Year 13 OCR Statistics: Core Knowledge Review | Year 13 OCR 统计核心知识点梳理

📚 Year 13 OCR Statistics: Core Knowledge Review | Year 13 OCR 统计核心知识点梳理

This revision guide summarises the essential statistical concepts covered in the Year 13 OCR A-Level Mathematics specification. Mastering these topics is crucial for success in the final examinations, as they form the basis of inferential statistics and probability modelling. We will review distributions, hypothesis testing, correlation, regression, and chi-squared tests.

本复习指南总结了 Year 13 OCR A-Level 数学大纲中的核心统计概念。掌握这些主题对期末考试取得成功至关重要，因为它们构成了推断统计和概率建模的基础。我们将回顾分布、假设检验、相关、回归和卡方检验。

1. Probability Distributions Overview | 概率分布总览

In Year 13, you need to be familiar with both discrete distributions (Binomial, Poisson) and continuous distributions (Normal). A discrete random variable takes countable values, while a continuous random variable can take any value within an interval. For discrete variables we use probability mass functions, and for continuous we use probability density functions (PDF).

在 Year 13，你需要熟悉离散分布（二项、泊松）和连续分布（正态）。离散随机变量取可数值，而连续随机变量可取区间内任何值。对离散变量我们使用概率质量函数，对连续变量使用概率密度函数（PDF）。

For a continuous random variable X, the total area under the PDF curve is 1, and probabilities are found by integration: P(a ≤ X ≤ b) = ∫_a^b f(x) dx.

对于连续随机变量 X，PDF 曲线下的总面积为 1，概率通过积分求得：P(a ≤ X ≤ b) = ∫_a^b f(x) dx。

2. Poisson Distribution | 泊松分布

P(X = r) = (e^−λ λ^r) / r!   for r = 0, 1, 2, …

The Poisson distribution with parameter λ > 0 applies when events occur independently at a constant average rate. The probability of exactly r events is given by the formula above, where e is Euler’s number.

泊松分布参数 λ > 0 适用于事件独立且以恒定平均率发生的情形。恰好发生 r 次的概率由上述公式给出，其中 e 为欧拉数。

Both the mean and variance of a Poisson distribution are λ. The sum of two independent Poisson variables X ~ Po(λ₁) and Y ~ Po(λ₂) is also Poisson: X + Y ~ Po(λ₁ + λ₂). In exams, you may need to use cumulative Poisson probability tables, or compute probabilities directly when λ is large.

泊松分布的均值和方差都是 λ。两个独立的泊松变量 X ~ Po(λ₁) 和 Y ~ Po(λ₂) 之和仍服从泊松分布：X + Y ~ Po(λ₁ + λ₂)。在考试中，你可能需要使用累积泊松概率表，或当 λ 较大时直接计算概率。

3. Continuous Random Variables & PDF | 连续型随机变量与概率密度函数

A continuous random variable X is defined by a probability density function f(x) ≥ 0, with total area ∫ f(x) dx = 1 over the domain. The cumulative distribution function is F(x) = P(X ≤ x) = ∫ f(t) dt from the lower bound to x.

连续随机变量 X 由概率密度函数 f(x) ≥ 0 定义，在其定义域上总面积为 ∫ f(x) dx = 1。累积分布函数为 F(x) = P(X ≤ x) = ∫ 从下限到 x 的 f(t) dt。

The median m satisfies F(m) = 0.5, and the mode is the value where f(x) is maximised. Percentiles can be found by solving F(p) = k/100. Remember that for a valid PDF, f(x) must be non‑negative and integrate to 1 over its support.

中位数 m 满足 F(m) = 0.5，众数是使 f(x) 最大的值。百分位数可通过解 F(p) = k/100 求得。请记住，有效的 PDF 须非负且在其支撑集上积分为 1。

4. The Normal Distribution | 正态分布

X ~ N(μ, σ²)   PDF: f(x) = (1/(σ√(2π))) exp(−(x − μ)² / (2σ²))

The normal distribution is the most important continuous distribution. Its curve is symmetric about the mean μ, with points of inflection at μ ± σ. The total area under the curve is 1, and the probability in any interval can be found by standardising.

正态分布是最重要的连续分布。其曲线关于均值 μ 对称，拐点位于 μ ± σ。曲线下总面积为 1，任意区间内的概率可通过标准化求得。

To find probabilities, we standardise to the standard normal Z ~ N(0,1) using Z = (X − μ) / σ. Then use the standard normal table. The table typically gives Φ(z) = P(Z ≤ z) for z ≥ 0. Inverse normal calculations find the value x such that P(X ≤ x) = p, using the percentage points of the normal distribution.

为求概率，我们通过 Z = (X − μ) / σ 标准化为标准正态 Z ~ N(0,1)，然后使用标准正态表。该表通常给出 Φ(z) = P(Z ≤ z)（z ≥ 0）。正态逆运算利用正态分布的百分点，求使得 P(X ≤ x) = p 的 x 值。

5. Normal Approximations | 正态近似

When n is

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