📚 IGCSE Edexcel Statistics: Formula and Theorem Quick Reference | IGCSE Edexcel 统计：公式定理速查手册
Welcome to the IGCSE Edexcel Statistics Formula and Theorem Quick Reference. This handbook compiles all essential formulae, definitions, and theorems you need for the Edexcel International GCSE in Statistics (4ST1). Use it as your go-to revision aid and during exam practice. Mastering these relationships is key to solving problems efficiently and confidently.
欢迎使用 IGCSE Edexcel 统计公式定理速查手册。这里汇编了 Edexcel 国际 GCSE 统计 (4ST1) 考试所需的所有核心公式、定义和定理。把它当作你的首选复习助手，平时练习时也可查阅。掌握这些关系是高效、自信解题的关键。
1. Measures of Central Tendency | 集中趋势的度量
The arithmetic mean (average) of a data set x₁, x₂, …, xₙ is the sum of all values divided by the number of values.
数据集的算术平均数（均值）是所有数值之和除以数值的个数。
Mean = Σx / n or x̄ = Σx / n
For a weighted mean, each value xᵢ is assigned a weight wᵢ. This is useful when values carry different importance.
加权平均数则为每个数值 xᵢ 赋予权重 wᵢ，这在数值重要性不同时很有用。
Weighted Mean = Σ(wᵢxᵢ) / Σwᵢ
The median is the middle value when data are arranged in order. For an odd-sized list, it is the ((n+1)/2)th value; for an even-sized list, it is the average of the (n/2)th and (n/2 + 1)th values.
中位数是数据排序后的中间值。若个数为奇数，则是第 (n+1)/2 个值；若为偶数，则是第 n/2 和 n/2+1 个值的平均数。
The mode is the most frequently occurring value. A dataset may have one mode, more than one mode (bimodal/multimodal), or no mode at all.
众数是出现频率最高的数值。一组数据可以有一个众数、多个众数（双峰/多峰），或者没有众数。
2. Measures of Dispersion | 离散度的度量
The range is the simplest measure of spread: it is the difference between the maximum and minimum values. While easy to compute, it is sensitive to outliers.
极差是最简单的离散度量：即最大值与最小值之差。虽然容易计算，但对异常值很敏感。
Range = xₘₐₓ − xₘᵢₙ
The interquartile range (IQR) measures the spread of the middle 50% of the data. It is the difference between the upper quartile Q₃ and the lower quartile Q₁. The IQR is resistant to outliers.
四分位距 (IQR) 衡量中间 50% 数据的分散程度，是上四分位数 Q₃ 与下四分位数 Q₁ 的差。IQR 对异常值具有较强的抗干扰性。
IQR = Q₃ − Q₁
Variance quantifies the average squared deviation from the mean. For a population of size N, the population variance is σ². For a sample of size n, the sample variance is s², which uses (n−1) as the denominator to give an unbiased estimate.
方差衡量各数据与均值偏差的平方的平均。对于大小为 N 的总体，总体方差为 σ²。对于大小为 n 的样本，样本方差 s² 使用 (n−1) 作为分母，以得到无偏估计。
σ² = Σ(x − μ)² / N | s² = Σ(x − x̄)² / (n−1)
The standard deviation is the positive square root of the variance. It returns the spread to the original units of measurement.
标准差是方差的正平方根，将离散度还原到原始测量单位。
σ = √σ² | s = √s²
Variance can also be computed using the shortcut formula: Var(X) = E(X²) − [E(X)]². For raw data, this translates to Σx²/n − (Σx/n)².
方差也可用简便公式计算：Var(X) = E(X²) − [E(X)]²。对于原始数据，即 Σx²/n − (Σx/n)²。
3. Working with Grouped Data | 分组数据的处理
When data are presented in a frequency table with class intervals, we use the class midpoint (x) as the representative value for each group. The estimated mean is obtained by summing the product of frequency f and midpoint x, divided by total frequency Σf.
当数据以组频数表呈现时，我们用组中值 (x) 代表各组数值。估算均值通过求频数 f 与组中值 x 的乘积之和除以总频数 Σf 得到。
Estimated Mean x̄ = Σfx / Σf
For grouped data, the variance can be estimated using the equivalent formula. Use the sum of fx² and Σf, then subtract the square of the estimated mean. Note whether you need the population or sample version: for sample variance, divide by (Σf − 1).
分组数据的方差也可用类似公式估算。利用 Σfx² 和 Σf，减去估算均值的平方。注意区分总体和样本：样本方差除以 (Σf − 1)。
σ² = (Σfx² / Σf) − (x̄)² | s² = [ Σf (x − x̄)² ] / (Σf − 1)
For the median and quartiles from grouped data, linear interpolation is applied using the cumulative frequency. The median position is at (Σf)/2. The formula locates the value within the median class interval.
