📚 Year 12 Cambridge Statistics: Core Knowledge Review | 剑桥Year12统计核心知识点梳理
Year 12 Cambridge Statistics builds the foundation for data handling, probability modelling, and statistical inference. This article distils the essential concepts of the Probability & Statistics 1 syllabus, offering clear explanations for effective revision.
剑桥 Year 12 统计课程为数据处理、概率建模和统计推断打下基础。本文提炼了概率与统计1大纲中的核心知识点，提供清晰的解释，助力高效复习。
1. Data Representation and Summary | 数据的表示与汇总
Data can be displayed using stem-and-leaf diagrams, box-and-whisker plots, histograms, and cumulative frequency graphs. Each representation highlights different features: skewness, outliers, median, and spread.
数据可通过茎叶图、箱线图、直方图和累积频率图来展示。每种表示方法突出不同的特征：偏态、异常值、中位数和分散程度。
A histogram uses area to represent frequency in each class; for unequal class widths, frequency density must be calculated.
直方图用面积表示各组的频数；当组距不相等时，必须计算频率密度。
Cumulative frequency curves allow estimation of medians, quartiles, and percentiles, and can be used to compare distributions.
累积频率曲线可用于估算中位数、四分位数和百分位数，并比较数据的分布情况。
2. Measures of Central Tendency | 集中趋势的度量
The mean, median, and mode describe the centre of a data set. The mean is sensitive to extreme values, while the median is robust and preferred for skewed data.
平均数、中位数和众数描述数据集的中心。平均数易受极端值影响，而中位数较为稳健，适用于偏态数据。
For grouped data, we estimate the mean using midpoints: x̄ = Σ(f × midpoint) / Σf. The modal class is the class with the highest frequency density.
对于分组数据，我们使用组中值估算平均数：x̄ = Σ(f × 组中值) / Σf。众数所在组是频率密度最高的组。
3. Measures of Dispersion | 离散程度的度量
d of data. The IQR is the difference between the upper and lower quartiles: IQR = Q₃ − Q₁.
极差、四分位距（IQR）、方差和标准差用于量化数据的分散程度。四分位距是上四分位数与下四分位数之差：IQR = Q₃ − Q₁。
Sample variance: s² = Σ(x − x̄)² / (n − 1)
样本方差：s² = Σ(x − x̄)² / (n − 1)
Standard deviation is the square root of variance. A larger standard deviation indicates greater variability around the mean.
标准差是方差的平方根。标准差越大，表明数据围绕平均数的波动越大。
4. Basic Probability | 概率基础
Probability measures the likelihood of an event, with values between 0 and 1. The sum of probabilities of all mutually exclusive outcomes in a sample space is 1.
概率衡量事件发生的可能性，取值在0到1之间。样本空间中所有互斥结果的概率之和为1。
For complementary events, P(A') = 1 − P(A). The addition rule for mutually exclusive events is P(A ∪ B) = P(A) + P(B).
对于互补事件，P(A') = 1 − P(A)。互斥事件的加法规则为 P(A ∪ B) = P(A) + P(B)。
For non‑mutually exclusive events, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
对于非互斥事件，P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
5. Conditional Probability and Independence | 条件概率与独立事件
Conditional probability P(A|B) is the probability of A occurring given that B has occurred. It is calculated using P(A|B) = P(A ∩ B) / P(B).
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，计算公式为 P(A|B) = P(A ∩ B) / P(B)。
are independent if the occurrence of one does not affect the probability of the other: P(A ∩ B) = P(A) × P(B), or P(A|B) = P(A).
两个事件独立，如果一个事件的发生不影响另一个事件的概率，即满足 P(A ∩ B) = P(A) × P(B)，或 P(A|B) = P(A)。
Tree diagrams are powerful tools for modelling conditional probabilities and combined outcomes.
树状图是处理条件概率和联合结果的有力工具。
6. Permutations and Combinations | 排列与组合
Permutations count the number of ways to arrange items when order matters. The number of permutations of n distinct items is n! (n factorial).
