📚 Year 11 CIE Statistics: Formula & Theorem Quick Reference | CIE 十一年级统计：公式定理速查手册
This handbook compiles the essential formulas, theorems, and key concepts required for the Year 11 CIE Statistics syllabus. Each topic is presented with clear English explanations immediately followed by their Chinese equivalents, making it ideal for quick revision and bilingual learners. Use this as your go-to reference for mastering statistical calculations, probability, and data analysis.
本手册汇编了 CIE 十一年级统计课程所需的基本公式、定理和核心概念。每个主题均以清晰的英文解释紧接着中文对应内容呈现，非常适合快速复习和双语学习者。将它作为掌握统计计算、概率和数据分析的首选参考。
1. Measures of Central Tendency | 集中趋势的度量
For ungrouped data, the arithmetic mean is the sum of all values divided by the number of values:
x̄ = ∑x / n
对于未分组数据，算术平均数是所有数值之和除以数值的个数：
x̄ = ∑x / n
For grouped data, the estimated mean uses midpoints (m) and frequencies (f):
x̄ = ∑fm / ∑f
对于分组数据，估计平均数使用组中值 (m) 和频数 (f)：
x̄ = ∑fm / ∑f
The median is the middle value when data are ordered. For ungrouped data with n values, position of median = (n+1)/2.
中位数是数据排序后位于中间的数值。对于未分组数据，中位数的位置为 (n+1)/2。
For grouped data, median = L + [(n/2 – CF)/f] × w, where L is the lower class boundary of the median group, CF is the cumulative frequency before the median group, f is the frequency of the median group, and w is the class width.
对于分组数据，中位数 = L + [(n/2 – CF)/f] × w，其中 L 为中位数组的下组界，CF 为低于中位数组的累积频数，f 为中位数组的频数，w 为组距。
The mode is the value that occurs most frequently. In grouped data, the modal class is the class with the highest frequency density.
众数是出现频率最高的数值。在分组数据中，众数组是频数密度最高的组。
2. Measures of Dispersion | 离散程度的度量
The range is the difference between the largest and smallest values:
Range = max – min
极差是最大值与最小值之差：
极差 = 最大值 – 最小值
The interquartile range is the difference between the upper quartile (Q₃) and lower quartile (Q₁):
IQR = Q₃ – Q₁
四分位距是上四分位数 (Q₃) 与下四分位数 (Q₁) 之差：
IQR = Q₃ – Q₁
For ungrouped data, the variance and standard deviation are given by:
s² = ∑(x – x̄)² / (n – 1)
s = √[∑(x – x̄)² / (n – 1)]
对于未分组数据，方差和标准差由以下公式给出：
s² = ∑(x – x̄)² / (n – 1)
s = √[∑(x – x̄)² / (n – 1)]
For grouped data, the estimated variance uses midpoints:
s² = ∑f(m – x̄)² / (∑f – 1)
对于分组数据，估计方差使用组中值：
s² = ∑f(m – x̄)² / (∑f – 1)
3. Frequency Distributions & Histograms | 频数分布与直方图
Frequency density is used when class widths are unequal to ensure the area of each bar is proportional to frequency.
Frequency density = Frequency / Class width
当组距不等时，使用频数密度以确保每个直条的面积与频数成正比。
频数密度 = 频数 / 组距
In a histogram, frequency is represented by the area of the bar:
Frequency = Frequency density × Class width
在直方图中，频数由直条的面积表示：
频数 = 频数密度 × 组距
When drawing a histogram, always label the vertical axis as ‘Frequency density’ and the horizontal axis with the variable and continuous scale.
绘制直方图时，始终将纵轴标记为“频数密度”，横轴使用变量及连续刻度。
4. Cumulative Frequency & Quartiles | 累积频数与四分位数
Cumulative frequency is the running total of frequencies. It is plotted against the upper class boundary to form a cumulative frequency curve (ogive).
累积频数是频数的累积总和，相对于上组界绘制形成累积频数曲线（拱形图）。
Quartiles are estimated from the cumulative frequency graph:
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Lower quartile Q₁: value at 25% of total frequency (n/4).
下四分位数 Q₁：占总频数 25% (n/4) 处的数值。
-
Median Q₂: value at 50% of total frequency (n/2).
中位数 Q₂：占总频数 50% (n/2) 处的数值。
-
Upper quartile Q₃: value at 75% of total frequency (3n/4).
上四分位数 Q₃：占总频数 75% (3n/4) 处的数值。
The interquartile range is Q₃ – Q₁, and the semi-interquartile range is (Q₃ – Q₁)/2.
四分位距为 Q₃ – Q₁，半四分位距为 (Q₃ – Q₁)/2。
Percentiles can also be read from the same curve; the k-th percentile is the value below which k% of the data fall.
百分位数也可从同一曲线读出；第 k 百分位数是有 k% 的数据低于其值的数值。
5. Probability Fundamentals | 概率基础
Probability of an event A is the ratio of favourable outcomes to total possible outcomes, assuming equally likely outcomes:
P(A) = n(A) / n(S)
事件 A 的概率是有利结果数与所有可能结果总数之比（假定等可能结果）：
P(A) = n(A) / n(S)
For any event A, 0 ≤ P(A) ≤ 1. The sum of probabilities of all possible mutually exclusive outcomes is 1.
对于任意事件 A，0 ≤ P(A) ≤ 1。所有可能互斥结果的概率之和为 1。
The probability that A does not occur (complement) is:
P(A’) = 1 – P(A)
A 不发生的概率（补集）为：
P(A’) = 1 – P(A)
For mutually exclusive events A and B, the probability that A or B occurs:
P(A ∪ B) = P(A) + P(B)
对于互斥事件 A 和 B，A 或 B 发生的概率：
P(A ∪ B) = P(A) + P(B)
For non-mutually exclusive events, we subtract the intersection:
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
对于非互斥事件，需减去交集的概率：
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
6. Conditional Probability & Tree Diagrams | 条件概率与树状图
Conditional probability of A given B is defined as:
P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.
