📚 Year 11 CAIE Statistics: Formula & Theorem Quick Reference Handbook | CAIE 统计公式定理速查手册
This quick reference handbook brings together the essential formulas and theorems needed for Year 11 CAIE Statistics. It covers measures of central tendency, dispersion, probability, statistical distributions, correlation and regression, sampling, and index numbers. Each formula is accompanied by a clear explanation in both English and Chinese to help you revise efficiently and build confidence for your examinations.
本速查手册汇集了 Year 11 CAIE 统计所需的核心公式与定理，涵盖集中趋势、离散程度、概率、统计分布、相关与回归、抽样以及指数等内容。每个公式均附有清晰的中英文解释，帮助你高效复习，从容应对考试。
1. Measures of Central Tendency | 集中趋势的度量
Measures of central tendency describe the centre of a data set. The three main statistics are the mean, median, and mode.
集中趋势度量描述数据集的中心位置，三种主要统计量是平均数、中位数和众数。
Mean (Arithmetic Average) | 算术平均数
The sample mean is the sum of all observations divided by the number of observations. It is the most common measure of location but can be affected by extreme values.
x̄ = Σxi / n
Where Σxi is the sum of the data values and n is the sample size.
其中 Σxi 是所有数据值的总和，n 为样本量。
Median | 中位数
The median is the middle value when the data are arranged in order. For an odd number of observations it is the central item; for an even number it is the average of the two central items. The median is not influenced by outliers and is often preferred for skewed data.
中位数是将数据按大小排序后位于中间的值。当观测个数为奇数时取中间项，偶数时取中间两项的平均值。中位数不受极端值影响，常用于偏态数据。
Mode | 众数
The mode is the value that occurs most frequently in the data set. A data set can have one mode (unimodal), two modes (bimodal), or more. The mode is the only measure of central tendency that can be used with categorical data.
众数是数据中出现次数最多的值。数据集可以有一个众数（单峰）、两个众数（双峰）或更多。众数是唯一可用于分类数据的集中趋势度量。
2. Measures of Dispersion | 离散程度的度量
Dispersion measures how spread out the data are. Key measures include the range, interquartile range, variance, and standard deviation.
离散程度衡量数据的分散情况。主要度量有极差、四分位距、方差和标准差。
Range | 极差
Range = maximum value – minimum value. It is simple to calculate but very sensitive to outliers.
极差 = 最大值 – 最小值。计算简单但对异常值十分敏感。
Interquartile Range (IQR) | 四分位距
IQR = Q₃ – Q₁, where Q₁ is the first quartile (25th percentile) and Q₃ is the third quartile (75th percentile). IQR measures the spread of the middle 50% of the data and is robust against outliers.
IQR = Q₃ – Q₁，其中 Q₁ 为第一四分位数（第25百分位数），Q₃ 为第三四分位数（第75百分位数）。IQR 衡量中间50%数据的分散程度，对异常值稳健。
Variance and Standard Deviation | 方差与标准差
The variance and standard deviation quantify how far each observation deviates from the mean. For a sample, use (n – 1) in the denominator to obtain an unbiased estimate of the population variance.
s² = Σ (xi – x̄)² / (n – 1)
s = √[ Σ (xi – x̄)² / (n – 1) ]
The standard deviation is the positive square root of the variance. A larger standard deviation indicates greater spread.
方差和标准差量化每个观测值与平均值的偏差。对于样本，分母使用 n – 1 可获得总体方差的无偏估计。标准差是方差的正平方根。标准差越大，数据越分散。
3. Grouped Data and Histograms | 分组数据与直方图
When data are presented in a grouped frequency table, additional care is needed to calculate statistics and to draw histograms correctly.
当数据以分组频数表呈现时，计算统计量和绘制直方图需要特别注意。
Frequency Density | 频数密度
In a histogram, the area of each bar is proportional to the frequency. Therefore the height of a bar represents frequency density.
Frequency density = Frequency ÷ Class width
Always use frequency density on the vertical axis, never the raw frequency, unless all class widths are equal.
