📚 Edexcel Year 1 Stats and Mechanics: Core Revision Guide | Edexcel Year 1 统计与力学核心复习指南
This revision guide covers the essential Statistics and Mechanics content for Edexcel Year 1 Maths. It pairs each English explanation with a Chinese translation, focusing on the formulas, key ideas and common exam pitfalls you need for AS-level papers.
本复习指南涵盖 Edexcel Year 1 数学中统计与力学的核心内容。每个英文解释都配有中文翻译，重点讲解公式、关键概念以及 AS 考试中常见的易错点。
1. Statistical Sampling and Data Types | 统计抽样与数据类型
A population is the whole set of items being studied, while a sample is a subset selected to represent it. A census collects data from every member, but it is often expensive and time-consuming, so sampling is usually preferred.
总体是研究对象的全体集合，而样本是从中选出的代表性子集。普查会收集每个成员的数据，但通常成本高、耗时长，因此抽样通常更常用。
Common sampling methods include simple random sampling, systematic sampling, stratified sampling, quota sampling and opportunity sampling. You must know the advantages and disadvantages of each, especially when a question asks for the most suitable method.
常见的抽样方法包括简单随机抽样、系统抽样、分层抽样、配额抽样和便利抽样。你必须了解每种方法的优缺点，尤其是当题目要求选择最合适的方法时。
- Simple random sampling: every member has an equal chance of selection; unbiased but requires a full sampling frame.
- 简单随机抽样：每个成员被选中的概率相等；无偏但需要完整的抽样框。
- Stratified sampling: population divided into groups; sample size from each group is proportional to group size; gives fair representation.
- 分层抽样：总体分成若干层；各层样本量与层的大小成比例；代表性较好。
- Opportunity sampling: uses people who are easily available; quick but likely biased.
- 便利抽样：使用容易接触到的人；快速但容易产生偏差。
Data can be qualitative or quantitative. Quantitative data may be discrete, taking only particular values, or continuous, taking any value in an interval.
数据可以是定性的或定量的。定量数据可以是离散的，只能取特定值；也可以是连续的，可以取区间内的任意值。
2. Measures of Location and Spread | 位置与离散度量
measures of location are the mean, median and mode. For a list of n values, the mean is x̄ median is the middle value when data are ordered, and the mode is the most frequent value.
三个主要的位置度量是平均数、中位数和众数。对于 n 个数据值，平均数 x̄ = Σxᵢ ÷ n。中位数是数据按大小排列后的中间值，众数是出现频率最高的值。
Quartiles divide ordered data into four equal parts. The lower quartile Q₁ is the 25th percentile, the median Q₂ is the 50th percentile, and the upper quartile Q₃ is the 75th percentile.
四分位数把排序后的数据分成四等份。下四分位数 Q₁ 是第 25 百分位数，中位数 Q₂ 是第 50 百分位数，上四分位数 Q₃ 是第 75 百分位数。
The interquartile range is IQR = Q₃ – Q₁. For variance, use σ² = Σ(xᵢ – x̄)² ÷ n for a population, or s² = Σ(xᵢ – x̄)² ÷ (n – 1) for a sample. Standard deviation is the square root of variance.
四分位距是 IQR = Q₃ – Q₁。对于方差，总体使用 σ² = Σ(xᵢ – x̄)² ÷ n，样本使用 s² = Σ(xᵢ – x̄)² ÷ (n – 1)。标准差是方差的平方根。
When data are grouped, use the midpoint of each class to estimate the mean and variance. For a grouped median, use linear interpolation within the median class interval.
当数据是分组数据时，使用每组的组中值来估计平均数与方差。对于分组数据的中位数，需要在中位数所在组内使用线性插值。
3. Data Representations and Outliers | 数据表示与异常值
Box plots show the minimum, Q₁, median, Q₃ and maximum. They are useful for comparing distributions and identifying skewness. A box plot can be drawn from raw data or from a cumulative frequency diagram.
箱线图显示最小值、Q₁、中位数、Q₃ 和最大值。它有助于比较分布并识别偏态。箱线图可以根据原始数据或累积频率图画出。
of each bar is proportional to frequency, so frequency density = frequency ÷ class width.
直方图用于连续数据且组距不一定相等。每个条形图的面积与频数成比例，因此频率密度 = 频数 ÷ 组距。
s an extreme data value. A common rule is that a value is an outlier if it is less than Q₁ – r greater than Q₃ + 1.5 × IQR. You must identify and interpret outliers in exam questions.
异常值是极端数据值。常用规则是：如果数值小于 Q₁ – 1.5 × IQR 或大于 Q₃ + 1.5 × IQR，则认为是异常值。考试中你必须识别并解释异常值。
Cumulative frequency curves can estimate the median and quartiles. When a cumulative frequency curve is plotted, the median is found at 50% of the total frequency.
