Statistics Formula & Theorem Quick Reference | 统计公式定理速查手册

📚 Statistics Formula & Theorem Quick Reference | 统计公式定理速查手册

This quick reference handbook provides a compact yet comprehensive summary of essential formulas and key theorems for Year 11 Cambridge Statistics. It is designed to help you revise efficiently, whether you are preparing for an end-of-topic test or the final examination. Each section is presented with a clear English explanation immediately followed by its Chinese equivalent, ensuring bilingual learners can grasp the concepts with confidence and precision.

本速查手册为 Year 11 剑桥统计课程提供简洁而全面的重要公式与核心定理总结，旨在帮助你高效复习，无论你是在准备单元测验还是期末考试。每个部分都先给出清晰的英文解释，紧接着提供对应的中文版本，确保双语学习者能够自信而精准地掌握这些概念。

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1. Measures of Central Tendency | 集中趋势的度量

The mean of a dataset is the arithmetic average, calculated by summing all values and dividing by the number of observations. For ungrouped data, the sample mean is denoted by x̄.

数据集的均值是算术平均数，通过将所有数值相加后除以观测值的个数计算得到。对于未分组数据，样本均值用 x̄ 表示。

Mean: x̄ = (Σxᵢ)/n , where n is the sample size.

均值：x̄ = (Σxᵢ)/n，其中 n 为样本容量。

The median is the middle value when the data are arranged in ascending 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. The mode is the value that occurs most frequently.

中位数是将数据按升序排列后位于中间的值。如果 n 为奇数，中位数是第 (n+1)/2 个数值；如果 n 为偶数，则是第 n/2 和第 (n/2 + 1) 个数值的平均值。众数是出现频率最高的数值。

For grouped data, the mean is estimated using midpoints of class intervals: x̄ uency and mᵢ the midpoint. The median class is found by cumulative frequency, and the mode is the class with the highest frequency density.

对于分组数据，均值可利用组中值进行估算：x̄ = (Σfᵢmᵢ) / (Σfᵢ)，其中 fᵢ 是频数，mᵢ 是组中值。中位数所在组通过累积频数确定，众数则是频数密度最高的组。

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2. Measures of Dispersion | 离差的度量

of spread, defined as the difference between the largest and smallest values: Range = Xₘₐₓ – Xₘᵢₙ. It is sensitive to outliers and does not use all data points.

极差是最简单的离散程度度量，定义为最大值与最小值之差：极差 = Xₘₐₓ – Xₘᵢₙ。它对异常值敏感，且未使用全部数据点。

Variance and standard deviation are far more useful. For a population, the variance σ² is the average of squared deviations from the mean. The standard deviation σ is its square root.

方差和标准差则有用得多。对于总体，方差 σ² 是各数据与均值离差平方的平均数。标准差 σ 是其平方根。

Population variance: σ² = Σ(xᵢ – μ)² / N ; Population standard deviation: σ = √σ²

总体方差：σ² = Σ(xᵢ – μ)² / N；总体标准差：σ = √σ²

In most exam contexts, we work with a sample. The sample variance s² uses (n – 1) in the denominator to give an unbiased estimate of the population variance.

在大多数考试情境中，我们处理的是样本。样本方差 s² 使用 (n – 1) 作为分母，以得到总体方差的无偏估计。

Sample variance: s² = Σ(xᵢ – x̄)² / (n – 1)

样本方差：s² = Σ(xᵢ – x̄)² / (n – 1)

For grouped data, replace xᵢ with class midpoint mᵢ and multiply by frequency: s² = Σfᵢ(mᵢ – x̄)² / (Σfᵢ – 1). The interquartile range (IQR = Q₃ – Q₁) is another robust measure, representing the middle 50% of data.

