AQA AS Mathematics (9660) MA02: Pure Statistics and Mechanics Unit 1 | AQA AS 数学（9660）MA02：纯统计与力学单元1

📚 AQA AS Mathematics (9660) MA02: Pure Statistics and Mechanics Unit 1 | AQA AS 数学（9660）MA02：纯统计与力学单元1

This article will guide you through the core content of AQA International AS Mathematics (9660) MA02, which covers statistics and mechanics in Unit 1. We will summarise key definitions, formulas and problem-solving strategies in a bilingual format so you can revise efficiently and accurately.

本文将带你系统梳理 AQA 国际 AS 数学（9660）MA02 的核心内容，涵盖 Unit 1 中的统计与力学主题。我们用中英双语概括关键定义、公式与解题策略，帮助你高效而准确地进行复习。

1. Statistical Sampling | 统计抽样

In statistics, the whole group you want to investigate is called the population. A census collects data from every member of the population, while a sample collects data from a subgroup. Sampling saves time and money but must aim to represent the population fairly.

在统计中，你想要研究的全体对象称为总体。普查（census）收集总体中每一个成员的数据；抽样（sample）仅从子集收集数据。抽样能节省时间和成本，但必须尽量使样本代表总体。

Before choosing a sample, you need a sampling frame: the list from which the sample is drawn. A sample chosen from an incorrect or incomplete frame may be biased.

在选择样本之前，需要先有抽样框（sampling frame），即从中抽取样本的名单。如果抽样框不正确或不完整，得到的样本可能会有偏。

• Simple random sampling: every member of the sampling frame has an equal chance of being chosen, for example using random numbers.

  简单随机抽样：抽样框中每个成员被选中的概率相同，例如使用随机数。

• Stratified sampling: the population is split into groups, called strata, and the number of items selected from each stratum is proportional to its size.

  分层抽样：将总体分成若干组（层），从每层中按该层规模比例抽取样本。

• Systematic sampling: choose every k-th member from an ordered list after choosing a random starting point.

  系统抽样：在有序名单上随机选择起点后，每隔 k 个成员抽取一个。

• Quota sampling: an interviewer selects a fixed quota from each category until the quota is filled.

  配额抽样：访问员从各个类别中按固定配额选取，直到名额填满。

• Opportunity sampling: choose people who are easiest to reach; this is often quick but may be unrepresentative.

  便利抽样：选择最容易接触的人；这种方法很快，但可能缺乏代表性。

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2. Data Presentation | 数据展示

Data can be discrete or continuous. Discrete data take exact values, often whole numbers, while continuous data can take any value in a range.

数据可分为离散型和连续型。离散数据取精确值（通常为整数）；连续数据可以在一个区间内取任意值。

For grouped continuous data, a histogram uses the area of each bar to represent frequency. This means you must calculate the frequency density.

对于分组的连续数据，直方图用每个矩形的面积表示频数，因此需要先计算频率密度。

Frequency density = frequency ÷ class width | 频率密度 = 频数 ÷ 组距

Bar charts are used for discrete or categorical data, and the height of each bar represents the frequency. Pie charts show proportions, while box plots and cumulative frequency graphs are useful for showing the spread of continuous data.

条形图用于离散数据或分类数据，条的高度表示频数。饼图显示比例，箱线图和累积频数图则适合展示连续数据的分布与离散程度。

A cumulative frequency graph is plotted by plotting cumulative frequency against the upper class boundary of each interval. It allows you to estimate the median, quartiles and percentiles. A box plot displays the minimum, lower quartile, median, upper quartile and maximum.

累积频数图以每个区间上界为横坐标、累积频数为纵坐标绘制。通过它可估计中位数、四分位数和百分位数。箱线图显示最小值、下四分位数、中位数、上四分位数和最大值。

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3. Measures of Location and Spread | 位置与离散程度的度量

Measures of location describe the centre or typical value of a data set. The main measures are the mean, median and mode.

位置度量描述数据集的中心或典型值，主要包括均值、中位数和众数。

x̄ = Σx ÷ n = (sum of data values) ÷ (number of values)

For grouped data, the mean is estimated using the midpoints of each class: mean ≈ Σfx ÷ Σf, where f is the frequency of each class.

