Formula & Theorem Quick Reference Handbook for Year 11 Cambridge Statistics | Year 11 剑桥统计公式定理速查手册

📚 Formula & Theorem Quick Reference Handbook for Year 11 Cambridge Statistics | Year 11 剑桥统计公式定理速查手册

This quick reference handbook gathers the core formulas, theorems and statistical notation required for the Year 11 Cambridge Statistics course. It is designed as a fast revision tool to support your understanding of data handling, probability, distributions and bivariate analysis. Keep it handy when working through past papers and end‑of‑topic tests.

本速查手册汇集了剑桥 Year 11 统计课程的核心公式、定理与统计符号，可作为快速复习工具，帮助你巩固数据处理、概率、分布以及双变量分析等关键内容。在做历年真题和单元测验时，请随手查阅。

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1. Measures of Central Tendency | 集中趋势度量

The mean of a set of ungrouped data x₁, x₂, …, xₙ is given by the total sum divided by the count.

未分组数据的均值等于总和除以数据个数。

x̄ = ∑x / n

For grouped data, the mean is estimated using class midpoints x and frequencies f.

对于分组数据，均值用组中值 x 与频数 f 估算。

x̄ = ∑fx / ∑f

undary of the median class, n total frequency, F cumulative frequency before the median class, fₘ frequency of the median class, and c the class width.

中位数是排序后中间位置的值。分组数据用线性插值计算：中位数 = L + ((n/2 – F) / fₘ) × c，其中 L 为中位数所在组下限，n 为总频数，F 为中位数组之前累计频数，fₘ 为中位数组频数，c 为组距。

the modal class, d₁ the excess frequency over the preceding class, and d₂ the excess over the following class.

众数是出现次数最多的值。分组数据的众数可用：众数 = L + (d₁ / (d₁ + d₂)) × c，其中 L 为众数组下限，d₁ 是众数组频数与前一组频数之差，d₂ 是众数组频数与后一组频数之差。

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2. Measures of Dispersion | 离散程度度量

Range = maximum value – minimum value.

极差 = 最大值 – 最小值。

Quartiles divide the ordered data into four equal parts. The interquartile range (IQR) is Q₃ – Q₁ and measures the spread of the middle 50% of the data.

四分位数将排序后的数据四等分。四分位距 IQR = Q₃ – Q₁，衡量中间50%数据的分散程度。

For a sample, the variance and standard deviation measure average squared deviation from the mean.

样本方差与标准差衡量各数据与均值之差的平方平均值。

s² = ∑(x - x̄)² / (n - 1)

s = √[∑(x - x̄)² / (n - 1)]

For grouped data, replace x with the class midpoint and multiply each deviation squared by the frequency f.

分组数据计算时，用组中值代替 x，并将每个离差平方乘以频数 f。

s² = ∑f(x - x̄)² / (∑f - 1)

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3. Frequency Distributions and Histograms | 频数分布与直方图

In a histogram with unequal class widths, the area of a bar represents frequency. Frequency density is used to draw the bars.

在组距不等的直方图中，条形面积代表频数，绘制时需要用到频数密度。

Frequency density = Frequency / Class width

频数密度 = 频数 / 组距

A cumulative frequency graph plots cumulative frequency against the upper class boundary. It can be used to estimate medians, quartiles and percentiles.

累计频数曲线图以累计频数对组上限作图，可用于估计中位数、四分位数和百分位数。

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4. Basic Probability | 概率基础

Probability is a measure of the chance of an event occurring, always between 0 and 1 inclusive.

概率衡量事件发生的可能性，取值总在 0 到 1 之间（含）。

0 ≤ P(A) ≤ 1

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

等可能结果下，P(A) = 有利结果数 / 所有可能结果总数。

The complement rule: P(A’) = 1 – P(A), where A’ means ‘not A’.

互补规则：P(A’) = 1 – P(A)，其中 A’ 表示“非 A”。

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5. Probability Rules | 概率法则

The general addition rule accounts for the overlap between events A and B.

一般加法法则需扣除事件 A 与 B 的交集部分。

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

If A and B are mutually exclusive (they cannot both happen), P(A ∩ B) = 0, so the addition rule simplifies.

若 A 与 B 互斥（不能同时发生），P(A ∩ B) = 0，加法法则简化为：

P(A ∪ B) = P(A) + P(B)

The multiplication rule connects joint probability with conditional probability. For independent events, the joint probability is the product of the individual probabilities.

乘法法则联系联合概率与条件概率。对于独立事件，联合概率等于各自概率的乘积。

P(A ∩ B) = P(A) × P(B | A)

If A and B are independent: P(A ∩ B) = P(A) × P(B)

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6. Conditional Probability | 条件概率

Conditional probability gives the probability of event A happening given that event B has occurred.