对于分组数据，将 xᵢ 替换为组中值 mᵢ 并乘以频数：s² = Σfᵢ(mᵢ – x̄)² / (Σfᵢ – 1)。四分位距（IQR = Q₃ – Q₁）是另一种稳健的度量，代表中间 50% 的数据。

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3. Frequency Distributions | 频数分布

A frequency distribution organises data into classes and records the number of observations in each class. Cumulative frequency is the running total of frequencies, often used to estimate the median and quartiles via graphs.

频数分布将数据分组，并记录每个组中的观测值个数。累积频数是频数的累加和，常用于通过图表估计中位数和四分位数。

widths, frequency density = frequency / class width must be plotted on the vertical axis.

直方图以图形方式显示频数分布，每个条形的面积与频数成比例。当组距不相等时，纵轴必须绘制频数密度 = 频数 / 组距。

Frequency polygons and cumulative frequency curves (ogives) provide alternative visual summaries. In an ogive, the median is found at the 50th percentile, Q₁ at the 25th, and Q₃ at the 75th.

频数折线图和累积频数曲线（拱形图）提供了其他可视化总结方式。在拱形图中，中位数位于第 50 百分位数，Q₁ 位于第 25，Q₃ 位于第 75。

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4. Probability Basics | 概率基础

equally likely outcomes, P(A) = number of favourable outcomes / total number of outcomes.

概率度量事件发生的可能性，范围从 0（不可能）到 1（必然）。对于具有等可能结果的有限样本空间，P(A) = 有利结果数 / 总结果数。

The complement rule states: P(not A) = 1 – nts are not mutually exclusive, the general addition rule applies: P(A or B) = P(A) + P(B) – P(A and B).

互补规则为：P(非 A) = 1 – P(A)。互斥事件 A 和 B 的加法规则是 P(A 或 B) = P(A) + P(B)。若事件不互斥，适用一般加法规则：P(A 或 B) = P(A) + P(B) – P(A 且 B)。

= P(A) × P(B). This multiplication rule also applies to tree diagrams for combined events.

独立事件满足 P(A 且 B) = P(A) × P(B)。这一乘法规则同样适用于组合事件的树状图。

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5. Conditional Probability | 条件概率

Conditional probability P(A|B) is the probability of event A occurring given that B has already occurred. It is defined as:

条件概率 P(A|B) 是在事件 B 已经发生的条件下，事件 A 发生的概率。其定义为：

P(A|B) = P(A and B) / P(B) , provided P(B) > 0.

P(A|B) = P(A 且 B) / P(B)，前提是 P(B) > 0。

es are extremely helpful in solving multi-stage problems. Remember that for independent events, P(A|B) = P(A).

条件概率使我们能够利用新信息更新预测。第二分支带有条件概率的树状图对于解决多阶段问题极有帮助。记住，对于独立事件，P(A|B) = P(A)。

Two-way tables and Venn diagrams also facilitate the calculation of joint and conditional probabilities. Always check that all probabilities sum to 1 in a given situation.

双向列联表和维恩图也有助于计算联合概率和条件概率。务必检查在给定情境中所有概率之和为 1。

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6. Permutations and Combinations | 排列与组合

Permutations count the number of ways to arrange items where order matters. The number of permutations of n distinct objects taken r at a time is given by:

排列计算的是顺序重要时排列物品的方法数。从 n 个不同物品中取出 r 个进行排列的数目表示为：

nPr = n! / (n – r)!

nPr = n! / (n – r)!

Combinations count selections where order does not matter. The number of ways to choose r objects from n is:

组合计算的是顺序不重要时的选择数。从 n 个物品中选出 r 个的方式数为：

nCr = n! / [r!(n – r)!] , sometimes written as (ⁿᵣ).

nCr = n! / [r!(n – r)!]，有时写作 (ⁿᵣ)。

Remember that 0! and combinations are fundamental in calculating probabilities in games of chance and sampling without replacement.

请记住 0! = 1，并且 nC0 = nCn = 1。排列和组合是计算机会游戏和不放回抽样概率的基础。

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7. Discrete Random Variables | 离散随机变量

ossible values of X and their probabilities P(X = x). The sum of all probabilities must equal 1.