对于分组数据的中位数和四分位数，需要使用累积频数进行线性插值。中位数位置是 (Σf)/2。利用公式在中位数组内定位具体数值。
Median = L + [ ( (Σf / 2) − Fₚᵣₑᵥ ) / fₘₑₔ ] × w
Here, L is the lower boundary of the median class, Fₚᵣₑᵥ is the cumulative frequency before the class, fₘₑₔ is the frequency of the median class, and w is the class width. The same principle applies to Q₁ and Q₃ using (Σf)/4 and 3(Σf)/4.
其中 L 是中位数组的下限，Fₚᵣₑᵥ 是该组之前的累积频数，fₘₑₔ 是中位数组的频数，w 为组距。同样的原则适用于 Q₁ 和 Q₃，分别使用 (Σf)/4 和 3(Σf)/4。
4. Effect of Linear Transformations | 线性变换的影响
When every data value is transformed by a linear coding y = (x − a)/b, the mean and standard deviation change in predictable ways. Coding is often used to simplify calculations.
当每个数据值通过线性编码 y = (x − a)/b 变换时，均值和标准差会以可预测的方式改变。编码常用于简化计算。
Mean of y: ȳ = (x̄ − a) / b
Standard deviation of y: s_y = sₓ / |b| (variance is divided by b²)
Addition or subtraction (shifting) changes the mean but does not affect the spread. Multiplication or division (scaling) changes both the location and the spread proportionally.
加减（平移）会改变均值但不影响离散度。乘除（缩放）则会按比例同时改变位置和离散度。
5. Probability Basics | 概率基础
The probability of an event A is a number between 0 and 1, written P(A). The addition rule for two events is:
事件 A 的概率是介于 0 和 1 之间的数，记作 P(A)。两个事件的加法法则为：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Mutually exclusive events cannot happen at the same time, so P(A ∩ B) = 0. Their addition rule simplifies to P(A ∪ B) = P(A) + P(B).
互斥事件不能同时发生，因此 P(A ∩ B) = 0。加法法则简化为 P(A ∪ B) = P(A) + P(B)。
Conditional probability P(A|B) is the probability that A occurs given that B has occurred. It is given by:
条件概率 P(A|B) 是在事件 B 已发生的条件下事件 A 发生的概率，计算公式为：
P(A|B) = P(A ∩ B) / P(B) (provided P(B) > 0)
Two events are independent if the occurrence of one does not affect the probability of the other. For independent events, P(A ∩ B) = P(A) × P(B) and P(A|B) = P(A). Tree diagrams help visualise combined probabilities.
若一事件的发生不影响另一事件的概率，则两事件独立。对于独立事件，P(A ∩ B) = P(A) × P(B) 且 P(A|B) = P(A)。树状图有助于将复合概率可视化。
6. Discrete Random Variables & Binomial Distribution | 离散随机变量与二项分布
A discrete random variable X takes a countable set of values with associated probabilities. The probability distribution lists all outcomes and satisfies Σ P(X=x) = 1.
离散随机变量 X 取可数个值，每个值对应一定的概率。概率分布列出所有结果且满足 Σ P(X=x) = 1。
Expected value: E(X) = μ = Σ x · P(X=x)
Variance: Var(X) = E(X²) − [E(X)]² = Σ x² P(X=x) − μ²
The binomial distribution models the number of successes in n independent trials, each with constant probability of success p. X ~ B(n, p). The probability of exactly r successes is:
二项分布描述 n 次独立试验中成功的次数，每次试验成功概率为 p。X ~ B(n, p)。恰好 r 次成功的概率为：
P(X = r) = nCr pr (1 − p)n−r where nCr = n! / [r! (n−r)!]
The mean and variance of a binomial distribution are simple functions of n and p:
二项分布的均值和方差是 n 和 p 的简单函数：
Mean: μ = np | Variance: σ² = np(1 − p) = npq (where q = 1 − p)
7. The Normal Distribution | 正态分布
The normal distribution is a continuous, symmetric, bell-shaped curve defined by its mean μ and variance σ². The total area under the curve is 1. Key intervals: about 68% of data lie within μ ± 1σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
正态分布是连续的、对称的钟形曲线，由其均值 μ 和方差 σ² 定义。曲线下总面积为 1。关键区间：约 68% 数据落在 μ ± 1σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。
To standardise any normal variable X, we convert to the standard normal Z ~ N(0, 1) using the formula:
要对任何正态变量 X 进行标准化，使用公式将其转换为标准正态 Z ~ N(0, 1)：
Z = (X − μ) / σ
Tables of the standard normal distribution give the cumulative probability Φ(z) = P(Z ≤ z). For probabilities greater than z, use
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