📚 A-Level Maths MA04 Statistics Unit 2 Key Points | A-Level数学MA04统计单元2核心知识点
Unit 2 of the International A-Level Mathematics (9660) Statistics module deepens your understanding of probability distributions, continuous data modelling and inferential statistics. This article recaps the essential concepts, formulas and conditions you need to master for the exam.
国际A-Level数学(9660)统计单元2加深了你对概率分布、连续数据建模和推断统计的理解。本文梳理了你必须掌握的核心概念、公式和条件，助你高效备考。
1. Discrete Probability Distributions Recap | 离散概率分布复习
A discrete random variable takes a countable number of values. The sum of all probabilities P(X = x) is 1. The expected value E(X) = Σ x·P(X = x) and variance Var(X) = E(X²) − [E(X)]².
离散随机变量取可数个值。所有概率P(X = x)之和为1。期望值E(X) = Σ x·P(X = x)，方差Var(X) = E(X²) − [E(X)]²。
For the binomial distribution B(n, p), the probability of exactly r successes is given by the formula below, where q = 1 − p.
对于二项分布B(n, p)，恰好r次成功的概率由下式给出，其中q = 1 − p。
P(X = r) = nCr × pr × qn−r
Its mean is np and variance is npq. These results are used in later approximations.
其均值为np，方差为npq。这些结果将在后面的近似中使用。
2. Poisson Distribution | 泊松分布
The Poisson distribution models the number of events occurring in a fixed interval, provided events happen independently at a constant average rate λ. The probability mass function is centered below.
泊松分布模型用于描述固定区间内事件发生的次数，前提是事件独立且以恒定平均速率λ发生。其概率质量函数居中展示如下。
P(X = r) = (λr × e−λ) / r!
The mean and variance of a Poisson variable are both λ. The distribution is positively skewed for small λ but becomes more symmetric as λ increases.
泊松变量的均值和方差均为λ。当λ较小时分布呈现正偏态，随着λ增大则趋于对称。
3. Poisson Approximation to Binomial | 泊松近似二项分布
When n is large and p is small, the binomial distribution B(n, p) can be approximated by a Poisson distribution with λ = np. The typical conditions are n ≥ 50 and np ≤ 5, or n > 20 and p < 0.1.
当n很大且p很小时，二项分布B(n, p)可用参数λ = np的泊松分布来近似。典型条件是n ≥ 50且np ≤ 5，或n > 20且p < 0.1。
This approximation simplifies calculations and is useful when binomial probabilities become computationally heavy.
此近似可简化计算，在二项概率计算量过大时尤为实用。
4. Continuous Random Variables | 连续随机变量
A continuous random variable takes any value in an interval. Its probability density function (pdf), f(x), satisfies f(x) ≥ 0 and the total area under the curve equals 1.
连续随机变量取某一区间内的任意值。其概率密度函数f(x)满足f(x) ≥ 0，且曲线下的总面积等于1。
Probabilities are found by integration: P(a < X < b) = ∫ab f(x) dx. There is no probability at a single point.
概率通过积分求得：P(a < X < b) = ∫ab f(x) dx。单点的概率为零。
5. Cumulative Distribution Functions | 累积分布函数
The cumulative distribution function (cdf), F(x), gives the probability that X takes a value less than or equal to x: F(x) = P(X ≤ x) = ∫−∞x f(t) dt.
累积分布函数F(x)给出X取值小于等于x的概率：F(x) = P(X ≤ x) = ∫−∞x f(t) dt。
The median m satisfies F(m) = 0.5. The lower and upper quartiles satisfy F(Q₁) = 0.25 and F(Q₃) = 0.75. The pdf can be recovered by differentiating the cdf: f(x) = F′(x).
中位数m满足F(m) = 0.5。下四分位数和上四分位数满足F(Q₁) = 0.25和F(Q₃) = 0.75。对cdf求导即可还原pdf：f(x) = F′(x)。
6. Normal Distribution Refresher | 正态分布回顾
The normal distribution N(μ, σ²) is symmetric and bell‑shaped. Standardising using Z = (X − μ) / σ gives Z ~ N(0, 1). Probabilities are found with normal tables.
正态分布N(μ, σ²)是对称的钟形曲线。标准化Z = (X − μ) / σ后得到Z ~ N(0, 1)。概率值可通过正态分布表查得。
You must be comfortable calculating probabilities such as P(X > a), P(a < X < b) and working backwards to find unknown μ or σ.