离散随机变量 X 取可数个值，每个值都有相应的概率。概率分布表列出 X 的所有可能取值及其概率 P(X = x)。所有概率之和必须等于 1。

d value (or mean) of X is E(X) = μ = Σ [x · P(X = x)]. The variance of X is Var(X) = Σ [(x – μ)² · P(X = x)], which simplifies to E(X²) – [E(X)]².

X 的期望值（或均值）为 E(X) = μ = Σ [x · P(X = x)]。X 的方差为 Var(X) = Σ [(x – μ)² · P(X = x)]，可简化为 E(X²) – [E(X)]²。

E(X) = Σ x·P(X=x) ; Var(X) = Σ x²·P(X=x) – [E(X)]²

E(X) = Σ x·P(X=x)；Var(X) = Σ x²·P(X=x) – [E(X)]²

= a² Var(X). These properties are especially useful in scaling and shifting distributions.

对于线性函数 Y = aX + b，以下规则成立：E(aX + b) = aE(X) + b，且 Var(aX + b) = a² Var(X)。这些性质在分布缩放和平移时尤其有用。

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8. Binomial Distribution | 二项分布

A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success, p. If X ~ B(n, p), then the probability of exactly r successes is:

二项分布模拟在固定次数的独立伯努利试验中成功的次数，每次试验的成功概率均为 p。若 X ~ B(n, p)，则恰好获得 r 次成功的概率为：

P(X = r) = nCr × pʳ × (1 – p)ⁿ⁻ʳ , for r = 0,1,2,…,n.

P(X = r) = nCr × pʳ × (1 – p)ⁿ⁻ʳ，其中 r = 0,1,2,…,n。

The mean and variance of a binomial random variable are:

二项随机变量的均值和方差为：

E(X) = np ; Var(X) = np(1 – p)

E(X) = np；Var(X) = np(1 – p)

The shape of a binomial distribution depends on n and p. When np and n(1 – p) are both large enough (typically > 5), a normal approximation can be used. For cumulative probabilities, remember to use tables or the sum of individual probabilities.

二项分布的形状取决于 n 和 p。当 np 和 n(1 – p) 都足够大（通常 > 5）时，可使用正态近似。对于累积概率，记得使用表格或单个概率之和。

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9. Normal Distribution | 正态分布

The normal distribution is a continuous probability distribution symmetrical about its mean μ. It is fully defined by μ and standard deviation σ. The notation is X ~ N(μ, σ²). The total area under the curve equals 1.

正态分布是一种连续型概率分布，关于其均值 μ 对称。它完全由 μ 和标准差 σ 确定。记作 X ~ N(μ, σ²)。曲线下的总面积为 1。

The standard normal distribution has mean 0 and standard deviation 1: Z ~ N(0, 1). Any normal variable can be standardised using the z-score:

标准正态分布均值为 0，标准差为 1：Z ~ N(0, 1)。任何正态变量都可以通过 z 分数进行标准化：

z = (x – μ) / σ

z = (x – μ) / σ

The empirical rule (68–95–99.7) states that about 68% of data lie within 1σ of μ, 95% within 2σ, and 99.7% within 3σ. Use standard normal tables or technology to find probabilities such as P(Z < z) or P(Z > z). Problems often require finding unknown μ or σ given probabilities.

经验法则（68–95–99.7）指出，大约 68% 的数据落在 μ ± 1σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。使用标准正态表或技术工具查找概率，如 P(Z < z) 或 P(Z > z)。题目常要求根据概率求未知的 μ 或 σ。

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10. Sampling and Estimation | 抽样与估计

Sampling is used to draw conclusions about a population without a full census. The sample mean x̄ is an unbiased estimator of the population mean μ. The standard error of the sample mean measures how much x̄ varies from sample to sample.

