📚 AS CCEA Statistics: Quick Reference Formula & Theorem Handbook | AS CCEA 统计：公式定理速查手册
This handbook collates the essential formulas, theorems, and key concepts required for the AS CCEA Statistics specification. It serves as a rapid revision tool, covering descriptive measures, probability, distributions, inference, and regression. Each section presents the core relationships in a clear, concise format to support your exam preparation.
本手册汇集了 AS CCEA 统计课程所需的基本公式、定理和关键概念，可作为快速复习工具，涵盖描述性度量、概率、分布、推断及回归。每一节以清晰简明的格式呈现核心关系，助力你的备考。
1. Descriptive Statistics: Location and Spread | 描述统计：位置与离散度量
For ungrouped data x₁, x₂, …, xₙ, the sample mean is the arithmetic average.
对于未分组数据 x₁, x₂, …, xₙ，样本均值即算术平均值。
x̄ = (∑xᵢ) / n
The median is the middle value when data are ordered; for an odd n it is the (n+1)/2 th value, for even n it is the average of the n/2 th and (n/2 + 1) th values.
中位数是将数据排序后的中间值；当 n 为奇数时取第 (n+1)/2 个数，n 为偶数时取第 n/2 个与第 (n/2 + 1) 个数的平均值。
The mode is the most frequently occurring value, particularly useful for discrete data. The lower quartile Q₁ is the median of the lower half, and the upper quartile Q₃ is the median of the upper half.
众数为出现频率最高的值，尤其适用于离散数据。下四分位数 Q₁ 是下半部分的中位数，上四分位数 Q₃ 是上半部分的中位数。
The interquartile range IQR = Q₃ − Q₁ measures spread resistant to outliers. Variance and standard deviation for a sample are given by:
四分位距 IQR = Q₃ − Q₁ 衡量离散度且不易受异常值影响。样本方差和标准差由下式给出：
s² = (∑(xᵢ − x̄)²) / (n−1) or s² = (∑xᵢ² − (∑xᵢ)²/n) / (n−1)
s = √s²
For grouped data, use midpoints mᵢ and frequencies fᵢ; then x̄ = (∑fᵢmᵢ) / ∑fᵢ and s² = (∑fᵢmᵢ² − (∑fᵢmᵢ)²/∑fᵢ) / (∑fᵢ − 1).
对于分组数据，使用组中值 mᵢ 和频数 fᵢ；则 x̄ = (∑fᵢmᵢ) / ∑fᵢ，s² = (∑fᵢmᵢ² − (∑fᵢmᵢ)²/∑fᵢ) / (∑fᵢ − 1)。
2. Data Presentation: Histograms and Box Plots | 数据展示：直方图与箱线图
In a histogram, the area of each bar is proportional to the frequency. For unequal class widths, frequency density must be used.
在直方图中，每个直条的面积与频数成正比。当组距不相等时，必须使用频数密度。
Frequency density = Frequency / Class width
Box plots display the five‑number summary: minimum, Q₁, median, Q₃, maximum. Outliers are often identified as values more than 1.5 × IQR below Q₁ or above Q₃.
箱线图展示五数概括：最小值、Q₁、中位数、Q₃、最大值。异常值通常指小于 Q₁ − 1.5×IQR 或大于 Q₃ + 1.5×IQR 的数值。
Stem‑and‑leaf diagrams preserve raw data while showing shape. The key indicates place value.
茎叶图既能保留原始数据又能展示分布形态，其图例说明数位。
3. Probability: Rules and Conditional Probability | 概率：法则与条件概率
For any events A and B:
对于任意事件 A 和 B：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B are mutually exclusive, P(A ∩ B) = 0 and the addition rule simplifies.
若 A 与 B 互斥，则 P(A ∩ B) = 0，加法法则可简化。
P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B)
Two events are independent if P(A ∩ B) = P(A) × P(B), equivalently P(B | A) = P(B).
若 P(A ∩ B) = P(A) × P(B)，或等价地 P(B | A) = P(B)，则两事件独立。
Conditional probability is defined as:
条件概率定义为：
P(A | B) = P(A ∩ B) / P(B), P(B) > 0
Probabilities can be found from tables, tree diagrams, or Venn diagrams.
