AS Mathematics: Statistics Key Points Review | AS 数学：统计考点精讲

📚 AS Mathematics: Statistics Key Points Review | AS 数学：统计考点精讲

Statistics in AS Mathematics builds the foundation for data analysis and probability. You will learn to summarise and display data, calculate measures of centre and spread, work with probability, and apply key distributions such as the binomial and normal. This concise revision guide covers the core topics you need to master, with parallel English and Chinese explanations.

AS 数学统计部分为数据分析与概率打下基础。你将学习如何汇总和展示数据，计算中心与离散的度量，运用概率知识，并应用二项分布、正态分布等重要分布。这份考点精讲以中英对照的方式覆盖了你需要掌握的核心主题。

1. Representing Data | 数据表示

Stem-and-leaf diagrams allow you to list data values in order while retaining the original precision. They are useful for small data sets and help identify the median, quartiles, and mode quickly.

茎叶图让你在保留原始数值的同时将数据按顺序列出。它适用于小数据集，并有助于快速识别中位数、四分位数和众数。

Box-and-whisker plots display the five-number summary: minimum, lower quartile (Q1), median (Q2), upper quartile (Q3) and maximum. They clearly reveal the spread and skewness of the data, and can be used to identify outliers.

箱线图展示五数概括：最小值、下四分位数(Q1)、中位数(Q2)、上四分位数(Q3)和最大值。它们清晰地反映数据的分散程度和偏态，并可用于识别异常值。

Histograms are used for grouped continuous data. The frequency is proportional to the area of each bar, so the vertical axis shows frequency density (frequency ÷ class width). Always check that the total area represents the total frequency.

直方图用于分组连续数据。频率与每个条形的面积成正比，因此纵轴表示频率密度（频率 ÷ 组距）。务必确保总面积代表总频率。

Cumulative frequency curves plot the running total of frequencies against the upper class boundaries. They are ideal for estimating medians, quartiles, and percentiles, and for finding the interquartile range (IQR).

累积频率曲线将累计频数相对于上组界绘制。它非常适合估算中位数、四分位数和百分位数，并用于求四分位距 (IQR)。

2. Measures of Central Tendency | 集中趋势的度量

The mean (x̄) is the arithmetic average: sum all data values and divide by the number of observations. For grouped data, use x̄ = Σfx / Σf, where f is the frequency and x is the class midpoint.

均值 (x̄) 是算术平均数：将所有数据值求和再除以观测值的个数。对于分组数据，使用 x̄ = Σfx / Σf，其中 f 为频数，x 为组中值。

The median is the middle value when data are ordered. For n observations, the median is at position (n+1)/2. In frequency tables, use cumulative frequencies to locate the median class or value.

中位数是将数据排序后位于中间的值。对于 n 个观测值，中位数的位置为 (n+1)/2。在频数表中，使用累积频数来确定中位数所在的组或值。

The mode is the most frequently occurring value or class. A data set may have no mode, one mode, or several modes. In a histogram, the modal class has the highest frequency density.

众数是出现最频繁的值或组。数据集可能没有众数、有一个众数或多个众数。在直方图中，众数所在的组具有最高的频率密度。

When the data are skewed, the median is a better measure of location than the mean, because the mean is pulled by extreme values.

当数据存在偏斜时，中位数是比均值更好的位置度量，因为均值会受到极端值的影响。

3. Measures of Spread | 离散程度的度量

The range is the difference between the maximum and minimum values. It is easy to calculate but sensitive to outliers.

极差是最大值与最小值之差。它易于计算，但对异常值敏感。

The interquartile range (IQR) = Q3 – Q1. It measures the spread of the middle 50% of the data and is resistant to outliers. It is used to identify an outlier as any value below Q1 − 1.5×IQR or above Q3 + 1.5×IQR.

四分位距 (IQR) = Q3 – Q1。它计算中间 50% 数据的分散程度，且不受异常值影响。通常将低于 Q1 − 1.5×IQR 或高于 Q3 + 1.5×IQR 的数值视为异常值。

Variance measures the average squared deviation from the mean. For a data set x₁,…,xₙ:

σ² = Σ(x − x̄)² / n

方差衡量各数据与均值偏差平方的平均值。对于数据集 x₁,…,xₙ：

σ² = Σ(x − x̄)² / n

The standard deviation is the square root of the variance: σ = √(σ²). It has the same units as the original data. For grouped data, the formula becomes σ² = Σf(x − x̄)² / Σf.

