📚 AS Cambridge Statistics Formula & Theorem Quick Reference Handbook | AS剑桥统计：公式定理速查手册
This article provides a compact yet comprehensive revision guide to the essential formulas and theorems required for the Cambridge AS Level Probability & Statistics 1 (Paper 5). Each section matches a key topic from the syllabus, paired with clear explanations and illustrative examples where appropriate. Use this handbook alongside past papers to strengthen your fluency in applying statistical techniques.
本文为剑桥AS阶段概率与统计1（Paper 5）提供一份凝练而全面的公式与定理速查手册。每个小节对应一个核心知识点，配合清晰解释与必要示例，帮助你在刷真题时快速巩固统计方法的灵活运用。
1. Data Representation & Frequency Density | 数据表示与频率密度
When constructing histograms for grouped continuous data, the area of each bar is proportional to the frequency. Frequency density is defined as frequency divided by class width. Always calculate class width from the given class boundaries, which may be expressed as inequalities.
在分组连续数据的直方图中，每个长方形的面积与频数成正比。频率密度等于频数除以组距。组距要由给定区间边界计算，注意边界可能以不等式形式给出。
- Frequency density = Frequency ÷ Class width
- 频率密度 = 频数 ÷ 组距
- Class width = Upper boundary − Lower boundary
- 组距 = 上界 − 下界
The cumulative frequency graph (ogive) plots cumulative frequency against the upper class boundary. From it you can estimate medians, quartiles and percentiles.
累积频率图以累积频率对组上界绘制，可用来估算中位数、四分位数和百分位数。
2. Measures of Central Tendency | 集中趋势度量
For ungrouped data, the mean is the sum of all values divided by the number of values. For grouped data, use the midpoints of intervals and multiply by frequencies.
对于不分组数据，均值是所有数据之和除以数据个数；对于分组数据，用各组中点值乘以频数后求平均。
- Ungrouped mean: x̄ = Σx / n
- 不分组均值：x̄ = Σx / n
- Grouped mean: x̄ = Σfx / Σf, where x is the midpoint of each class
- 分组均值：x̄ = Σfx / Σf，其中x为各组中点
- Median: the middle value when data are ordered. For a frequency table, identify the position n/2 in cumulative frequency.
- 中位数：有序数据的中间值。在频数表中利用累计频率找到n/2的位置。
- Mode: the value with the highest frequency. In grouped data, a modal class is the class with the highest frequency density.
- 众数：出现频率最高的值。分组数据中，众数所在组是频率密度最高的组。
3. Measures of Dispersion & Box Plots | 离散程度度量与箱形图
Dispersion tells you how spread out the data are. Variance and standard deviation are the most important measures for further statistical inference.
离散程度反映数据的分散程度。方差和标准差是最重要的进一步统计推断基础。
- Variance for a population: σ² = Σ(x − μ)² / N
- 总体方差：σ² = Σ(x − μ)² / N
- Sample variance (unbiased estimate): s² = Σ(x − x̄)² / (n − 1)
- 样本方差（无偏估计）：s² = Σ(x − x̄)² / (n − 1)
- Standard deviation is the square root of variance: s = √s²
- 标准差是方差的平方根：s = √s²
ta are summarized in a frequency table, use the equivalent formulas: s² = (Σfx² / Σf) − (x̄)² for grouped data, and then multiply by n/(n−1) if it is a sample.
对于频数表，使用公式：方差 = Σfx² / Σf − x̄²，若为样本可再乘n/(n−1)。
Interquartile range (IQR) = Q₃ − Q₁. On a box‑and‑whisker plot, outliers can be identified using the 1.5 × IQR rule.
四分位距IQR = Q₃ − Q₁。箱形图中用1.5倍IQR规则识别异常值。
4. Basic Probability Rules | 概率基本规则
Probability measures the chance of an event on a scale from 0 to 1. The sum of probabilities of all mutually exclusive outcomes in a sample space is 1.
