📚 IGCSE CAIE Statistics: Formula & Theorem Quick Reference | IGCSE CAIE 统计：公式定理速查手册
This quick reference guide brings together all the essential formulas and theorems you will need for the IGCSE CAIE Statistics examination. It covers measures of central tendency, dispersion, probability rules, distributions, correlation, and data representation tools. Keep this handbook handy when revising or solving past papers to ensure you can instantly recall the correct formula and apply it with confidence.
本速查手册汇集了 IGCSE CAIE 统计学科考试所需的所有关键公式与定理，涵盖集中趋势、离散程度、概率定律、分布、相关性以及数据表示方法。复习或练习真题时随身参考本手册，有助于你快速准确地调用正确公式，自信作答。
1. Measures of Central Tendency | 集中趋势度量
The mean for ungrouped data is calculated by summing all observations and dividing by the number of observations. For grouped data, use the sum of the product of frequency and mid-point divided by total frequency.
平均数 用于未分组数据时，将所有观测值相加再除以观测值个数。对于分组数据，用频数乘以组中值的总和除以总频数。
x̄ = ∑x / n (ungrouped) x̄ = ∑fx / ∑f (grouped)
The median is the middle value when data are arranged in order. For grouped data, linear interpolation within the median class is used.
中位数 是数据排序后位于中间的值。对于分组数据，需在中位数所在组内进行线性插值。
Median = L + [ (n/2 − cf) / f ] × w
- L = lower boundary of median class | 中位数所在组的下限
- cf = cumulative frequency before the median class | 中位数所在组之前的累积频数
- f = frequency of median class | 中位数所在组的频数
- w = class width | 组距
The mode is the most frequently occurring value. For grouped data, the modal class is the class with the highest frequency density.
众数 是出现次数最多的值。对于分组数据，众数所在组是频数密度最高的组。
2. Measures of Dispersion | 离散程度度量
The range is the difference between the largest and smallest observations. The interquartile range (IQR) is the difference between the upper quartile and lower quartile: IQR = Q₃ − Q₁. It measures the spread of the middle 50% of the data.
极差 为最大值与最小值之差。四分位距 (IQR) 等于上四分位数与下四分位数之差：IQR = Q₃ − Q₁，反映中间50%数据的散布程度。
Quartiles for grouped data are found using interpolation, similar to the median, with positions at n/4 and 3n/4.
分组数据中的四分位数采用插值法计算，方法与中位数类似，位置分别为 n/4 和 3n/4。
Q₁ = L₁ + [ (n/4 − cf₁) / f₁ ] × w
Q₃ = L₃ + [ (3n/4 − cf₃) / f₃ ] × w
Percentiles can also be determined using the same interpolation approach, adjusting the position to k% of n. The p-th percentile position is at (p/100) × n.
百分位数同样可用插值法确定，将位置调整为 n 的 k%。第 p 百分位数的位置为 (p/100) × n。
3. Variance and Standard Deviation | 方差与标准差
Variance measures the average squared deviation from the mean. For a population, use σ² = ∑(x − μ)² / N. For a sample, use s² = ∑(x − x̄)² / (n − 1).
方差衡量各个数据与平均数之差的平方的平均值。总体方差 σ² = ∑(x − μ)² / N，样本方差 s² = ∑(x − x̄)² / (n − 1)。
σ² = ∑(x − μ)² / N s² = ∑(x − x̄)² / (n − 1)
For grouped data, replace x with the class mid-point and multiply by frequency. A computational formula often simplifies the calculation: σ² = (∑fx² / ∑f) − (x̄)².
分组数据中，用组中值代替 x 并乘以频数。常用计算简式：σ² = (∑fx² / ∑f) − (x̄)²。
The standard deviation is the positive square root of variance: σ = √σ². Both are expressed in the same units as the original data.