标准差是方差的平方根：σ = √(σ²)，它与原始数据单位相同。对于分组数据，公式变为 σ² = Σf(x − x̄)² / Σf。

An alternative form for variance, which is often easier for calculation, is σ² = Σx²/n − x̄² (or Σfx²/Σf − x̄² for grouped data).

方差的另一个常用计算公式是 σ² = Σx²/n − x̄²（分组数据用 Σfx²/Σf − x̄²），该式在手工计算时更为简便。

4. Basic Probability Rules | 基础概率规则

Probability is a measure of how likely an event is, given by P(A) = number of favourable outcomes / total number of outcomes, provided all outcomes are equally likely.

概率是对事件发生可能性的度量，P(A) = 有利结果数 / 总结果数，前提是所有结果等可能发生。

For any event A, 0 ≤ P(A) ≤ 1. The probability of the complement A’ is P(A’) = 1 − P(A).

对于任意事件 A，0 ≤ P(A) ≤ 1。补事件 A’ 的概率为 P(A’) = 1 − P(A)。

Addition Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive (cannot occur together), then P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

加法规则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。如果 A 与 B 互斥（不能同时发生），则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。

Multiplication Rule for independent events: A and B are independent if P(A ∩ B) = P(A) × P(B). Independence means the occurrence of one does not affect the probability of the other.

独立事件的乘法规则：如果 P(A ∩ B) = P(A) × P(B)，则 A 与 B 独立。独立意味着一个事件的发生不影响另一个事件的概率。

5. Conditional Probability | 条件概率

The conditional probability of A given B is denoted P(A|B) and is defined as:

P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.

事件 B 发生的条件下事件 A 的条件概率记为 P(A|B)，定义为：

P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0。

Rearranging gives the general multiplication rule: P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(A). This is essential for solving problems involving sequential events.

移项可得一般乘法规则：P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(A)。这对于解决包含连续事件的问题至关重要。

Tree diagrams are extremely useful for conditional probability. Each branch shows a conditional probability, and the probability of a path is found by multiplying along the branches. The sum of probabilities on all branches from a single point must equal 1.

树状图在处理条件概率时非常有用。每个分支表示条件概率，某条路径的概率等于该路径上各分支概率的乘积。从同一点出发的所有分支概率之和必为 1。

When events are not independent, P(A|B) ≠ P(A). Also, remember that the multiplication rule for independent events is a special case where P(A|B) = P(A).

当事件不独立时，P(A|B) ≠ P(A)。注意，独立事件的乘法规则是条件概率的特殊情况，即此时 P(A|B) = P(A)。

6. Permutations and Combinations | 排列与组合

The number of ways to arrange n distinct objects in order is n! (n factorial). For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

将 n 个不同物体按顺序排列的方法数是 n!（n 的阶乘）。例如，5! = 5 × 4 × 3 × 2 × 1 = 120。

When selecting r objects from n distinct objects and the order matters, the number of permutations is:

ⁿPᵣ = n! / (n − r)!

从 n 个不同物体中选出 r 个且考虑顺序，排列数为：

ⁿPᵣ = n! / (n − r)!

When the order does not matter, the number of combinations is:

ⁿCᵣ = n! / [r!(n − r)!]

当顺序不重要时，组合数为：

ⁿCᵣ = n! / [r!(n − r)!]

These are essential in probability when calculating the number of ways events can occur, e.g., choosing successes from trials without regard to order (binomial coefficient).

这些公式在概率计算中至关重要，例如在不考虑顺序的情况下从试验中选择成功次数（二项式系数）。

7. Discrete Random Variables | 离散随机变量

A discrete random variable X can take a countable number of values. Its probability distribution is given by P(X = x) for each possible value x, with ΣP(X = x) = 1.