抽样用于在没有进行全数调查的情况下对总体做出推断。样本均值 x̄ 是总体均值 μ 的无偏估计量。样本均值的标准误差衡量 x̄ 在不同样本间的变异程度。

Standard error of mean: SE(x̄) = σ / √n (if σ known) or s / √n (if σ estimated)

均值的标准误差：SE(x̄) = σ / √n（若 σ 已知）或 s / √n（若 σ 估计）

tates that for a large enough sample size (usually n ≥ 30), the sampling distribution of x̄ ximately normal, regardless of the population shape, with mean μ and standard deviation σ/√n. This underpins confidence intervals and hypothesis tests.

中心极限定理 (CLT) 指出，当样本容量足够大（通常 n ≥ 30）时，无论总体分布形状如何，x̄ 的抽样分布近似正态，均值为 μ，标准差为 σ/√n。这是置信区间和假设检验的基础。

A (1 – α) confidence interval for μ when σ is known is given by: x̄ ± zₐ/₂ × (σ/√n). If σ is unknown and n < 30, use the t-distribution with n – 1 degrees of freedom.

当 σ 已知时，μ 的 (1 – α) 置信区间为：x̄ ± zₐ/₂ × (σ/√n)。若 σ 未知且 n < 30，使用自由度为 n – 1 的 t 分布。

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11. Correlation and 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) is given by:

相关性衡量两个变量之间线性关系的强度和方向。积矩相关系数 (PMCC) 由下式给出：

r = Sₓᵧ / √Sₓₓ × Sᵧᵧ , where Sₓₓ = Σx² – (Σx)²/n, Sᵧᵧ = Σy² – (Σy)²/n, Sₓᵧ = Σxy – (Σx)(Σy)/n.

r = Sₓᵧ / √Sₓₓ × Sᵧᵧ，其中 Sₓₓ = Σx² – (Σx)²/n，Sᵧᵧ = Σy² – (Σy)²/n，Sₓᵧ = Σxy – (Σx)(Σy)/n。

r ranges from –1 (perfect negative) to +1 (perfect positive). A value near 0 suggests no linear correlation. The coefficient of determination r² indicates the proportion of variance in y explained by x.

r 的取值范围从 –1（完全负相关）到 +1（完全正相关）。接近 0 的值表明无线性相关。决定系数 r² 表示 y 的方差中被 x 解释的比例。

The least squares regression line of y on x has the form y = a + bx, where:

y 对 x 的最小二乘回归线具有形式 y = a + bx，其中：

b = Sₓᵧ / Sₓₓ ; a = ȳ – b x̄

b = Sₓᵧ / Sₓₓ；a = ȳ – b x̄

This line minimises the sum of squared vertical distances from data points. Use it for interpolation within the data range; extrapolation beyond the range is unreliable.

这条线最小化数据点到直线的垂直距离的平方和。可用于数据范围内的内插；超出范围的外推不可靠。

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12. Index Numbers | 指数

Index numbers measure relative changes in price, quantity, or value over time. A simple price index for a single item compares the current price to a base period price:

指数衡量价格、数量或价值随时间发生的相对变化。单个商品的价格指数将当前价格与基期价格进行比较：

Price relative = (Pₙ / P₀) × 100

价比 = (Pₙ / P₀) × 100

P₀ × Q₀) × 100. Paasche index uses current-period quantities: P = Σ(Pₙ × Qₙ) / Σ(P₀ × Qₙ) × 100.

拉斯佩尔指数和帕舍指数等综合指数将多种商品汇总。拉斯佩尔指数以基期数量为权数：L = Σ(Pₙ × Q₀) / Σ(P₀ × Q₀) × 100。帕舍指数以现期数量为权数：P = Σ(Pₙ × Qₙ) / Σ(P₀ × Qₙ) × 100。

pplications. When interpreting index numbers, remember that percentage change = (new index – old index) / old index × 100.

消费者价格指数 (CPI) 和零售价格指数 (RPI) 是重要的实际应用。在解释指数时，记住百分比变化 = (新指数 – 旧指数) / 旧指数 × 100。

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