标准差 是方差的正平方根：σ = √σ²。两者都与原始数据单位一致。
| Ungrouped data | 未分组数据 s = √[ ∑(x − x̄)² / (n − 1) ] |
Grouped data | 分组数据 s = √[ ∑f(x − x̄)² / ∑f ] |
4. Basic Probability Rules | 基本概率定律
The probability of an event A is P(A) = number of favourable outcomes / total number of outcomes, provided all outcomes are equally likely. Probability always lies between 0 and 1 inclusive.
事件 A 的概率 P(A) = 有利结果数 / 所有结果总数，前提是所有结果等可能发生。概率值总在 0 到 1 之间（含）。
0 ≤ P(A) ≤ 1
The addition rule for any two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events where A ∩ B = ∅, this simplifies to P(A ∪ B) = P(A) + P(B).
任何两事件的加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若 A 与 B 互斥（A ∩ B = ∅），则简化为 P(A ∪ B) = P(A) + P(B)。
The multiplication rule for independent events: P(A ∩ B) = P(A) × P(B). The complement rule states P(A’) = 1 − P(A).
独立事件的乘法法则：P(A ∩ B) = P(A) × P(B)。补集法则：P(A’) = 1 − P(A)。
5. Conditional Probability and Independence | 条件概率与独立性
Conditional probability is the probability of an event B occurring given that event A has already occurred. It is defined as P(B|A) = P(A ∩ B) / P(A), provided P(A) > 0.
条件概率是指在事件 A 已经发生的条件下事件 B 发生的概率。定义为 P(B|A) = P(A ∩ B) / P(A)，其中 P(A) > 0。
Two events A and B are independent if and only if P(A ∩ B) = P(A) × P(B), or equivalently P(B|A) = P(B) and P(A|B) = P(A). This means the occurrence of one does not affect the probability of the other.
两事件 A 和 B 相互独立 的充要条件是 P(A ∩ B) = P(A) × P(B)，或等价地 P(B|A) = P(B) 且 P(A|B) = P(A)。即一事件的发生不影响另一事件的概率。
Use a tree diagram to organize sequential events: multiply along branches for intersections and add probabilities for unions of disjoint paths.
使用树状图梳理连续事件：沿着分支相乘得到交集概率，互斥路径的概率相加得到并集概率。
6. Permutations and Combinations | 排列与组合
A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is nPr = n! / (n − r)!.
排列 是按特定顺序排列对象。从 n 个物体中取出 r 个进行排列的方法数为 nPr = n! / (n − r)!。
A combination is a selection of objects without regard to order. The number of combinations of n objects taken r at a time is nCr = n! / [r! (n − r)!]. This is also written as (n r).
组合 是不考虑顺序的选择。从 n 个物体中取出 r 个进行组合的方法数为 nCr = n! / [r! (n − r)!]，也写作 (n r)。
| P(n, r) = n! / (n − r)! | C(n, r) = n! / (r! (n − r)!) |
Remember that n! = n × (n−1) × … × 1, and 0! = 1. Factorial notation is fundamental for counting problems.
记住 n! = n × (n−1) × … × 1，且 0! = 1。阶乘符号是计数问题的基础。
7. Binomial Distribution | 二项分布
The binomial distribution models the number of successes in n independent trials, each with success probability p. The random variable X ~ B(n, p) has probability mass function:
二项分布用于描述 n 次独立试验中成功的次数，每次成功概率为 p。随机变量 X ~ B(n, p) 的概率质量函数为：
P(X = r) = nCr × pʳ × (1 − p)ⁿ⁻ʳ
where r = 0, 1, 2, …, n. The mean of the binomial distribution is μ = np, and the variance is σ² = np(1 − p).
其中 r 取 0, 1, 2, …, n。二项分布的均值 μ = np，方差 σ² = np(1 − p)。
Mean = np Variance = np(1 − p) Standard deviation = √[np(1 − p)]
To find cumulative probabilities P(X ≤ k) or P(X ≥ k), sum the individual probabilities or use statistical tables provided in the exam.