概率可从表格、树状图或文氏图中求得。
4. Discrete Random Variables: Expectation and Variance | 离散随机变量：期望与方差
For a discrete random variable X with probability function P(X = x) = p(x):
对于具有概率函数 P(X = x) = p(x) 的离散随机变量 X：
E(X) = μ = ∑ x·p(x)
Var(X) = E((X − μ)²) = ∑ (x − μ)²·p(x) = E(X²) − [E(X)]²
Linear transformations follow:
线性变换满足：
E(aX + b) = a E(X) + b
Var(aX + b) = a² Var(X)
For independent random variables X and Y, E(X + Y) = E(X) + E(Y) and Var(X + Y) = Var(X) + Var(Y). The standard deviation is σ = √Var(X).
对于独立的随机变量 X 和 Y，有 E(X + Y) = E(X) + E(Y) 且 Var(X + Y) = Var(X) + Var(Y)。标准差 σ = √Var(X)。
5. Binomial Distribution: B(n, p) | 二项分布：B(n, p)
A random variable X follows a binomial distribution if there are n fixed independent trials, each with constant probability of success p. Conditions: fixed n, independence, two outcomes, constant p.
若试验次数 n 固定，各次试验独立且每次成功概率 p 恒定，则随机变量 X 服从二项分布。条件：n 固定、独立、两个结果、p 恒定。
X ~ B(n, p) P(X = r) = C(n, r) pr (1 − p)n−r
where C(n, r) = n! / [r! (n−r)!] is the binomial coefficient.
其中 C(n, r) = n! / [r! (n−r)!] 为二项式系数。
E(X) = np, Var(X) = np(1 − p) = npq
Cumulative probabilities P(X ≤ r) can be read from tables. When using tables for P(X ≥ r), remember P(X ≥ r) = 1 − P(X ≤ r−1).
累积概率 P(X ≤ r) 可从表中查得。使用表格计算 P(X ≥ r) 时，注意 P(X ≥ r) = 1 − P(X ≤ r−1)。
For large n, a normal approximation may be used if np > 5 and nq > 5, with continuity correction.
若 n 较大且 np > 5 且 nq > 5，可使用正态近似并作连续性校正。
6. Normal Distribution: Standardising and Using Tables | 正态分布：标准化与查表
The normal distribution with mean μ and variance σ² is denoted N(μ, σ²). The standard normal distribution Z ~ N(0, 1) uses the transformation:
均值为 μ、方差为 σ² 的正态分布记作 N(μ, σ²)。标准正态分布 Z ~ N(0, 1) 通过以下变换得到：
Z = (X − μ) / σ
Probabilities are found using the standard normal table, which typically gives P(Z < z). For any X ~ N(μ, σ²),
概率通过标准正态表求得，该表通常提供 P(Z < z)。对于任意 X ~ N(μ, σ²)，
P(X < a) = P(Z < (a − μ) / σ)
To find a value k given a probability, work backwards: first find z from the table such that P(Z < z) equals the given probability, then X = μ + zσ.
由给定概率反求 k 值时，先从表中查出满足 P(Z < z) = 概率的 z 值，然后 X = μ + zσ。
The distribution is symmetric about the mean, so P(Z < −z) = 1 − P(Z < z). Percentage points are often required for 90%, 95%, 99% intervals.
正态分布关于均值对称，故 P(Z < −z) = 1 − P(Z < z)。常见须记住 90%、95%、99% 区间的百分点。
7. Sampling and the Distribution of the Sample Mean | 抽样与样本均值的分布
If X ~ N(μ, σ²) and a sample of size n is taken, the sample mean X̄ follows:
若 X ~ N(μ, σ²) 且抽取样本量为 n，则样本均值 X̄ 服从：
X̄ ~ N(μ, σ² / n)
The Central Limit Theorem (CLT) states that for a non‑normal population with mean μ and variance σ², as n increases, the distribution of X̄ approaches N(μ, σ² / n). In practice, n ≥ 30 is usually sufficient.