你需要熟练计算P(X > a)、P(a < X < b)等概率，并能逆向求解未知的μ或σ。
7. Normal Approximation to Binomial | 二项分布的正态近似
If a binomial variable X ~ B(n, p) satisfies np ≥ 5 and nq ≥ 5, we can approximate it by a normal distribution with μ = np and σ² = npq. A continuity correction (±0.5) must be applied.
若二项变量X ~ B(n, p)满足np ≥ 5且nq ≥ 5，我们可用均值为np、方差为npq的正态分布近似，同时必须进行连续性校正(±0.5)。
| Binomial probability | Normal approximation with continuity correction |
| P(X = a) | P(a − 0.5 < Y < a + 0.5) |
| P(X ≥ a) | P(Y > a − 0.5) |
| P(X ≤ a) | P(Y < a + 0.5) |
Continuity correction bridges the discrete binomial to the continuous normal distribution.
连续性校正弥补了离散二项分布与连续正态分布之间的差异。
8. Normal Approximation to Poisson | 泊松分布的正态近似
Similarly, a Poisson variable X ~ Po(λ) with large λ (typically λ > 10) can be approximated by N(λ, λ). The continuity correction is used in the same way.
类似地，当泊松变量X ~ Po(λ)的λ较大（通常λ > 10）时，可用N(λ, λ)近似，并以同样方式使用连续性校正。
Z = (X − λ) / √λ
After applying the half‑unit correction, the standardised value is used with normal tables.
执行0.5单位校正后，将标准化值配合正态分布表使用。
9. Sampling and the Central Limit Theorem | 抽样与中心极限定理
A sample of size n yields a sample mean x̄. If the population has mean μ and variance σ², the sampling distribution of x̄ has mean μ and standard error σ/√n.
容量为n的样本得到样本均值x̄。若总体的均值为μ、方差为σ²，则x̄的抽样分布具有均值μ和标准误σ/√n。
The Central Limit Theorem states that for large n (typically n ≥ 30), the distribution of x̄ is approximately normal regardless of the population’s shape.
中心极限定理指出，当n较大（通常n ≥ 30）时，无论总体是何形状，x̄的分布均近似正态。
10. Confidence Intervals for the Mean | 均值的置信区间
A 95% confidence interval for the population mean μ, when the population variance σ² is known, is constructed as:
当总体方差σ²已知时，总体均值μ的95%置信区间构造如下：
x̄ ± 1.96 × (σ / √n)
The value 1.96 is the z‑score cutting off 2.5% in each tail of N(0,1). For a 99% CI, replace 1.96 by 2.576.
1.96是N(0,1)双侧尾部各截尾2.5%的z分数。99%置信区间则将1.96替换为2.576。
This interval means we are 95% confident that μ lies within the calculated range.
这个区间意味着我们有95%的信心认为μ落在计算出的范围内。
11. Hypothesis Testing Concepts | 假设检验概念
A hypothesis test compares observed data against a null hypothesis H₀. The alternative hypothesis H₁ can be one‑tailed or two‑tailed. The significance level α is the maximum probability of wrongly rejecting H₀ (Type I error).
假设检验将观测数据与零假设H₀进行比较。备择假设H₁可以是单侧或双侧的。显著性水平α是错误拒绝H₀的最大概率（第一类错误）。
A Type II error occurs when H₀ is false but is not rejected. The power of a test is 1 − P(Type II error). Increasing the sample size improves power.
当H₀为假却未被拒绝时发生第二类错误。检验功效为1 − P(第二类错误)。增大样本量可提升功效。
12. Hypothesis Tests for the Mean | 均值的假设检验
For a test on the population mean with known variance, we compute the test statistic:
对于已知方差的总体均值检验，我们计算检验统计量：
Z = (x̄ − μ₀) / (σ/√n)
Compare this Z value to the critical value from normal tables. Alternatively, find the p‑value and reject H₀ if p‑value < α.
将此Z值与正态分布表中的临界值比较。或者，求出p值，若p值 < α则拒绝H₀。
Always state a conclusion in context: there is (or is not) sufficient evidence at the α% level to support the claim.
始终在上下文中给出结论：在α%的水平下，有（或没有）充分证据支持该声明。
Published by TutorHao | Statistics Revision Series | aleveler.com
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