📚 AS CIE Statistics: Formula & Theorem Quick Reference Handbook | AS CIE 统计：公式定理速查手册
This concise handbook gathers all essential formulas and theorems required for the Cambridge International AS Level Statistics syllabus. It covers measures of central tendency, dispersion, probability, permutations and combinations, discrete random variables, the binomial distribution, and the normal distribution. Each entry is paired with a straightforward English explanation followed by its Chinese equivalent, helping bilingual learners quickly locate and revise critical content.
本速查手册整理了剑桥国际 AS 水平统计课程所需的所有核心公式与定理，涵盖集中趋势量数、离散程度量数、概率、排列与组合、离散随机变量、二项分布以及正态分布。每条内容均配有清晰的英文解释及对应的中文说明，方便双语学习者快速查找并复习关键知识点。
1. Measures of Central Tendency | 集中趋势量数
For ungrouped data, the arithmetic mean of the values x₁, x₂, …, xₙ is defined as Σx / n, where n is the number of observations.
对于未分组数据，数值 x₁, x₂, …, xₙ 的算术平均数定义为 Σx / n，其中 n 为观测值的个数。
For grouped data, the mean is estimated using midpoints: x̄ = Σfx / Σf, where f is the frequency of each class and x is the class midpoint.
对于分组数据，平均数用组中点进行估计：x̄ = Σfx / Σf，其中 f 为各组的频数，x 为组中点。
The median of ungrouped data is the middle value when the data are arranged in order. 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.
未分组数据的中位数是将数据排序后位于中间的值。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 与第 (n/2 + 1) 个值的平均数。
For grouped data, the median is found by linear interpolation: Median = L + ( (n/2 – F) / fₘ ) × c, where L is the lower boundary of the median class, n is total frequency, F is the cumulative frequency before the median class, fₘ is the frequency of the median class, and c is the class width.
对于分组数据，中位数通过线性插值求得：Median = L + ( (n/2 – F) / fₘ ) × c，其中 L 为中位数所在组的下限，n 为总频数，F 为中位数所在组之前的累计频数，fₘ 为中位数所在组的频数，c 为组距。
The mode is the value that occurs most frequently. For grouped data, the modal class is the class with the highest frequency density.
众数是出现次数最多的值。对于分组数据，众数所在的组是频率密度最高的组。
2. Measures of Spread | 离散程度量数
The range is the difference between the largest and smallest observations: Range = x_max – x_min.
全距是最大观测值与最小观测值之差：Range = x_max – x_min。
The interquartile range (IQR) is Q₃ – Q₁, where Q₁ is the lower quartile and Q₃ is the upper quartile. For discrete data, Q₁ is the (n+1)/4 th value and Q₃ is the 3(n+1)/4 th value when ordered.
四分位距 (IQR) 为 Q₃ – Q₁，其中 Q₁ 为下四分位数，Q₃ 为上四分位数。对于离散数据，排序后 Q₁ 为第 (n+1)/4 个值，Q₃ 为第 3(n+1)/4 个值。
For grouped data, quartiles are obtained by linear interpolation similar to the median, using cumulative frequency curves or formula.
对于分组数据，四分位数通过类似于中位数的线性插值法求得，可利用累计频率曲线或公式计算。
The variance of a population is σ² = Σ(x – μ)² / N or using the computational formula σ² = (Σx² / N) – μ². For a sample, the unbiased estimate of population variance is s² = Σ(x – x̄)² / (n – 1).
总体方差为 σ² = Σ(x – μ)² / N，或使用计算式 σ² = (Σx² / N) – μ²。对于样本，总体方差的无偏估计为 s² = Σ(x – x̄)² / (n – 1)。
The standard deviation is the square root of the variance: σ = √σ² or s = √s².
标准差是方差的平方根：σ = √σ² 或 s = √s²。
For grouped data, s² = Σf(x – x̄)² / Σf, and the computational form is s² = (Σfx² / Σf) – x̄², where f is the frequency and x is the midpoint.
对于分组数据，s² = Σf(x – x̄)² / Σf，计算式为 s² = (Σfx² / Σf) – x̄²，其中 f 为频数，x 为组中点。
3. Probability | 概率
The probability of an event A, P(A), satisfies 0 ≤ P(A) ≤ 1. The sample space S has P(S) = 1.
事件 A 的概率 P(A) 满足 0 ≤ P(A) ≤ 1。样本空间 S 的概率为 P(S) = 1。
For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B). In general, P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
对于互斥事件 A 和 B，有 P(A ∪ B) = P(A) + P(B)。一般地，P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。
Complement rule: P(A’) = 1 – P(A), where A’ is the event that A does not occur.
互补律：P(A’) = 1 – P(A)，其中 A’ 表示 A 不发生的事件。
Conditional probability: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. This is the probability of A given B.
条件概率：P(A|B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。它表示在 B 发生的情况下 A 发生的概率。
Two events are independent if P(A ∩ B) = P(A) × P(B) or equivalently P(A|B) = P(A).
如果 P(A ∩ B) = P(A) × P(B) 或等价地 P(A|B) = P(A)，则称两个事件独立。
4. Permutations and Combinations | 排列与组合
The number of ways to arrange n distinct objects in a line is n! = n × (n-1) × … × 1.
n 个不同物体排成一排的排列数为 n! = n × (n-1) × … × 1。
The number of permutations of n distinct objects taken r at a time is ⁿPᵣ = n! / (n – r)!.