离散随机变量 X 可取有限个或可列无限个值。其概率分布由每个可能取值 x 的 P(X = x) 给出，且满足 ΣP(X = x) = 1。

The expected value (mean) of X is a measure of centre:

E(X) = μ = Σ [x × P(X = x)]

X 的期望（均值）是中心位置的度量：

E(X) = μ = Σ [x × P(X = x)]

The variance of X is given by:

Var(X) = E(X²) − [E(X)]² = Σ[x²P(X=x)] − μ²

X 的方差为：

Var(X) = E(X²) − [E(X)]² = Σ[x²P(X=x)] − μ²

For any constants a and b, the linear transformation properties hold: E(aX + b) = aE(X) + b, and Var(aX + b) = a² Var(X).

对于任意常数 a 和 b，线性变换的性质成立：E(aX + b) = aE(X) + b，且 Var(aX + b) = a² Var(X)。

8. The Binomial Distribution | 二项分布

A binomial distribution arises when there are a fixed number n of independent trials, each with two outcomes (success/failure) and a constant probability of success p. We write X ~ B(n, p).

当存在固定次数 n 的独立试验，每次试验只有两种结果（成功/失败），且成功的概率 p 保持不变时，随机变量服从二项分布，记为 X ~ B(n, p)。

The probability of exactly r successes is:

P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, for r = 0, 1, 2, …, n.

恰好获得 r 次成功的概率为：

P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，其中 r = 0, 1, 2, …, n。

The mean and variance of the binomial distribution are E(X) = np and Var(X) = np(1 − p). These can be derived from the discrete random variable formulas or remembered directly.

二项分布的均值与方差分别为 E(X) = np 和 Var(X) = np(1 − p)。这些可以由离散随机变量的公式推导得出，也可直接记忆。

To find probabilities such as P(X ≤ k) or P(X ≥ k), you often use cumulative binomial tables or the formula combined with addition. Remember P(X ≥ k) = 1 − P(X ≤ k − 1).

计算形如 P(X ≤ k) 或 P(X ≥ k) 的概率时，常借助累积二项分布表，或利用公式求和。注意 P(X ≥ k) = 1 − P(X ≤ k − 1)。

9. The Normal Distribution | 正态分布

The normal distribution is a continuous distribution with a symmetric bell-shaped curve. It is fully described by its mean μ and variance σ²: X ~ N(μ, σ²). The total area under the curve is 1.

正态分布是一种连续的、具有对称钟形曲线的分布。它完全由其均值 μ 和方差 σ² 描述，记作 X ~ N(μ, σ²)。曲线下的总面积为 1。

To find probabilities for any normal distribution, we convert to the standard normal variable Z:

Z = (X − μ) / σ, where Z ~ N(0, 1).

要计算任意正态分布的概率，需将其转换为标准正态变量 Z：

Z = (X − μ) / σ，其中 Z ~ N(0, 1)。

Standard normal tables give Φ(z) = P(Z ≤ z) for z ≥ 0. By symmetry, P(Z ≤ −z) = 1 − Φ(z). For a range, P(a < X < b) = P( (a−μ)/σ < Z < (b−μ)/σ ) = Φ(z_b) − Φ(z_a).

标准正态分布表给出当 z ≥ 0 时的 Φ(z) = P(Z ≤ z)。由对称性，P(Z ≤ −z) = 1 − Φ(z)。对于区间概率，P(a < X < b) = P( (a−μ)/σ < Z < (b−μ)/σ ) = Φ(z_b) − Φ(z_a)。

The inverse normal method is used to find an unknown value x when given a probability. First find the z-value such that P(Z ≤ z) = given probability, then back-transform: x = μ + zσ.

反向正态方法用于已知概率求未知值 x。首先找到满足 P(Z ≤ z) = 给定概率的 z 值，然后回代：x = μ + zσ。

If the population is normally distributed, the sample mean also follows a normal distribution, but this is covered in further topics. For now, master the standard normal calculations and the use of symmetry.

若总体服从正态分布，样本均值同样服从正态分布，但这属于后续内容。当前重点掌握标准正态计算及对称性的使用。

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AS Mathematics: Statistics Key Points Review | AS 数学：统计 考点精讲