概率用0到1之间的数值衡量事件发生的可能性。样本空间中所有互斥结果的概率之和为1。
- Complement rule: P(A') = 1 − P(A)
- 互补事件：P(A') = 1 − P(A)
- Addition rule for mutually exclusive events: P(A or B) = P(A) + P(B)
- 互斥事件的加法法则：P(A或B) = P(A) + P(B)
- General addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- 一般加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Conditional probability: P(A|B) = P(A ∩ B) / P(B)
- 条件概率：P(A|B) = P(A ∩ B) / P(B)
- Multiplication rule: P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)
- 乘法法则：P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)
- Independent events: A and B are independent if P(A|B) = P(A) or equivalently P(A ∩ B) = P(A)P(B).
- 独立事件：若P(A|B) = P(A)或P(A ∩ B) = P(A)P(B)，则A与B独立。
5. Permutations & Combinations | 排列与组合
Counting techniques are often needed to work out total possible outcomes before applying probability formulas.
在应用概率公式前，常需借助计数技巧求出所有可能结果的总数。
- Permutations of n distinct objects taken r at a time: ⁿPᵣ = n! / (n − r)!
- 从n个不同对象中取出r个的排列数：ⁿPᵣ = n! / (n − r)!
- Combinations (selections) of n distinct objects taken r at a time: ⁿCᵣ = n! / [r!(n − r)!]
- 从n个不同对象中取出r个的组合数：ⁿCᵣ = n! / [r!(n − r)!]
- Special cases: ⁿC₀ = 1, ⁿCₙ = 1
- 特殊情况：ⁿC₀ = 1，ⁿCₙ = 1
- can be done in m ways and a second in n ways, the two together can be done in m × n ways.
- 乘法原理：若一件事有m种做法，另一件事有n种做法，则两件事依次进行共有m×n种做法。
6. Discrete Random Variables | 离散随机变量
ach with a specified probability. The probability distribution must satisfy ΣP(X = x) = 1.
离散随机变量X取可数个值，每个值有明确的概率。概率分布须满足ΣP(X=x) = 1。
- Expected value (mean): E(X) = μ = Σ[x × P(X = x)]
- 期望（均值）：E(X) = μ = Σ[x × P(X=x)]
- Variance: Var(X) = Σ[(x − μ)² P(X = x)] = Σ[x²P(X = x)] − μ²
- 方差：Var(X) = Σ[(x − μ)² P(X=x)] = Σ[x²P(X=x)] − μ²
- Standard deviation: σ = √Var(X)
- 标准差：σ = √Var(X)
- E(aX + b) = aE(X) + b, where a and b are constants
- E(aX + b) = aE(X) + b（a、b为常数）
- Var(aX + b) = a²Var(X), adding b does not change variance.
- Var(aX + b) = a²Var(X)，加常数b不改变方差。
- For any function g(X): E[g(X)] = Σ[g(x) P(X = x)]
- 对任意函数g(X)：E[g(X)] = Σ[g(x) P(X=x)]
7. The 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. If X ~ B(n, p), then:
二项分布描述在固定次数、独立、每次成功概率p不变的试验中成功次数的分布。若X ~ B(n, p)，则有：
- Probability of exactly r successes: P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ
- 恰好r次成功的概率：P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ
- Mean: E(X) = np
- 均值：E(X) = np
- Variance: Var(X) = np(1 − p)
- 方差：Var(X) = np(1 − p)
- Cumulative probabilities are found from tables or by summing individual terms.
- 累积概率可查表或逐个概率加总获得。
Conditions: fixed n, independent trials, two outcomes (success/failure), constant p.
