AS Edexcel Statistics: Formula & Theorem Quick Reference Handbook | AS Edexcel 统计：公式定理速查手册

📚 AS Edexcel Statistics: Formula & Theorem Quick Reference Handbook | AS Edexcel 统计：公式定理速查手册

This article serves as a rapid reference guide for all essential formulas and theorems covered in the AS Level Edexcel Statistics syllabus. From measures of central tendency to hypothesis testing, each formula is presented with its application context to support effective revision.

本文为 AS Edexcel 统计课程必备公式与定理的速查手册，涵盖从集中趋势度量到假设检验的所有核心内容，每个公式均附应用情境，助力高效复习。

1. Measures of Location and Spread | 位置与离散度量公式

Measures of location summarise the centre of a data set, while measures of spread describe its variability. All formulas apply to sample data unless stated otherwise.

位置度量用于描述数据集中趋势，离散度量则反映数据的变异性。以下公式均针对样本数据（如未特别说明）。

Mean x̄ = Σx / n

The sample mean is the sum of all observations divided by the number of observations.

样本均值等于所有观测值总和除以观测值个数。

Median = middle value when data are ordered

If n is odd, the median is the ((n+1)/2)th value; if n is even, it is the average of the (n/2)th and (n/2 + 1)th values.

将数据排序后，若n为奇数，中位数为第((n+1)/2)个值；若n为偶数，则为第(n/2)个与第(n/2+1)个值的平均数。

Lower quartile Q₁, Median Q₂, Upper quartile Q₃

Q₁ is the median of the lower half of the data, Q₃ is the median of the upper half. The interquartile range IQR = Q₃ − Q₁.

下四分位数 Q₁ 是数据下半部分的中位数，上四分位数 Q₃ 是上半部分的中位数。四分位距 IQR = Q₃ − Q₁。

Range = max − min

Range is the simplest measure of spread, easily affected by outliers.

极差是最简单的离散度量，易受离群值影响。

Variance: s² = Σ(x − x̄)² / (n − 1)

An equivalent computational formula: s² = ( Σx² − (Σx)²/n ) / (n − 1). The divisor n−1 is used for a sample.

等价的简便计算公式：s² = ( Σx² − (Σx)²/n ) / (n − 1)。样本方差除数为 n−1。

Standard deviation: s = √(s²)

The standard deviation is the square root of the variance, measured in the same units as the original data.

标准差是方差的平方根，单位与原始数据相同。

For grouped data with frequencies f: x̄ = Σfx / Σf, and s² = ( Σfx² − (Σfx)²/Σf ) / (Σf − 1).

对于分组数据（频数为f）：x̄ = Σfx / Σf，s² = ( Σfx² − (Σfx)²/Σf ) / (Σf − 1)。

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2. Representations of Data | 数据表示关键公式

Graphical and numerical representations help reveal patterns in data. Key concepts include frequency density and criteria for identifying outliers.

图形与数值表示有助于揭示数据规律，关键概念包含频数密度和离群值识别标准。

Frequency density = frequency / class width

Used in histograms with unequal class widths; area of each bar is proportional to frequency.

用于组距不等的直方图：每个条形面积与频数成正比，频数密度 = 频数 / 组距。

Outlier boundaries: Q₁ − 1.5 × IQR and Q₃ + 1.5 × IQR

Data points below lower fence or above upper fence are considered outliers and are often marked with separate symbols on a box plot.

数据点小于下边界 Q₁−1.5IQR 或大于上边界 Q₃+1.5IQR 被视为离群值，在箱线图上通常用独立符号标出。

Outliers using mean and standard deviation: x ± 2s or x ± 3s

Alternatively, if approximately normally distributed, values more than 2 or 3 standard deviations from the mean may be flagged as outliers.

另一种方法：若数据近似正态分布，超过均值 ±2s 或 ±3s 可被标记为离群值。

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3. Correlation | 相关分析与公式

The product moment correlation coefficient (PMCC) quantifies the strength and direction of a linear relationship between two variables x and y.

积矩相关系数（PMCC）量化两个变量 x 和 y 之间线性关系的强度与方向。

r = S_xy / √(S_xx S_yy)

Where the summary statistics are defined as:

其中各汇总统计量定义如下：

S_xx = Σ(x − x̄)² = Σx² − (Σx)²/n

S_yy = Σ(y − ȳ)² = Σy² − (Σy)²/n

S_xy = Σ(x − x̄)(y − ȳ) = Σxy − (Σx)(Σy)/n

The correlation coefficient r always lies between −1 and +1. r = +1 indicates perfect positive linear correlation, r = −1 indicates perfect negative linear correlation, and r = 0 suggests no linear correlation.

相关系数 r 的取值范围为 [−1, +1]。r = +1 表示完全正线性相关，r = −1 表示完全负线性相关，r = 0 表示无线性相关。

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4. Probability Laws | 概率基本法则

Probability quantifies uncertainty on a scale from 0 to 1. Understanding the basic rules is essential for solving combined events problems.