排列计算了在顺序重要时物品的排列方式数量。n个不同物品的排列总数为 n!（n的阶乘）。
When selecting r items from n without replacement and order matters: ⁿPᵣ = n! / (n − r)!.
当从n个物品中不重复地选取r个且顺序重要时：ⁿPᵣ = n! / (n − r)!。
Combinations are used when order does not matter: ⁿCᵣ = n! / [r!(n − r)!]. This is often read as 'n choose r'.
组合用于顺序不重要的情况：ⁿCᵣ = n! / [r!(n − r)!]，常读作“从n中选r”。
Combinations appear in binomial expansions and binomial probability.
组合出现在二项式展开和二项概率中。
7. Discrete Random Variables | 离散随机变量
es with associated probabilities. The probability distribution must satisfy ΣP(X = x) = 1.
离散随机变量X取可数个值，每个值对应一个概率。概率分布必须满足 ΣP(X = x) = 1。
The expected value E(X) is the long‑run average: E(X) = Σ x·P(X = x). It represents the mean of the distribution.
期望值 E(X) 是长期的平均值：E(X) = Σ x·P(X = x)，代表了分布的均值。
Variance measures spread: Var(X) = E(X²) − [E(X)]², where E(X²) = Σ x²·P(X = x).
方差衡量分散程度：Var(X) = E(X²) − [E(X)]²，其中 E(X²) = Σ x²·P(X = x)。
8. Binomial Distribution | 二项分布
A binomial distribution models the number of successes in n independent trials, each with constant probability of success p. We write X ~ B(n, p).
二项分布用于描述在n次独立试验中成功的次数，每次试验的成功概率p恒定。记作 X ~ B(n, p)。
P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ
P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ
n by E(X) = np and Var(X) = np(1 − p). These formulas simplify calculations significantly.
其均值和方差为 E(X) = np，Var(X) = np(1 − p)。这些公式可以大大简化计算。
Use binomial cumulative tables or a calculator to find P(X ≤ k) and related probabilities.
使用二项累积分布表或计算器求 P(X ≤ k) 及相关概率。
9. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution with a bell‑shaped curve, defined by its mean μ and standard deviation σ. We write X ~ N(μ, σ²).
正态分布是一种连续型概率分布，具有钟形曲线，由其均值μ和标准差σ决定。记作 X ~ N(μ, σ²)。
The total area under the curve is 1, and the distribution is symmetric about the mean. Probabilities correspond to areas under the curve.
曲线下的总面积为1，分布关于均值对称。概率与曲线下的面积相对应。
We use standardisation to convert any normal variable to the standard normal Z ~ N(0, 1) using:
我们通过标准化将任意正态变量转换为标准正态变量 Z ~ N(0, 1)：
Z = (X − μ) / σ
Z = (X − μ) / σ
Then standard normal tables give Φ(z), the cumulative probability P(Z ≤ z).
然后利用标准正态分布表查找 Φ(z)，即累积概率 P(Z ≤ z)。
10. Working with the Normal Distribution | 正态分布的应用
Finding probabilities often requires sketching the bell curve, shading the relevant area, and translating the problem into a standard normal calculation.
求概率时通常需要画出钟形曲线，标记相关区域，并将问题转化为标准正态计算。
To find an unknown mean or standard deviation, set up an equation using the Z‑score and the given probability, then solve for the parameter.
若要求未知的均值或标准差，可利用Z分数和已知概率建立方程，然后求解参数。
Inverse normal calculations determine the value of X corresponding to a given cumulative probability, often used for finding quartiles and percentiles.
逆正态计算可确定对应给定累积概率的X值，常用于求四分位数和百分位数。
Remember that the normal distribution can approximate the binomial distribution when n is large and p is close to 0.5, using continuity correction.
当n较大且p接近0.5时，正态分布可近似二项分布，此时需要使用连续性校正。
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