从 n 个不同物体中取出 r 个进行排列的排列数为 ⁿPᵣ = n! / (n – r)!。
When some objects are identical, the number of distinct permutations of n objects, where there are n₁ identical of one type, n₂ identical of another type, etc., is n! / (n₁! n₂! …).
当存在相同物体时，n 个物体中有 n₁ 个相同的第一类，n₂ 个相同的第二类等，则不同的排列数为 n! / (n₁! n₂! …)。
The number of combinations of n distinct objects taken r at a time is ⁿCᵣ = n! / [r! (n – r)!], which is also written as (n choose r).
从 n 个不同物体中取出 r 个进行组合的组合数为 ⁿCᵣ = n! / [r! (n – r)!]，也可记作 binomial coefficient。
ⁿCᵣ = ⁿCₙ₋ᵣ, and ⁿC₀ = ⁿCₙ = 1.
ⁿCᵣ = ⁿCₙ₋ᵣ，且 ⁿC₀ = ⁿCₙ = 1。
5. Discrete Random Variables | 离散随机变量
A discrete random variable X takes values x₁, x₂, … with probabilities P(X = xᵢ). The sum of all probabilities is 1.
离散随机变量 X 的取值为 x₁, x₂, …，对应的概率为 P(X = xᵢ)。所有概率之和为 1。
The expected value (mean) of X is E(X) = Σ xᵢ P(X = xᵢ) = Σ x p.
X 的期望值（均值）为 E(X) = Σ xᵢ P(X = xᵢ) = Σ x p。
For a function g(X), E[g(X)] = Σ g(xᵢ) P(X = xᵢ). Thus E(aX + b) = aE(X) + b for constants a and b.
对于函数 g(X)，有 E[g(X)] = Σ g(xᵢ) P(X = xᵢ)。因此，对于常数 a 和 b，E(aX + b) = aE(X) + b。
The variance of X is Var(X) = E[(X – μ)²] = E(X²) – [E(X)]², where μ = E(X). The computational form Var(X) = Σ x² p – μ² is often used.
X 的方差为 Var(X) = E[(X – μ)²] = E(X²) – [E(X)]²，其中 μ = E(X)。计算中常用 Var(X) = Σ x² p – μ²。
For constants a and b, Var(aX + b) = a² Var(X). The standard deviation is sd(X) = √Var(X).
对于常数 a 和 b，有 Var(aX + b) = a² Var(X)。标准差为 sd(X) = √Var(X)。
If X and Y are independent random variables, E(XY) = E(X)E(Y) and Var(X ± Y) = Var(X) + Var(Y).
若 X 与 Y 是独立的随机变量，则 E(XY) = E(X)E(Y) 且 Var(X ± Y) = Var(X) + Var(Y)。
6. The Binomial Distribution | 二项分布
A binomial experiment consists of n independent identical trials, each with two outcomes (success or failure). The probability of success, p, remains constant. X counts the number of successes obtained.
二项试验由 n 次独立的相同试验组成，每次试验有两种结果（成功或失败）。成功的概率 p 保持不变。X 表示获得的成功次数。
If X ~ B(n, p), then P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ, for r = 0, 1, 2, …, n.
若 X ~ B(n, p)，则 P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ，r 取 0, 1, 2, …, n。
The expected value is E(X) = np, and the variance is Var(X) = np(1 – p).
期望值为 E(X) = np，方差为 Var(X) = np(1 – p)。
The mode is the most likely number of successes; it is either floor[(n+1)p] or floor[(n+1)p] – 1, depending on whether (n+1)p is an integer.
众数为最可能的成功次数；它通常是 floor[(n+1)p] 或 floor[(n+1)p] – 1，取决于 (n+1)p 是否为整数。
7. The Normal Distribution | 正态分布
A continuous random variable X follows a normal distribution with mean μ and variance σ², written as X ~ N(μ, σ²). Its probability density function is symmetric and bell-shaped.
连续随机变量 X 服从均值为 μ、方差为 σ² 的正态分布，记作 X ~ N(μ, σ²)。其概率密度函数是对称的钟形曲线。
The standard normal distribution has mean 0 and variance 1: Z ~ N(0, 1). Any normal variable X can be standardised: Z = (X – μ) / σ.
标准正态分布的均值为 0，方差为 1：Z ~ N(0, 1)。任一正态变量 X 可标准化为 Z = (X – μ) / σ。
To find P(X < a) for X ~ N(μ, σ²), first calculate z = (a - μ) / σ and then use the standard normal probability table for Φ(z). For P(X > a) = 1 – Φ(z).
对于 X ~ N(μ, σ²)，求 P(X < a) 时，先计算 z = (a - μ) / σ，再查标准正态概率表得到 Φ(z)。而 P(X > a) = 1 – Φ(z)。
For probabilities in an interval: P(a < X < b) = Φ((b - μ)/σ) - Φ((a - μ)/σ).
区间概率：P(a < X < b) = Φ((b - μ)/σ) - Φ((a - μ)/σ)。
The symmetry property of the standard normal distribution gives Φ(-z) = 1 – Φ(z).
标准正态分布的对称性给出 Φ(-z) = 1 – Φ(z)。
When finding an unknown mean or standard deviation, set up an equation using Z-scores and given probabilities, then solve using inverse normal table values.
当需要求未知的均值或标准差时，利用 Z 分数与给定的概率建立方程，然后借助逆正态分布表值求解。
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