求累积概率 P(X ≤ k) 或 P(X ≥ k) 时，可以将单个概率相加，或使用考试提供的统计表。
8. Normal Distribution | 正态分布
The normal distribution is a continuous, bell-shaped distribution defined by its mean μ and standard deviation σ. The random variable X ~ N(μ, σ²) is standardised to the standard normal variable Z using:
正态分布是一种连续的钟形分布，由其均值 μ 和标准差 σ 定义。随机变量 X ~ N(μ, σ²) 通过以下变换转化为标准正态变量 Z：
Z = (X − μ) / σ
Z follows N(0, 1²). The standard normal table gives the probability Φ(z) = P(Z ≤ z) for right-hand side or left-hand side areas. Use symmetry: P(Z > z) = 1 − Φ(z) and P(Z < −z) = P(Z > z).
Z 服从 N(0, 1²)。标准正态表给出概率 Φ(z) = P(Z ≤ z)。利用对称性：P(Z > z) = 1 − Φ(z)，P(Z < −z) = P(Z > z)。
To find a probability for X, first convert the value to a Z-score, then look up in the table. To find an unknown mean or standard deviation, set up an equation using standardised values.
计算 X 的概率时，先将数值转换为 Z 分数，再查表。要求未知的均值或标准差时，利用标准化值建立方程。
9. Correlation and Regression | 相关与回归
Product moment correlation coefficient (PMCC), r, measures the strength and direction of a linear relationship between two variables x and y. Its value ranges from −1 to +1.
积矩相关系数 (PMCC) r 衡量两个变量 x 与 y 之间线性关系的强度和方向，取值范围为 −1 至 +1。
r = Sxy / √(Sxx × Syy)
where Sxy = ∑(x − x̄)(y − ȳ) = ∑xy − (∑x∑y)/n,
Sxx = ∑(x − x̄)² = ∑x² − (∑x)²/n,
Syy = ∑(y − ȳ)² = ∑y² − (∑y)²/n.
其中 Sxy = ∑(x − x̄)(y − ȳ) = ∑xy − (∑x∑y)/n，
Sxx = ∑(x − x̄)² = ∑x² − (∑x)²/n，
Syy = ∑(y − ȳ)² = ∑y² − (∑y)²/n。
The least squares regression line of y on x has the equation y = a + bx, where:
y 对 x 的最小二乘回归线 方程为 y = a + bx，其中：
b = Sxy / Sxx a = ȳ − b x̄
This line can be used to predict y for a given x within the data range (interpolation), but extrapolation beyond the data should be treated with caution.
该直线可用于在数据范围内由给定 x 预测 y（内插），向数据范围之外的外推应谨慎对待。
10. Graphical Data Representation | 图形数据表示
Histograms display grouped continuous data. The vertical axis represents frequency density, calculated as frequency / class width. The area of each bar is proportional to the frequency.
直方图 用于显示分组的连续数据。纵轴表示频数密度，计算公式为 频数 / 组距。每个柱形的面积与频数成正比。
Frequency density = Frequency / Class width
A cumulative frequency curve (ogive) plots the running total of frequencies against the upper class boundaries. The median and quartiles can be read directly from the graph.
累积频率曲线（ogive）将逐次累积的频数对应上限标绘成图。中位数与四分位数可直接从图上读取。
A box-and-whisker plot (box plot) shows the five-number summary: minimum, Q₁, median, Q₃, maximum. It helps identify skewness and outliers.
箱线图 展示五个关键数值：最小值、下四分位数、中位数、上四分位数、最大值，有助于识别偏态和异常值。
Scatter diagrams are used to visualise the relationship between two variables; the pattern suggests correlation. A line of best fit is drawn by eye or calculated via regression.
散点图 用于可视化两变量之间的关系，点的分布形态可显示相关性。最佳拟合线通过目测或回归计算得出。
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