概率将不确定性量化在 0 到 1 之间，掌握基本法则是解决复合事件问题的基础。

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若A与B互斥，则P(A ∩ B)=0，得P(A ∪ B) = P(A) + P(B)。

Conditional probability: P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0

Conditional probability is the probability of event A occurring given that event B has occurred.

条件概率表示在事件 B 已发生的条件下事件 A 发生的概率。

Multiplication rule: P(A ∩ B) = P(A) P(B | A) = P(B) P(A | B)

For independent events, P(A | B) = P(A) and P(B | A) = P(B), so P(A ∩ B) = P(A) P(B).

乘法法则：若A与B独立，则P(A ∩ B) = P(A) P(B)。

Complement rule: P(A’) = 1 − P(A)

The complement of A is the event that A does not happen.

互补规则：事件A不发生（A’）的概率为1−P(A)。

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5. Discrete Random Variables | 离散随机变量

A discrete random variable X takes a countable number of possible values with associated probabilities. The probability distribution defines P(X = x) for each value x.

离散随机变量 X 可取有限或可数个值，每个值对应一个概率 P(X = x)，构成概率分布。

Expected value: E(X) = μ = Σ x P(X = x)

The expected value is the long‑run average of the outcomes.

期望值即试验多次趋于稳定时的平均结果。

Variance: Var(X) = Σ x² P(X = x) − [E(X)]²

Var(X) = E(X²) − [E(X)]². An alternative expression: Var(X) = Σ (x − μ)² P(X = x).

方差公式：Var(X) = E(X²) − [E(X)]² 等价于 Σ (x − μ)² P(X = x)。

Linear transformations: E(aX + b) = aE(X) + b

Var(aX + b) = a² Var(X)

Adding a constant b shifts the expectation but does not affect the variance; multiplying by a scales both expectation and variance accordingly.

线性变换性质：加常数 b 平移期望但不改变方差；乘以常数 a 会同时缩放期望并使方差乘以 a²。

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6. Binomial Distribution | 二项分布

A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p.

二项分布用于描述在固定次数独立试验中成功的次数，每次试验成功概率恒为 p。

X ~ B(n, p) ⇒ P(X = x) = ⁿC_x p^x (1 − p)^{n − x}

The binomial coefficient ⁿC_x = n! / [x! (n − x)!] counts the number of ways x successes can occur in n trials.

二项分布概率公式：P(X = x) = C(n,x) pˣ (1-p)ⁿ⁻ˣ，其中组合数 ⁿC_x 表示 n 次试验中成功 x 次的方式数。

Expectation and variance: E(X) = np, Var(X) = np(1 − p)

These are derived directly from the properties of the binomial distribution.

这两条性质可由二项分布的定义导出。

For cumulative probabilities P(X ≤ x) or P(X ≥ x), use statistical tables or a calculator. When p > 0.5, the table can be used by counting failures.

累积概率 P(X ≤ x) 或 P(X ≥ x) 可使用统计表或计算器求得。当 p > 0.5 时，可通过转换为失败次数来查表。

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7. Normal Distribution | 正态分布

The normal distribution is a continuous probability distribution modelled by a symmetric bell‑shaped curve. It is defined by two parameters: mean μ and variance σ².

正态分布是一种连续型概率分布，呈对称钟形曲线，由均值 μ 和方差 σ² 两个参数决定。

X ~ N(μ, σ²) ⇒ Standardised score: Z = (X − μ) / σ

After standardisation, Z follows the standard normal distribution N(0, 1), whose probabilities are tabulated.

标准化得分 Z = (X − μ)/σ 服从标准正态分布 N(0,1)，对应的概率值可查表获取。

P(Z < z) = Φ(z) is read from the standard normal table

The table typically gives the cumulative probability to the left of a positive z‑value. Symmetry is used for negative z: Φ(−z) = 1 − Φ(z).

标准正态表通常给出左侧累积概率 Φ(z)；对于负 z 值利用对称性：Φ(−z) = 1 − Φ(z)。

To find an unknown mean or standard deviation, set up an equation using the standardisation formula and the corresponding z‑value from the table.

若要求未知的均值或标准差，可依据标准化公式建立方程，并使用查表所得的 z 值求解。

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8. Hypothesis Testing for Binomial Proportion | 二项分布假设检验

Hypothesis testing provides a formal procedure to decide whether sample data support a claimed population parameter. In AS Statistics, the test concerns a binomial probability p.

假设检验提供一套规范流程，判断样本数据是否支持关于总体参数的声称。AS 统计部分只涉及二项分布概率 p 的检验。

Null hypothesis H₀: p = p₀

The null hypothesis is the statement assumed to be true, stating that the proportion of success is p₀.

原假设 H₀ 假定成功概率等于特定值 p₀，作为待检验的声明。

Alternative hypothesis H₁: p > p₀, p < p₀, or p ≠ p₀

One‑tailed tests use < or > based on the direction of interest; two‑tailed tests use ≠.

备择假设 H₁ 根据实际关注方向设定为 p > p₀（右尾）、p < p₀（左尾）或 p ≠ p₀（双尾）。

Test statistic: X ~ B(n, p₀) under H₀

The evidence is summarised by the observed number of successes

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