中心极限定理（CLT）指出，对于均值为 μ、方差为 σ² 的非正态总体，当 n 足够大时，X̄ 近似服从 N(μ, σ² / n)。实践中 n ≥ 30 通常足够。
The standard error of the mean is SE = σ / √n. If σ is unknown, the sample standard deviation s is used, but then the t‑distribution may be required (AS often restricts to known σ or large n).
均值的标准误为 SE = σ / √n。若 σ 未知，可用样本标准差 s 替代，但此时需用 t 分布（AS 常限于已知 σ 或大样本情形）。
8. Confidence Intervals for the Population Mean | 总体均值的置信区间
A 95% confidence interval for the mean μ, when the population is normal and σ is known, is:
当总体为正态且 σ 已知时，均值 μ 的 95% 置信区间为：
x̄ ± 1.96 × (σ / √n)
More generally, a C% confidence interval uses the appropriate z‑value, z*, from the standard normal table such that P(−z* < Z < z*) = C/100.
更一般地，置信水平为 C% 的区间使用相应的标准正态表 z* 值，满足 P(−z* < Z < z*) = C/100。
Confidence interval: x̄ ± z* × (σ / √n)
The width of the interval decreases with larger n or smaller σ. The stated confidence level is the long‑run proportion of intervals that capture μ.
区间宽度随 n 增大或 σ 减小而变窄。所述置信水平是指长期中能捕获 μ 的区间比例。
9. Hypothesis Testing: Mean and Binomial | 假设检验：均值与二项分布检验
A hypothesis test assesses evidence against a null hypothesis H₀. The alternative H₁ is either one‑tailed or two‑tailed.
假设检验用于评估反对原假设 H₀ 的证据。备择假设 H₁ 可为单侧或双侧。
For testing a mean μ with known σ, the test statistic is:
对已知 σ 的均值进行检验时，检验统计量为：
Z = (x̄ − μ₀) / (σ / √n)
Compare Z with critical values from N(0,1). For a two‑tailed test at the 5% level, reject H₀ if |Z| > 1.96.
将 Z 与 N(0,1) 的临界值比较。在 5% 显著性水平的双侧检验中，若 |Z| > 1.96，则拒绝 H₀。
For a binomial proportion test, suppose X ~ B(n, p) and H₀: p = p₀. The p‑value is P(X ≥ observed | p₀) for H₁: p > p₀, or the appropriate tail probability. Alternatively, find critical regions using tables.
对于二项分布比例检验，设 X ~ B(n, p) 且 H₀: p = p₀。若 H₁: p > p₀，则 p 值为 P(X ≥ 观测值 | p₀)，或采用相应的尾部概率。也可用表格确定临界域。
Conclusions are stated in context: ‘reject H₀’ or ‘do not reject H₀’. Never say ‘accept H₀’.
结论应结合情境陈述：“拒绝 H₀”或“不拒绝 H₀”，切忌说“接受 H₀”。
10. Correlation and Regression | 相关与回归
The product moment correlation coefficient (PMCC) r measures the strength of linear association between two variables x and y:
积差相关系数（PMCC）r 衡量两个变量 x 与 y 之间线性相关性的强度：
r = Sxy / √(Sxx Syy)
where Sxx = ∑(x − x̄)² = ∑x² − (∑x)²/n, and Syy similarly, Sxy = ∑(x − x̄)(y − ȳ) = ∑xy − (∑x)(∑y)/n.
其中 Sxx = ∑(x − x̄)² = ∑x² − (∑x)²/n，Syy 类同，Sxy = ∑(x − x̄)(y − ȳ) = ∑xy − (∑x)(∑y)/n。
The least‑squares regression line of y on x is:
y 对 x 的最小二乘回归直线为：
y = a + bx
where b = Sxy / Sxx and a = ȳ − b x̄. This line minimises the sum of squared vertical deviations.
其中 b = Sxy / Sxx，a = ȳ − b x̄。该直线使垂直偏差的平方和最小。
Interpret r close to +1 or −1 as strong linear correlation; r = 0 indicates no linear correlation. The regression line can be used for interpolation but should be extrapolated with caution.
r 接近 +1 或 −1 表示强线性相关；r = 0 表明无线性相关。回归直线可用于内插，外推则需谨慎。
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