条件概率表示在事件 B 已发生的条件下事件 A 发生的概率。

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

probability or to test for independence: events A and B are independent if and only if P(A | B) = P(A) or equivalently P(B | A) = P(B).

该公式可变形用于求联合概率或检验独立性：A 与 B 独立当且仅当 P(A | B) = P(A) 或等价地 P(B | A) = P(B)。

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7. Discrete Random Variables | 离散随机变量

each with an assigned probability p(x) = P(X = x). The sum of all probabilities equals 1.

离散随机变量 X 取可数个值，每个值有对应的概率 p(x) = P(X = x)，且所有概率之和为 1。

∑p(x) = 1

The expected value (mean) of X is a weighted average of the possible values.

X 的期望值（均值）是可能取值按概率加权的平均值。

E(X) = μ = ∑ x·p(x)

Variance measures the spread of the distribution around the mean.

方差衡量分布围绕均值的离散程度。

Var(X) = σ² = ∑(x - μ)² p(x) = E(X²) – [E(X)]²

For a linear transformation Y = aX + b, the expectation and variance follow simple rules.

对于线性变换 Y = aX + b，期望与方差遵循简单规律。

E(aX + b) = a E(X) + b

Var(aX + b) = a² Var(X)

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8. Binomial Distribution | 二项分布

The binomial distribution models the number of successes r in a fixed number n of independent trials, each with the same probability of success p.

二项分布描述在 n 次独立重复试验中成功次数 r 的规律，每次试验的成功概率 p 相同。

X ~ B(n, p)

obability of exactly r successes is given by the binomial probability formula, where q = 1 – p.

恰好 r 次成功的概率由二项概率公式给出，其中 q = 1 – p。

P(X = r) = C(n, r) · pʳ qⁿ⁻ʳ

The mean and variance of a binomial distribution are simple functions of the parameters.

二项分布的均值和方差是参数的简单函数。

μ = np

σ² = npq

The conditions for a binomial model are: fixed number of trials, independent trials, constant probability of success p, and only two outcomes (success/failure) per trial.

二项模型的条件为：试验次数固定、各次试验相互独立、每次试验成功概率 p 不变、每次试验只有两个结果（成功/失败）。

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9. Normal Distribution | 正态分布

The normal distribution is a continuous, symmetric, bell‑shaped curve fully defined by its mean μ and standard deviation σ.

正态分布是一种连续、对称的钟形曲线，完全由均值 μ 和标准差 σ 决定。

X ~ N(μ, σ²)

To find probabilities, the variable X is converted to a standard normal score Z, which has mean 0 and standard deviation 1.

为求概率，将变量 X 转化为标准正态计分 Z，其均值为 0，标准差为 1。

z = (x - μ) / σ

The empirical rule (68–95–99.7%) provides quick estimates: approximately 68% of data lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.

经验法则（68–95–99.7%）提供快速估计：约 68% 的数据落在均值 ±1σ 内，95% 落在 ±2σ 内，99.7% 落在 ±3σ 内。

ndard normal tables (or calculator functions) are used to find Φ(z) = P(Z < z) for given z‑values.

标准正态分布表（或计算器功能）用于求出 Φ(z) = P(Z < z) 对应给定 z 值的概率。

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10. Correlation and Regression | 相关与回归

Pearson’s product‑moment correlation coefficient r measures the strength and direction of a linear relationship between two variables x and y. It always lies between –1 and 1.

皮尔逊积矩相关系数 r 衡量两个变量 x 与 y 之间线性关系的强度与方向，取值总在 –1 到 1 之间。

r = Σ[(x - x̄)(y - ȳ)] / √[Σ(x - x̄)² Σ(y - ȳ)²]

A convenient calculation form uses totals of xy, x, y, x², y².

另一种计算形式使用各总和量：

r = (nΣxy – Σx Σy) / √[(nΣx² – (Σx)²) (nΣy² – (Σy)²)]

The equation of the least‑squares regression line of y on x is used to predict y from x.

y 对 x 的最小二乘回归直线方程用于由 x 预测 y。

y = a + bx

The slope b and intercept a are calculated from the data means and sums of squares.

斜率 b 和截距 a 由数据均值和平方和算得。

b = Σ[(x - x̄)(y - ȳ)] / Σ(x - x̄)²

a = ȳ – b x̄

A value of r close to +1 or – tion. The coefficient of determination r² shows the proportion of variation in y explained by x.

r 接近 +1 或 –1 表示

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