累积频率曲线可以估计中位数和四分位数。绘制累积频率曲线后，中位数在总频数的 50% 处读取。
4. Probability and Venn Diagrams | 概率与维恩图
For two events A and B, the addition rule is P(A ∪ B) = P(A) + P(B) – P(A ∩ B). If A and B are mutually exclusive, then P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
对于两个事件 A 和 B，加法公式为 P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。如果 A 和 B 互斥，则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。
Conditional probability is the probability of A given B: P(A | B) = P(A ∩ B) ÷ P(B). Rearranging gives the multiplication rule P(A ∩ B) = P(A | B) × P(B).
条件概率是在 B 发生的前提下 A 发生的概率：P(A | B) = P(A ∩ B) ÷ P(B)。移项可得乘法公式 P(A ∩ B) = P(A | B) × P(B)。
Two events are independent if P(A ∩ B) = P(A) × P(B), or equivalently if P(A | B) = P(A). Tree diagrams are essential for multi-stage probability problems.
如果 P(A ∩ B) = P(A) × P(B)，或等价地 P(A | B) = P(A)，则两个事件独立。树形图是解决多阶段概率问题的关键。
Venn diagrams help visualise probabilities and solve problems involving complements, intersections and unions. Always check that probabilities sum to 1.
维恩图有助于可视化概率，并解决涉及补集、交集和并集的问题。一定要检查概率之和是否为 1。
5. Discrete Random Variables and Binomial Distribution | 离散随机变量与二项分布
A discrete random variable X takes a countable number of values. A probability distribution lists each value with its probability, and the probabilities must sum to 1.
离散随机变量 X 取可数个值。概率分布列出每个值及其概率，所有概率之和必须为 1。
ted value E(X) = Σ xᵢ P(X = xᵢ) is the mean of X. The variance is Var(X) = Σ xᵢ² P(X = xᵢ) – [E(X)]², and the standard deviation is √Var(X).
期望值 E(X) = Σ xᵢ P(X = xᵢ) 是 X 的平均数。方差 Var(X) = Σ xᵢ² P(X = xᵢ) – [E(X)]²，标准差是 √Var(X)。
ution models n independent trials, each with the same probability of success p. We write X ~ B(n, p). The probability of exactly r successes is P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ.
二项分布描述 n 次独立试验，每次试验成功的概率 p 相同。记作 X ~ B(n, p)。恰好 r 次成功的概率为 P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ。
For a binomial distribution, the mean is E(X) = np and the variance is Var(X) = np(1 – fixed number of trials, independent trials, two outcomes, constant probability of success.
对于二项分布，平均数 E(X) = np，方差 Var(X) = np(1 – p)。你必须检查二项分布的四个条件：试验次数固定、试验独立、两种结果、成功概率恒定。
6. Binomial Hypothesis Testing | 二项假设检验
is H₁: p < p₀, p > p₀ or p ≠ p₀ depending on whether the test is one-tailed or two-tailed.
对二项概率 p 进行假设检验时，使用原假设 H₀: p = p₀ 和备择假设 H₁: p < p₀、p > p₀ 或 p ≠ p₀，具体取决于检验是单尾还是双尾。
The significance level α is the probability of rejecting H₀ when it is true. Common values are 5% or 1%. The critical region is the set of outcomes that lead to rejecting H₀.
显著性水平 α 是当 H₀ 为真时拒绝 H₀ 的概率。常用值为 5% 或 1%。拒绝域是导致拒绝 H₀ 的所有结果的集合。
l distribution. Compare the observed value with the critical value, or compare the p-value with α.
要找拒绝域，使用二项分布计算 P(X ≤ c) 或 P(X ≥ c)。将观测值与临界值比较，或将 p 值与 α 比较。
Write a conclusion in context: if the test statistic is in the critical region, reject H₀; otherwise, do not reject H₀. Never say you prove H₁ or H₀ is true.
写出与背景相关的结论：如果检验统计量落在拒绝域内，则拒绝 H₀；否则不拒绝 H₀。永远不要说证明了 H₁ 或 H₀ 为真。
7. Kinematics: Constant Acceleration (SUVAT) | 运动学：匀加速运动
In mechanics, displacement s is distance in a given direction, velocity v is rate of change of displacement, and acceleration a is rate of change of velocity. For constant acceleration, five SUVAT equations link s, u, v, a, t.