在直方图中，每一条的面积与频数成正比，因此条形的高度表示频数密度。纵轴上必须使用频数密度，除非所有组距都相等。
Estimating the Mean from Grouped Data | 由分组数据估算平均数
Use the class midpoints (xi) as representative values for each interval. Then apply the formula:
x̄ ≈ Σ fi xi / Σ fi
Where fi is the frequency of each class and xi is the class midpoint.
用组中值 xi 表示每个区间，再代入公式：x̄ ≈ Σ fi xi / Σ fi，其中 fi 为组频数。
4. Cumulative Frequency and Quartiles | 累积频数与四分位数
Cumulative frequency graphs (ogives) are used to estimate medians, quartiles, and percentiles visually.
累积频数曲线图可用于直观地估算中位数、四分位数和百分位数。
Finding Quartiles from a Cumulative Frequency Curve | 从累积频数曲线找四分位数
Plot cumulative frequency against the upper class boundary. To find Q₁, Q₂ (median) and Q₃, locate the positions (n+1)/4, (n+1)/2, and 3(n+1)/4 on the cumulative frequency axis, then read the corresponding values on the horizontal axis. Interpolation may be used for grouped data without a graph.
以累积频数对组上界作图。从累积频数轴上找到 (n+1)/4、 (n+1)/2 和 3(n+1)/4 的位置，再读取对应的横轴数值，即可得到 Q₁、Q₂ 和 Q₃。对于分组数据也可不画图而用线性插值法。
Box-and-Whisker Plot | 箱线图
A box plot displays the five-number summary: minimum, Q₁, median, Q₃, and maximum. Outliers are often marked as individual points if they lie beyond 1.5 × IQR from the quartiles.
箱线图展示五数概括：最小值、Q₁、中位数、Q₃ 和最大值。异常值通常被定义为超出上下四分位数1.5倍 IQR 的点，并单独标出。
5. Basic Probability Rules | 基本概率法则
Probability quantifies the chance of an event occurring. It ranges from 0 (impossible) to 1 (certain). Fundamental rules govern combinations of events.
概率量化事件发生的可能性，范围从0（不可能）到1（必然）。基本法则用于处理事件的组合。
Addition Rule | 加法法则
For any two events A and B:
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
If A and B are mutually exclusive, P(A ∩ B) = 0 and the rule simplifies to P(A ∪ B) = P(A) + P(B).
对于任意事件 A 和 B：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若 A 与 B 互斥，P(A ∩ B) = 0，则简化为 P(A ∪ B) = P(A) + P(B)。
Multiplication Rule and Conditional Probability | 乘法法则与条件概率
The probability that both A and B occur is:
P(A ∩ B) = P(A) × P(B | A)
Where P(B | A) is the conditional probability of B given A. If A and B are independent, P(B | A) = P(B) and therefore P(A ∩ B) = P(A) × P(B).
A 和 B 同时发生的概率为：P(A ∩ B) = P(A) × P(B | A)，其中 P(B | A) 是在 A 发生的条件下 B 发生的概率。若 A 与 B 独立，则 P(B | A) = P(B)，因此 P(A ∩ B) = P(A) × P(B)。
Conditional Probability Formula | 条件概率公式
P(B | A) = P(A ∩ B) / P(A), provided P(A) > 0
This rearranges the multiplication rule and is essential when calculating probabilities from tables or tree diagrams.
该式由乘法法则变形得到，在借助表格或树形图计算概率时十分关键。
6. Permutations and Combinations | 排列与组合
Counting techniques are used to find the number of ways events can occur without listing all outcomes. The factorial function n! = n × (n – 1) × … × 1 is fundamental.
计数方法用于计算事件发生的可能方式总数。阶乘 n! = n × (n – 1) × … × 1 是最基本的工具。
Permutations | 排列
A permutation is an arrangement of objects where order matters. The number of ways to arrange r objects from n distinct objects is:
nPr = n! / (n – r)!
For example, the number of ways to arrange 3 letters from the word ‘STATS’ requires care with repeated items, so the general formula for permutations with identical objects is n! / (p! q! …) where p, q, … are the frequencies of repeated items.
排列与顺序有关。从 n 个不同物品中选取 r 个进行排列的方法数为 nPr = n! / (n – r)! 。若物品有重复，需除以各重复次数的阶乘：n! / (p! q! …)。
Combinations | 组合
A combination is a selection of objects where order does not matter. The number of ways to choose r objects from n distinct objects is:
nCr = n! / [r! (n – r)!]
