Edexcel Maths: Statistics Key Topics Explained | Edexcel 数学：统计考点精讲

📚 Edexcel Maths: Statistics Key Topics Explained | Edexcel 数学：统计考点精讲

Mastering statistics in Edexcel Mathematics requires a solid understanding of core concepts, from data summaries to probability distributions. This revision guide breaks down the essential topics, providing clear explanations and worked examples to help you excel in your exam.

掌握 Edexcel 数学统计部分需要扎实理解核心概念，从数据汇总到概率分布。本复习指南分解基本主题，提供清晰解释和例题，助你在考试中脱颖而出。

1. Measures of Location and Spread | 位置与散布度量

The measures of central tendency include the mean, median, and mode. The mean (x̄ for a sample, μ for a population) is calculated as the sum of all data values divided by the number of data points. The median is the middle value when the data is ordered, and the mode is the most frequent value. For grouped data, we use interpolation to estimate the median.

集中趋势的度量包括均值、中位数和众数。均值（样本使用 x̄，总体使用 μ）计算为所有数据值之和除以数据个数。中位数是数据排序后的中间值，众数是出现频率最高的值。对于分组数据，我们使用插值法估算中位数。

The spread of data is described by the range, interquartile range (IQR), variance, and standard deviation. The variance (σ² for population, s² for sample) is the average of squared deviations from the mean. The standard deviation (σ or s) is the square root of the variance. A key formula for sample variance is:

数据的散布程度由极差、四分位距 (IQR)、方差和标准差描述。方差（总体 σ²，样本 s²）是各数据与均值偏差平方的平均值。标准差 (σ 或 s) 是方差的平方根。样本方差的一个关键公式为：

s² = Σ(x – x̄)² / (n – 1)

Note that when finding quartiles (Q₁, Q₂, Q₃) for discrete data, different exam boards may use slight variations, but Edexcel typically uses the (n+1)/4, (n+1)/2, and 3(n+1)/4 positions to locate quartiles in an ordered list. Outliers are often defined as values below Q₁ – 1.5×IQR or above Q₃ + 1.5×IQR.

注意，在求离散数据的四分位数（Q₁, Q₂, Q₃）时，不同考试局可能有细微差异，但 Edexcel 通常使用 (n+1)/4、(n+1)/2 和 3(n+1)/4 的位置在有序列表中确定四分位数。异常值通常定义为低于 Q₁ – 1.5×IQR 或高于 Q₃ + 1.5×IQR 的值。

2. Probability Basics | 概率基础

Probability is a measure of the likelihood of an event, ranging from 0 to 1. The sample space is the set of all possible outcomes. For any event A, P(A) = number of favorable outcomes / total number of outcomes, provided all outcomes are equally likely. The complement rule states P(not A) = 1 – P(A).

概率是对事件发生可能性的度量，范围从 0 到 1。样本空间是所有可能结果的集合。对于任何事件 A，P(A) = 有利结果数 / 总结果数，前提是所有结果等可能。互补规则指出 P(非 A) = 1 – P(A)。

Two events A and B are mutually exclusive if they cannot occur at the same time, meaning P(A ∩ B) = 0. The addition rule for mutually exclusive events is P(A ∪ B) = P(A) + P(B). For non-mutually exclusive events, we use P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

如果两个事件 A 和 B 不能同时发生，则它们互斥，意味着 P(A ∩ B) = 0。互斥事件的加法规则是 P(A ∪ B) = P(A) + P(B)。对于非互斥事件，我们使用 P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。

Independent events satisfy P(A ∩ B) = P(A) × P(B). Conditional probability is given by P(A|B) = P(A ∩ B) / P(B), which leads to the multiplication rule for dependent events. Tree diagrams are an excellent tool for handling sequential events.

独立事件满足 P(A ∩ B) = P(A) × P(B)。条件概率由 P(A|B) = P(A ∩ B) / P(B) 给出，并导出相关事件的乘法规则。树状图是处理序贯事件的极好工具。

3. Discrete Random Variables | 离散随机变量

A discrete random variable X takes on a countable number of values. Its probability distribution is defined by a table listing each value x and the corresponding probability P(X = x). The sum of all probabilities must equal 1. The expected value (mean) is E(X) = Σ x·P(X = x). The variance is Var(X) = E(X²) – [E(X)]², where E(X²) = Σ x²·P(X = x).