对于分组数据，均值可用每组组中值估计：均值 ≈ Σfx ÷ Σf，其中 f 为每组的频数。

Measures of spread include the range, interquartile range, variance and standard deviation. The range is maximum minus minimum. The interquartile range is Q₃ − Q₁, where Q₁ is the lower quartile and Q₃ is the upper quartile.

离散程度的度量包括极差、四分位距、方差和标准差。极差等于最大值减最小值；四分位距为 Q₃ − Q₁，其中 Q₁ 为下四分位数，Q₃ 为上四分位数。

Variance and standard deviation measure how far the data values are from the mean. A larger standard deviation means the data are more spread out.

方差和标准差度量数据值与均值的偏离程度。标准差越大，数据越分散。

Population variance σ² = Σ(x − μ)² ÷ N; sample variance s² = Σ(x − x̄)² ÷ (n − 1)

The standard deviation is the square root of the variance.

标准差是方差的平方根。

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4. Probability | 概率

Probability measures the chance of an event occurring on a scale from 0 to 1. The sample space is the set of all possible outcomes of an experiment.

概率用 0 到 1 之间的数表示事件发生的可能性。样本空间是一次试验所有可能结果的集合。

For two events A and B, the addition rule gives the probability that A or B occurs:

对任意两个事件 A 和 B，加法法则给出了 A 或 B 发生的概率：

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

If A and B are mutually exclusive, they cannot happen at the same time, so P(A ∩ B) = 0.

如果 A 与 B 互斥，则它们不可能同时发生，因此 P(A ∩ B) = 0。

If A and B are independent, the occurrence of one does not affect the probability of the other. For independent events:

如果 A 与 B 相互独立，则一个事件的发生不会影响另一个事件的概率。对独立事件：

P(A ∩ B) = P(A) × P(B)

Conditional probability is the probability of A given that B has already occurred:

条件概率是在已知 B 发生的条件下 A 发生的概率：

P(A | B) = P(A ∩ B) ÷ P(B)

You can solve probability problems using Venn diagrams, two-way tables and tree diagrams. Always state the sample space clearly when the problem is not straightforward.

你可以在概率问题中使用韦恩图、双因素表和树形图。当问题较复杂时，先明确写出样本空间。

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5. Discrete Random Variables | 离散随机变量

A discrete random variable X takes a countable set of values x, with probabilities P(X = x). The set of these probabilities is called the probability distribution of X.

离散随机变量 X 取可数多个值 x，对应概率为 P(X = x)。所有概率的集合称为 X 的概率分布。

For a valid probability distribution, each probability must be between 0 and 1, and the sum of all probabilities must equal 1:

一个有效的概率分布要求每个概率介于 0 和 1 之间，且所有概率之和等于 1：

Σ P(X = x) = 1

The expected value E(X) is the mean of the distribution. It is calculated by multiplying each value by its probability and adding the results.

期望值 E(X) 是分布的均值，计算方法是把每个取值乘以对应概率再相加。

E(X) = Σ x P(X = x)

The variance measures the spread of X around the mean. A useful formula is:

方差度量 X 在均值周围的离散程度，一个常用公式是：

Var(X) = E(X²) − μ², where μ = E(X)

For a linear transformation Y = aX + b, the expected value and variance transform in simple ways:

对于线性变换 Y = aX + b，期望和方差的变换规律为：

E(aX + b) = aE(X) + b; Var(aX + b) = a²Var(X)

Remember that variance is not expressed in the same units as the data; use standard deviation when you need a measure in the original units.

注意方差的单位不是原始数据的单位；如果要用原始单位，应使用标准差。

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6. Binomial Distribution | 二项分布

The binomial distribution models the number of successes in a fixed number of independent trials. For a binomial experiment to apply, there must be a fixed number n of trials, each trial has exactly two outcomes, the probability p of success is constant, and the trials are independent.

二项分布用于建模固定次数独立试验中的成功次数。使用二项分布必须满足：试验次数 n 固定；每次试验只有两种结果；成功概率 p 恒定；各次试验相互独立。

X ~ B(n, p)

If X counts the number of successes in n trials, then the probability of exactly r successes is:

如果 X 表示 n

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