Combinations are widely used in probability distributions such as the binomial distribution.
组合与顺序无关。从 n 个不同物品中选取 r 个的组合数为 nCr = n! / [r! (n – r)!]。组合在二项分布等概率分布中广泛应用。
7. Probability Distributions | 概率分布
Two key distributions in Year 11 CAIE Statistics are the binomial distribution for discrete data and the normal distribution for continuous data.
Year 11 CAIE 统计中两个关键分布是离散的二项分布和连续的正态分布。
Binomial Distribution | 二项分布
If X ~ B(n, p), then X counts the number of successes in n independent trials, each with success probability p. The probability of exactly r successes is:
P(X = r) = nCr pr (1 – p)n – r
The mean (expected value) is μ = np, and the variance is σ² = np(1 – p). These are useful for modelling success/failure experiments.
若 X ~ B(n, p)，X 表示 n 次独立试验中成功的次数，每次成功概率为 p。恰好成功 r 次的概率为 P(X = r) = nCr pr (1 – p)n – r。均值 μ = np，方差 σ² = np(1 – p)。
Normal Distribution | 正态分布
If X ~ N(μ, σ²), the probability density is a bell-shaped curve symmetric about μ. To find probabilities, convert X to the standard normal variable Z:
Z = (X – μ) / σ, Z ~ N(0, 1)
Use standard normal tables to find P(Z < z). For binomial distributions with large n, a normal approximation can be used with a continuity correction (adding or subtracting 0.5 to/from the boundary).
若 X ~ N(μ, σ²)，其概率密度呈钟形，关于 μ 对称。求概率时需将 X 转化为标准正态变量 Z：Z = (X – μ) / σ，Z ~ N(0, 1)。查标准正态表可得 P(Z < z)。对于样本量大的二项分布，可使用正态近似，并加上连续性校正（边界 ±0.5）。
8. Correlation and Regression | 相关与回归
Correlation measures the strength and direction of a linear relationship between two variables, while regression gives the equation of the line that best fits the data.
相关衡量两变量间线性关系的强度与方向，回归则给出最佳拟合直线方程。
Pearson’s Product-Moment Correlation Coefficient | 皮尔逊积矩相关系数
Given paired data (xi, yi), the Pearson correlation coefficient r is:
r = Σ (xi – x̄)(yi – ȳ) / √[ Σ (xi – x̄)² Σ (yi – ȳ)² ]
r lies between –1 and 1. Values close to +1 indicate strong positive correlation, close to –1 indicate strong negative correlation, and near 0 indicate weak or no linear correlation.
r 介于 –1 与 1 之间。接近 +1 表示强正相关，接近 –1 表示强负相关，接近 0 表示弱线性相关或无关。
Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
When data are ranked, use the rank differences di = rank(xi) – rank(yi):
rs = 1 – [6 Σ di²] / [n(n² – 1)]
Spearman’s coefficient is a non-parametric measure and is less affected by outliers than Pearson’s r.
当数据用等级表示时，计算等级差 di = rank(xi) – rank(yi)，代入公式：rs = 1 – [6 Σ di²] / [n(n² – 1)]。斯皮尔曼系数属于非参数度量，受异常值影响小于皮尔逊 r。
Linear Regression Equation | 线性回归方程
The least squares regression line of y on x is:
y = a + bx
where b = Σ (xi – x̄)(yi – ȳ) / Σ (xi – x̄)² and a = ȳ – b x̄.
The line can be used to predict values of y for given x, but extrapolation beyond the data range should be handled with caution.
y 对 x 的最小二乘回归直线为 y = a + bx，其中 b = Σ (xi – x̄)(yi – ȳ) / Σ (xi – x̄)²，a = ȳ – b x̄。该直线可用于由 x 预测 y，但外推至数据范围外需谨慎。
9. Sampling and Standard Error | 抽样与标准误
When a sample is taken from a population, statistics calculated from the sample (e.g. the sample mean) vary from sample to sample. The standard error quantifies this variability.
当从总体中抽样时，由样本计算的统计量（
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