📚 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。
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
The median is the middle value when data are ordered. For grouped data, use linear interpolation: median = L + ((n/2 – F) / fₘ) × c, where L is the lower boundary 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 mode is the most frequent value. For grouped data, mode = L + (d₁ / (d₁ + d₂)) × c, where L is the lower boundary of 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。
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)
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。
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
For equally 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”。
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)
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
The formula can be rearranged to find the joint 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)。
7. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable set of values, 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)
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)
The probability 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）。
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。
Standard 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。
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 –1 indicates a strong linear correlation; r = 0 indicates no linear correlation. The coefficient of determination r² shows the proportion of variation in y explained by x.
r 接近 +1 或 –1 表示
Published by TutorHao | Year 11 统计 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导