离散随机变量 X 取可数个值。其概率分布由列出每个值 x 及相应概率 P(X = x) 的表格定义。所有概率之和必须等于 1。期望值（均值）为 E(X) = Σ x·P(X = x)。方差为 Var(X) = E(X²) – [E(X)]²，其中 E(X²) = Σ x²·P(X = x)。

For a linear transformation Y = aX + b, we have E(Y) = aE(X) + b and Var(Y) = a²Var(X). The standard deviation is the square root of the variance. The cumulative distribution function F(x) = P(X ≤ x) is often useful for finding probabilities of inequalities.

对于线性变换 Y = aX + b，有 E(Y) = aE(X) + b，且 Var(Y) = a²Var(X)。标准差是方差的平方根。累积分布函数 F(x) = P(X ≤ x) 在求不等式概率时常常很有用。

4. Binomial Distribution | 二项分布

The binomial distribution models the number of successes in a fixed number of independent trials, with each trial having the same probability of success p. If X ~ B(n, p), then the probability of exactly r successes is given by:

二项分布用于建模固定次数的独立试验中成功的次数，每次试验有相同的成功概率 p。若 X ~ B(n, p)，则恰好有 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)!]

其中 ⁿCᵣ = n! / [r!(n-r)!]

We can find binomial probabilities using the formula, tables, or a calculator. The mean of a binomial distribution is E(X) = np, and the variance is Var(X) = np(1-p).

我们可以使用公式、表格或计算器求二项概率。二项分布的均值为 E(X) = np，方差为 Var(X) = np(1-p)。

When n is large and p is close to 0.5, the binomial distribution can be approximated by a normal distribution with μ = np and σ² = np(1-p), provided np > 5 and n(1-p) > 5. A continuity correction is applied.

当 n 较大且 p 接近 0.5 时，二项分布可用均值为 μ = np、方差为 σ² = np(1-p) 的正态分布近似，前提是 np > 5 且 n(1-p) > 5，并应用连续性校正。

5. Normal Distribution | 正态分布

The normal distribution is a continuous probability distribution with a bell-shaped curve. It is defined by its mean μ and standard deviation σ. We write X ~ N(μ, σ²). To find probabilities, we standardise to the standard normal Z ~ N(0,1²) using:

正态分布是具有钟形曲线的连续概率分布。它由其均值 μ 和标准差 σ 定义。记作 X ~ N(μ, σ²)。为了求概率，我们使用下式将其标准化为标准正态 Z ~ N(0,1²)：

Z = (X – μ) / σ

Probability tables give Φ(z) = P(Z < z) for z ≥ 0. For negative z-values, we use symmetry: Φ(-z) = 1 - Φ(z). To find a value x given a probability, we work backwards from the table to find z and then convert using x = μ + zσ.

概率表给出当 z ≥ 0 时的 Φ(z) = P(Z < z)。对于负 z 值，我们利用对称性：Φ(-z) = 1 - Φ(z)。给定概率求数值 x 时，我们从表中反向寻找 z，然后使用 x = μ + zσ 转换。

It is crucial to sketch the normal curve and shade the required area to avoid errors. For P(a < X < b), we compute Φ((b-μ)/σ) - Φ((a-μ)/σ). The total area under the curve is 1.

至关重要的一点是画出正态曲线并给所需区域涂上阴影，以避免错误。对于 P(a < X < b)，我们计算 Φ((b-μ)/σ) - Φ((a-μ)/σ)。曲线下的总面积为 1。

6. Correlation and Regression | 相关与回归

Scatter diagrams show the relationship between two variables. Correlation measures the strength and direction of a linear relationship. The product moment correlation coefficient (PMCC), denoted by r, is calculated as:

散点图展示两个变量之间的关系。相关性衡量线性关系的强度和方向。积矩相关系数 (PMCC) 记作 r，计算公式为：

r = Sxy / √(Sxx · Syy)

where Sxy = Σ(x – x̄)(y – ȳ), Sxx = Σ(x – x̄)², Syy = Σ(y – ȳ)².

其中 Sxy = Σ(x – x̄)(y – ȳ), Sxx = Σ(x – x̄)², Syy = Σ(y – ȳ)²。

Values of r range from -1 to +1. A value close to +1 indicates strong positive correlation; close to -1 indicates strong negative

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Edexcel Maths: Statistics Key Topics Explained | Edexcel 数学：统计 考点精讲