📚 GCSE CIE Statistics: Formula & Theorem Quick Reference Guide | GCSE CIE 统计：公式定理速查手册
This quick reference guide summarises the essential formulas, theorems and statistical techniques required for the CIE GCSE Statistics syllabus. It covers measures of central tendency and dispersion, probability rules, discrete random variables, the binomial and normal distributions, sampling, estimation, hypothesis testing and chi‑squared tests. Each entry is presented with a concise explanation in English followed by a parallel Chinese translation to support bilingual learners.
本速查手册总结了CIE GCSE统计课程所需的关键公式、定理和统计方法。内容涵盖集中趋势与离散程度的度量、概率法则、离散随机变量、二项分布与正态分布、抽样与估计、假设检验以及卡方检验。每个条目均先提供简洁的英文解释，再配以中文翻译，方便中英双语学习者使用。
1. Measures of Central Tendency | 集中趋势的度量
The mean (arithmetic average) for ungrouped data is x̄ = ∑x / n, where ∑x is the sum of all observations and n is the sample size. For a population, the mean is denoted by μ.
未分组数据的均值（算术平均数）为 x̄ = ∑x / n ，其中 ∑x 表示所有观测值的总和，n 为样本容量。总体的均值用 μ 表示。
For data presented in a frequency table, the mean is calculated using x̄ = ∑fx / ∑f, where f is the frequency of each value x. If data are grouped, x represents the class midpoint.
对于频数表呈现的数据，均值用 x̄ = ∑fx / ∑f 计算，其中 f 为每个数值 x 的频数。若数据已分组，x 代表组中值。
The median is the middle value when data are arranged in order. For n observations, the position of the median is (n + 1)/2. If n is even, the median is the average of the two middle values.
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。若 n 为偶数，中位数为中间两个数值的平均值。
The mode is the value that occurs most frequently. A data set may have one mode, more than one mode (bimodal or multimodal) or no mode at all.
众数是出现频数最高的数值。一组数据可能有一个众数、多个众数（双峰或多峰）或没有众数。
2. Measures of Dispersion | 离散程度的度量
The range is the difference between the largest and smallest values: Range = Maximum – Minimum. It is simple but sensitive to outliers.
极差是最大值与最小值之差：极差 = 最大值 – 最小值。计算简单，但易受异常值影响。
The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁): IQR = Q₃ – Q₁. It measures the spread of the middle 50% of the data.
四分位距（IQR）是上四分位数（Q₃）与下四分位数（Q₁）之差：IQR = Q₃ – Q₁。它衡量中间50%数据的离散程度。
Variance measures the average squared deviation from the mean. For a population, σ² = ∑(x – μ)² / N. For a sample, the unbiased estimate of the population variance uses s² = ∑(x – x̄)² / (n – 1).
方差衡量数据与均值之差的平方的平均值。总体方差为 σ² = ∑(x – μ)² / N 。样本中，总体方差的无偏估计使用 s² = ∑(x – x̄)² / (n – 1) 。
The standard deviation is the positive square root of the variance: σ = √(σ²) or s = √(s²). It is expressed in the original units of the data.
标准差是方差的正平方根： σ = √(σ²) 或 s = √(s²)。其单位与原数据相同。
For grouped data, the variance is estimated using class midpoints x and frequencies f: s² = [∑fx² – (∑fx)²/∑f] / (∑f – 1).
对于分组数据，方差利用组中值 x 和频数 f 进行估计： s² = [∑fx² – (∑fx)²/∑f] / (∑f – 1) 。
3. Percentiles and Linear Interpolation for Grouped Data | 分组数据的百分位数与线性插值
For grouped data, the p-th percentile is found using linear interpolation within the class that contains the position (p/100) × n. The formula is: Percentile = L + [( (p/100)n – F ) / f ] × w, where L is the lower class boundary, F is the cumulative frequency before the class, f is the class frequency and w is the class width.
对于分组数据，第p百分位数通过在其所在组内线性插值求得，该组位置为 (p/100) × n 。公式为：百分位数 = L + [( (p/100)n – F ) / f ] × w ，其中 L 为组下限，F 为该组之前的累积频数，f 为该组频数，w 为组距。
The median is the 50th percentile, the lower quartile is the 25th percentile, and the upper quartile is the 75th percentile. The same interpolation method applies.
中位数即第50百分位数，下四分位数为第25百分位数，上四分位数为第75百分位数。可使用相同的插值法求得。
4. Probability Rules | 概率法则
For two mutually exclusive events A and B, the addition rule states: P(A ∪ B) = P(A) + P(B). They cannot happen at the same time.