在力学中，位移 s 是带有方向的距离，速度 v 是位移的变化率，加速度 a 是速度的变化率。对于匀加速运动，五个 SUVAT 方程把 s、u、v、a、t 联系起来。
The SUVAT equations are:
SUVAT 方程为：
v = u + at
s = ut + ½at²
s = ½(u + v)t
v² = u² + 2as
s = vt – ½at²
f a particle moves upwards and gravity acts downwards, take upward as positive and set a = –9.8 m s⁻².
使用 SUVAT 前一定要先规定正方向。如果粒子向上运动而重力向下，则取向上为正方向，并令 a = –9.8 m s⁻²。
Velocity–time graphs are useful: the gradient gives acceleration, and the area under the graph gives displacement. For a straight line, the area is a trapezium, which links to the SUVAT equations.
速度–时间图很有用：斜率给出加速度，图线下面积给出位移。对于直线，面积是梯形，这与 SUVAT 方程一致。
8. Forces and Newton's Laws | 力与牛顿定律
Newton' mains at rest or moves with constant velocity unless acted on by a resultant force. Newton' second law states F = ma, where F is the resultant force, m is mass and a is acceleration.
牛顿第一定律指出，除非受到合外力作用，物体将保持静止或匀速直线运动。牛顿第二定律指出 F = ma，其中 F 是合外力，m 是质量，a 是加速度。
Newton's third law states that when two objects interact, the forces they exert on each other are equal in magnitude and opposite in direction.
牛顿第三定律指出，两个物体相互作用时，彼此施加的力大小相等、方向相反。
parallel to the surface and opposes motion, with F ≤ μR or F = μR at limiting equilibrium.
重力 W = mg，其中 g = 9.8 m s⁻²。法向反作用力 R 垂直于接触面。摩擦力平行于接触面并阻碍运动，在极限平衡时 F ≤ μR 或 F = μR。
eparately to each particle or to the whole system. Tension in an inextensible string is the same throughout, and the acceleration of connected particles is equal in magnitude.
对于连接粒子问题，先画受力图，标出所有力，并分别对每个粒子或整体应用 F = ma。不可伸长的绳中张力处处相同，连接粒子的加速度大小相等。
9. Variable Acceleration and Calculus | 变加速度与微积分
ivative of displacement: v = ds/dt. Acceleration is the derivative of velocity: a = dv/dt.
当加速度不恒定时，位移、速度和加速度都是时间 t 的函数。速度是位移的导数：v = ds/dt。加速度是速度的导数：a = dv/dt。
y, s = ∫ v dt. Remember to include the constant of integration and use initial conditions.
反过来使用积分：速度是加速度的积分，v = ∫ a dt；位移是速度的积分，s = ∫ v dt。记住要加上积分常数并利用初始条件确定。
Distance travelled is not always the same as displacement. To find total distance when velocity changes sign, integrate |v| or split the motion into intervals where v is positive and negative.
路程与位移并不总是相同。当速度改变符号时，求总路程可以对 |v| 积分，或把运动分成速度为正值和负值的区间分别计算。
m displacement occurs when v = ds/dt = 0. Always check the nature of the stationary point.
速度的最大值或最小值出现在 a = dv/dt = 0 时。位移的最大值或最小值出现在 v = ds/dt = 0 时。一定要检验驻点的性质。
10. Modelling Assumptions and Exam Tips | 建模假设与考试技巧
Mechanics questions often ask about modelling assumptions. Common assumptions include: a particle has no size but has mass; a rod is rigid and one-dimensional; a string is light and inextensible; a pulley is smooth and light; air resistance is negligible.
力学题经常考建模假设。常见假设包括：质点有质量但无大小；杆是刚性的且为一维；绳是轻质且不可伸长的；滑轮光滑且轻质；空气阻力可忽略。
In statistics, always state definitions clearly, give units where appropriate, and interpret results in context. For hypothesis tests, use the word 'significant' only when rejecting H₀, and include the significance level.
在统计中，要清晰地说明定义，适当给出单位，并结合背景解释结果。在假设检验中，只有拒绝 H₀ 时才使用“显著”一词，并注明显著性水平。
Before attempting a mechanics problem, draw a clear diagram, list known quantities, choose a positive direction and select the correct SUVAT equation. In statistics, organise data carefully and show working for interpolation or standard deviation.
在做力学题之前，先画清晰的示意图，列出已知量，选择正方向，并选择合适的 SUVAT 方程。在统计中，要认真整理数据，并在插值或标准差计算中展示步骤。
Finally, practise past Edexcel papers under timed conditions. Common errors include forgetting units, using the wrong critical region, confusing displacement with distance and mixing up one-tailed and two-tailed tests.
最后，要在计时条件下练习 Edexcel 历年真题。常见错误包括忘记单位、使用错误的拒绝域、混淆位移与路程、弄混单尾与双尾检验。
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