要求根据给定概率求未知的均值或标准差。这时需要建立方程 Φ(z) = 已知概率，反查表得到 z 值，再解出 μ 或 σ。
The symmetry of the normal curve gives Φ(-z) = 1 – Φ(z). This is particularly useful when calculating probabilities on the left tail.
正态曲线的对称性给出 Φ(-z) = 1 – Φ(z)，这在计算左尾概率时非常有用。
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