对于两个互斥事件 A 与 B ，加法法则为： P(A ∪ B) = P(A) + P(B) 。它们不能同时发生。
For any two events, the general addition rule is P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
对于任意两个事件，一般加法法则为 P(A ∪ B) = P(A) + P(B) – P(A ∩ B) 。
The complement rule: P(A’) = 1 – P(A), where A’ is the event that A does not occur.
互补事件法则： P(A’) = 1 – P(A) ，其中 A’ 表示 A 不发生的事件。
Conditional probability is defined as P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0. This is the probability of A given that B has occurred.
条件概率的定义为 P(A | B) = P(A ∩ B) / P(B) ，前提是 P(B) > 0 。它表示在事件 B 已发生的条件下事件 A 发生的概率。
The multiplication rule for dependent events: P(A ∩ B) = P(A) × P(B | A). For independent events, P(A | B) = P(A) and P(A ∩ B) = P(A) × P(B).
相关事件的乘法法则： P(A ∩ B) = P(A) × P(B | A) 。对于独立事件，有 P(A | B) = P(A) 且 P(A ∩ B) = P(A) × P(B) 。
5. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of values with associated probabilities. The sum of all probabilities ∑P(X = x) = 1.
离散随机变量 X 取可数个值，每个值对应一定的概率。所有概率之和 ∑P(X = x) = 1 。
The expected value (mean) of X is E(X) = ∑x·P(X = x). It represents the long‑run average outcome.
X 的期望值（均值）为 E(X) = ∑x·P(X = x) ，代表长期平均结果。
The variance of X is Var(X) = ∑(x – μ)²·P(X = x) = E(X²) – [E(X)]², where μ = E(X).
X 的方差为 Var(X) = ∑(x – μ)²·P(X = x) = E(X²) – [E(X)]² ，其中 μ = E(X) 。
For a linear transformation Y = aX + b, the expectation and variance are: E(Y) = aE(X) + b and Var(Y) = a²Var(X).
对于线性变换 Y = aX + b ，期望和方差分别为： E(Y) = aE(X) + b ， Var(Y) = a²Var(X) 。
6. Binomial Distribution | 二项分布
If a trial has two outcomes (success and failure) with constant probability of success p, and n independent trials are performed, then the random variable X = number of successes follows a binomial distribution: X ~ B(n, p).
若一次试验只有两种结果（成功与失败），成功的概率恒定为 p ，且进行 n 次独立试验，则随机变量 X = 成功次数服从二项分布： X ~ B(n, p) 。
The probability of exactly r successes is given by P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, where q = 1 – p and ⁿCᵣ = n! / [r!(n – r)!].
恰好取得 r 次成功的概率为 P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ ，其中 q = 1 – p ， ⁿCᵣ = n! / [r!(n – r)!] 。
The mean and variance of a binomial distribution are E(X) = np and Var(X) = npq.
二项分布的均值与方差为 E(X) = np ， Var(X) = npq 。
A normal approximation to the binomial can be used when np > 5 and nq > 5. The approximating normal distribution has mean np and variance npq. A continuity correction of ±0.5 is applied.
当 np > 5 且 nq > 5 时，可用正态分布近似二项分布。近似正态分布的均值为 np ，方差为 npq 。需进行 ±0.5 的连续性校正。
7. Normal Distribution | 正态分布
The normal distribution is a continuous distribution with a bell‑shaped curve, defined by its mean μ and standard deviation σ: X ~ N(μ, σ²).
正态分布是一种连续型分布，呈钟形曲线，由其均值 μ 和标准差 σ 决定： X ~ N(μ, σ²) 。
Standardisation converts any normal variable to the standard normal Z ~ N(0, 1²) using: Z = (X – μ) / σ.
标准化可将任何正态变量转化为标准正态变量 Z ~ N(0, 1²) ，公式为： Z = (X – μ) / σ 。
The standard normal table gives cumulative probabilities Φ(z) = P(Z ≤ z). Probabilities for X are found by standardising: P(X ≤ x) = Φ( (x – μ) / σ ).
标准正态表提供累积概率 Φ(z) = P(Z ≤ z) 。求 X 的概率可通过标准化得到： P(X ≤ x) = Φ( (x – μ) / σ ) 。
For inverse problems, the z‑value corresponding to a given cumulative probability is read from the table, and X = μ + zσ.