使用条件：固定n，各次独立，两种结果（成功/失败），p恒定。
8. The Geometric Distribution | 几何分布
The geometric distribution models the number of trials up to and including the first success in a sequence of independent Bernoulli trials. If X ~ Geo(p), then:
几何分布描述在独立伯努利试验序列中，首次成功所需的试验次数。若X ~ Geo(p)，则：
- Probability that the first success occurs on the r‑th trial: P(X = r) = (1 − p)ʳ⁻¹ p, for r = 1, 2, 3, …
- 第一次成功发生在第r次试验的概率：P(X = r) = (1 − p)ʳ⁻¹ p，r = 1,2,3,…
- Mean: E(X) = 1/p
- 均值：E(X) = 1/p
- Variance: Var(X) = (1 − p) / p²
- 方差：Var(X) = (1 − p) / p²
- P(X > r) = (1 − p)ʳ, P(X ≤ r) = 1 − (1 − p)ʳ
- P(X > r) = (1 − p)ʳ, P(X ≤ r) = 1 − (1 − p)ʳ
The distribution is memoryless: P(X > s + t | X > s) = P(X > t).
几何分布具有无记忆性：P(X > s + t | X > s) = P(X > t)。
9. The Normal Distribution | 正态分布
The normal distribution N(μ, σ²) is continuous, bell‑shaped and symmetric about the mean μ. Standardising transforms any normal variable X to the standard normal Z ~ N(0, 1²).
正态分布N(μ, σ²)是连续的钟形分布，关于均值μ对称。标准化可将任意正态变量X转化为标准正态变量Z ~ N(0, 1²)。
- Standardisation formula: Z = (X − μ) / σ
- 标准化公式：Z = (X − μ) / σ
- Use the standard normal table to find probabilities for Z and then relate them back to X.
- 使用标准正态分布表查找Z的概率，再反求X的概率。
- Symmetry: P(Z < −a) = P(Z > a) = 1 − Φ(a)
- 对称性：P(Z < −a) = P(Z > a) = 1 − Φ(a)
- 68-95-99.7 rule: about 68% of data lie within 1σ of μ, 95% within 2σ, and 99.7% within 3σ.
- 68-95-99.7规则：约68%的数据落在μ±1σ内，95%在μ±2σ内，99.7%在μ±3σ内。
- When calculating “greater than” or “between” probabilities, always draw a diagram.
- 计算“大于”或“介于”概率时一定要画图辅助。
10. Linear Transformations of Data | 数据的线性变换
Adding a constant or multiplying by a constant changes the mean and variance in predictable ways. This is particularly useful when coding data to simplify calculations.
加减常数或乘以常数会以可预测的方式改变均值和方差。这在进行数据编码简化计算时特别有用。
| Original (原) | Transformation (变换) | New Mean (新均值) | New Variance (新方差) | New Standard Deviation (新标准差) |
|---|---|---|---|---|
| X | Y = aX + b | E(Y) = aE(X) + b | Var(Y) = a²Var(X) | σ_Y = |a| σ_X |
Note that adding a constant shifts the mean but does not affect variance or standard deviation. Multiplying by a negative constant flips the distribution but variance remains positive.
注意加常数仅平移均值，不影响方差和标准差；乘以负常数会翻转分布，但方差恒为正。
11. Correlation & Regression | 相关与回归
Correlation measures the strength and direction of a linear relationship between two variables. The product moment correlation coefficient (PMCC) r is given by:
相关衡量两个变量之间线性关系的强弱和方向。积矩相关系数r的公式为：
r = [nΣxy − (Σx)(Σy)] / √[ (nΣx² − (Σx)²)(nΣy² − (Σy)²) ]
- means perfect positive correlation, r = −1 perfect negative, r = 0 no linear correlation.
- −1 ≤ r ≤ 1；r=1完全正相关，r=−1完全负相关，r=0无线性相关。
- Regression line of y on x (used to predict y from x): y = a + bx where
- y对x的回归线（用于由x预测y）：y = a + bx，其中
b = [nΣxy − (Σx)(Σy)] / [nΣx² − (Σx)²]
a = ȳ − bx̄
- The regression line always passes through (x̄, ȳ).
- 回归直线必经过均值点(x̄, ȳ)。
- Do not use the regression line for prediction far outside the range of data (extrapolation).
- 不可以将回归直线用于数据范围之外的预测（外推）。
For regression of x on y, swap the roles of x and y. The two regression lines intersect at the mean point.
x对y的回归线同理交换变量角色。两条回归线相交于均值点。
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