Year 1 Maths Stats and Mechanics: Complete Revision Guide | Year 1 数学统计与力学完整复习指南

📚 Year 1 Maths Stats and Mechanics: Complete Revision Guide | Year 1 数学统计与力学完整复习指南

This guide covers the essential content for Year 1 Statistics and Mechanics in A Level Mathematics. It brings together statistical sampling, data representation, probability, the binomial distribution, hypothesis testing, and the core mechanics topics of modelling, constant acceleration, forces, and variable acceleration. Work through each section, practise the standard methods, and make sure you can explain every result in context.

本指南涵盖 A Level 数学 Year 1 统计与力学的核心内容。它整合了统计抽样、数据表示、概率、二项分布、假设检验，以及力学中的建模、匀加速运动、力和变加速度等核心主题。请逐节学习，练习标准方法，并确保能够在实际情境中解释每个结果。

1. Statistical Sampling and Types of Data | 统计抽样与数据类型

In statistics, the population is the whole set of items you want to study, while a sample is a subset selected from the population. A census examines every member, but it is often expensive, time-consuming or impossible. Sampling aims to produce a representative subset so that conclusions can be generalised with known uncertainty.

在统计学中，总体是你想研究的全部对象，样本是从总体中选出的子集。普查会调查每一个成员，但通常成本高、耗时长或不可能实现。抽样的目的是获得一个有代表性的子集，使结论可以推广，并且不确定性可控。

Common sampling methods include simple random sampling, systematic sampling, stratified sampling, opportunity sampling and quota sampling. Random methods reduce bias; stratified sampling ensures key groups are represented in proportion to their size in the population.

常见的抽样方法包括简单随机抽样、系统抽样、分层抽样、便利抽样和配额抽样。随机方法可以减少偏差；分层抽样确保关键群体按其在总体中的比例被抽取。

Data can be qualitative or quantitative. Quantitative data are discrete if they can only take particular values, such as shoe size, and continuous if they can take any value in an interval, such as height or time.

数据可以是定性或定量的。定量数据如果只能取特定值，例如鞋码，就是离散的；如果在一个区间内可以取任意值，例如身高或时间，就是连续的。

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

Measures of location summarise the centre of a data set. The mean is the arithmetic average, the median is the middle value when data are ordered, and the mode is the most frequent value. The median and mode are less affected by extreme values than the mean.

位置度量概括数据集的中心。平均数是算术平均值，中位数是排序后位于中间的值，众数是出现频率最高的值。与平均数相比，中位数和众数受极端值影响较小。

For raw data, the mean is calculated as x̄ = Σx / n. The variance is the average squared deviation from the mean, and the standard deviation is its square root. For a frequency table, use x̄ = Σfx / Σf and variance = Σfx² / Σf − x̄².

对于原始数据，平均数计算为 x̄ = Σx / n。方差是数据与平均数之差的平方平均值，标准差是方差的平方根。对于频数表，使用 x̄ = Σfx / Σf 和方差 = Σfx² / Σf − x̄²。

x̄ = Σx / n, σ² = Σ(x − x̄)² / n = Σx² / n − x̄²

For grouped data, use the midpoint of each class as x. Remember that grouped estimates are approximate because the raw values are unknown.

对于分组数据，使用每组的组中值作为 x。请记住，分组数据的估计值是近似的，因为原始数值未知。

3. Representing Data and Outliers | 数据表示与异常值

Box plots show the lowest value, lower quartile Q₁, median Q₂, upper quartile Q₃ and highest value. Outliers are values that lie far from the main body of data. Common fences are Q₁ − 1.5 × IQR and Q₃ + 1.5 × IQR, where IQR = Q₃ − Q₁.

箱线图显示最小值、下四分位数 Q₁、中

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