对于逆向求解问题，先从表中查出给定累积概率对应的 z 值，再用 X = μ + zσ 计算。
8. Sampling and Estimation | 抽样与估计
If a random sample of size n is drawn from a normal population with mean μ and variance σ², the sample mean x̄ follows a normal distribution: x̄ ~ N(μ, σ²/n).
若从均值为 μ、方差为 σ² 的正态总体中抽取容量为 n 的随机样本，则样本均值 x̄ 服从正态分布： x̄ ~ N(μ, σ²/n) 。
When the population variance is unknown, the standard error of the mean is estimated as s / √n, where s is the sample standard deviation. For large samples, the Central Limit Theorem ensures x̄ is approximately normal even if the population is not normal.
当总体方差未知时，均值的标准误差用 s / √n 估计，其中 s 为样本标准差。对于大样本，由中心极限定理可知，即使总体非正态， x̄ 的分布也近似正态。
A 95% confidence interval for the population mean μ (when σ is known) is x̄ ± 1.96 × (σ/√n). For a 99% confidence interval, the multiplier is 2.576.
当 σ 已知时，总体均值 μ 的95%置信区间为 x̄ ± 1.96 × (σ/√n) 。99%置信区间的乘数为 2.576 。
For a population proportion p, the sample proportion p̂ is an unbiased estimator. An approximate confidence interval is p̂ ± z* √[p̂(1 – p̂)/n], where z* is the appropriate critical value from the standard normal distribution.
对于总体比例 p ，样本比例 p̂ 是无偏估计量。近似置信区间为 p̂ ± z* √[p̂(1 – p̂)/n] ，其中 z* 是标准正态分布对应的临界值。
9. Hypothesis Testing: Binomial Test | 假设检验：二项检验
A binomial test is used when investigating a claimed population proportion p. The null hypothesis H₀: p = p₀ is tested against a one‑ or two‑tailed alternative H₁. The test statistic is the observed number of successes X.
当检验声称的总体比例 p 时使用二项检验。零假设 H₀: p = p₀ 与单侧或双侧备择假设 H₁ 进行对比。检验统计量为观测到的成功次数 X 。
The p‑value is the probability of obtaining a result at least as extreme as X, assuming H₀ is true. Probabilities are calculated using the binomial distribution B(n, p₀). If the p‑value is less than the significance level α (commonly 0.05), H₀ is rejected.
p 值是在 H₀ 为真的条件下，得到至少与 X 一样极端的结果的概率。概率使用二项分布 B(n, p₀) 计算。若 p 值小于显著性水平 α（通常为0.05），则拒绝 H₀。
Alternatively, critical regions determined by the significance level can be used: find the values of X that lead to rejection. If the observed X falls in the critical region, H₀ is rejected.
另一种方法是利用显著性水平确定临界域：找出导致拒绝 H₀ 的 X 值。若观测到的 X 落入临界域，则拒绝 H₀。
10. Hypothesis Testing: z‑test for a Mean | 假设检验：均值的z检验
When testing a population mean μ with a known population standard deviation σ, the test statistic is Z = (x̄ – μ₀) / (σ/√n), where μ₀ is the hypothesised mean. Z follows the standard normal distribution under H₀.
当已知总体标准差 σ 检验总体均值 μ 时，检验统计量为 Z = (x̄ – μ₀) / (σ/√n) ，其中 μ₀ 为假设的均值。在 H₀ 下，Z 服从标准正态分布。
For a two‑tailed test at significance level α, the critical values are ±zα/2 (e.g. ±1.96 for α = 0.05). H₀ is rejected if |Z| > zα/2.
对于显著性水平为 α 的双侧检验，临界值为 ±zα/2（例如 α = 0.05 时为 ±1.96）。若 |Z| > zα/2，则拒绝 H₀。
For a one‑tailed test, the critical value is zα (or –zα for a left‑tailed test). The conclusion is based on whether the test statistic falls in the rejection region.
对于单侧检验，临界值为 zα（左侧检验为 –zα）。结论取决于检验统计量是否落入拒绝域。
11. Chi‑Squared Tests | 卡方检验
The chi‑squared statistic is used for goodness‑of‑fit tests and tests for independence in contingency tables. The formula is χ² = ∑ (O – E)² / E, where O is the observed frequency and E is the expected frequency.
卡方统计量用于拟合优度检验和列联表独立性检验。其公式为 χ² = ∑ (O – E)² / E ，其中 O 为观测频数，E 为期望频数。
For a goodness‑of‑fit test, expected frequencies are calculated based on the hypothesised distribution. The degrees of freedom are (number of categories – 1). The null hypothesis is rejected if χ²
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