在给定 B 发生的条件下 A 的条件概率定义为：
P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。
Events A and B are independent if and only if:
P(A ∩ B) = P(A) × P(B)
或者等价的 P(A | B) = P(A)。事件 A 和 B 独立当且仅当：
P(A ∩ B) = P(A) × P(B)
或等价的 P(A | B) = P(A)。
Tree diagrams help model sequential events. Multiply probabilities along branches for ‘and’, add probabilities of different branches for ‘or’.
树状图有助于模拟顺序事件。沿分支相乘概率得到“且”的概率，不同分支相加得到“或”的概率。
When sampling without replacement, probabilities change at each step; the branches must be updated accordingly.
当无放回抽样时，每一步概率发生变化；分支必须相应更新。
7. Permutations & Combinations | 排列与组合
The factorial of a positive integer n is:
n! = n × (n – 1) × (n – 2) × … × 3 × 2 × 1
正整数 n 的阶乘为：
n! = n × (n – 1) × (n – 2) × … × 3 × 2 × 1
The number of permutations (arrangements) of r objects chosen from n distinct objects is:
ⁿPᵣ = n! / (n – r)!
从 n 个不同对象中选取 r 个的排列（有序安排）数为：
ⁿPᵣ = n! / (n – r)!
The number of combinations (selections) of r objects from n distinct objects, where order does not matter, is:
ⁿCᵣ = n! / [r! (n – r)!]
从 n 个不同对象中选取 r 个的组合（无序选择）数为：
ⁿCᵣ = n! / [r! (n – r)!]
Useful properties: nC₀ = 1, nCₙ = 1, nCᵣ = nCₙ₋ᵣ. Also, permutations with repeated items: total permutations of n items with n₁ identical of one type, n₂ of another, … is n!/(n₁! n₂! …).
常用性质：ⁿC₀ = 1，ⁿCₙ = 1，ⁿCᵣ = ⁿCₙ₋ᵣ。另外，含有重复项的排列：n 个物品中有 n₁ 个相同的第一类，n₂ 个相同的第二类……，总排列数为 n!/(n₁! n₂! …)。
8. Scatter Graphs & Correlation | 散点图与相关性
Scatter diagrams display the relationship between two variables. Correlation may be positive, negative, or zero.
散点图展示两个变量之间的关系。相关性可以是正相关、负相关或零相关。
Spearman’s rank correlation coefficient is used when data are ranked or non-linear:
rₛ = 1 – [6∑d²] / [n(n² – 1)]
where d is the difference between the ranks of each pair, and n is the number of pairs.
斯皮尔曼等级相关系数用于有序或非线性数据：
rₛ = 1 – [6∑d²] / [n(n² – 1)]
其中 d 为每对数据的秩次差，n 为数据对的数量。
The coefficient rₛ ranges from -1 (perfect negative) to +1 (perfect positive). A value near 0 suggests no rank correlation.
系数 rₛ 的取值范围从 -1（完全负相关）到 +1（完全正相关）。接近 0 表示无等级相关。
For Pearson’s product-moment correlation coefficient (usually given in exams if required), it measures linear correlation between quantitative variables.
皮尔逊积矩相关系数（考试中通常会给出公式）衡量定量变量间的线性相关程度。
9. Discrete Random Variables & Expectation | 离散随机变量与期望
A discrete random variable X takes values x₁, x₂, … with probabilities P(X = xᵢ) = pᵢ, where ∑pᵢ = 1.
离散随机变量 X 取值为 x₁, x₂, …，其概率为 P(X = xᵢ) = pᵢ，且 ∑pᵢ = 1。
The expected value (mean) of X is:
E(X) = μ = ∑ [xᵢ · pᵢ]
X 的期望值（均值）为：
E(X) = μ = ∑ [xᵢ · pᵢ]
The variance of X can be computed using:
Var(X) = ∑ [(xᵢ – μ)² pᵢ] = E(X²) – [E(X)]²
X 的方差可通过以下公式计算：
Var(X) = ∑ [(xᵢ – μ)² pᵢ] = E(X²) – [E(X)]²
For any constants a and b, E(aX + b) = aE(X) + b, and Var(aX + b) = a² Var(X).
对于任意常数 a 和 b，E(aX + b) = aE(X) + b，且 Var(aX + b) = a² Var(X)。
The standard deviation of X is σ = √Var(X).
X 的标准差为 σ = √Var(X)。
10. Sampling Methods & Estimation | 抽样方法与估计
A simple random sample ensures every member of the population has an equal chance of being selected. Other methods include stratified sampling, systematic sampling, and quota sampling.
简单随机抽样确保总体中每个个体被选中的概率相等。其他方法包括分层抽样、系统抽样和配额抽样。
In stratified sampling, the population is divided into groups (strata) and random samples are taken from each stratum proportionally to its size.
在分层抽样中，总体被划分为若干层（层），然后按各层大小比例从每层中随机抽取样本。
The sample mean x̄ is an unbiased estimator of the population mean μ. For a sample of size n from a large population, the standard error of the mean is σ/√n.
样本均值 x̄ 是总体均值 μ 的无偏估计量。对于来自大总体的容量为 n 的样本，样本均值的标准误为 σ/√n。
A confidence interval for the population mean (when σ known) is:
x̄ ± z × (σ/√n)
总体均值的置信区间（当 σ 已知时）为：
x̄ ± z × (σ/√n)
where z is the critical value from the normal distribution (e.g. 1.96 for 95% confidence).
其中 z 是标准正态分布的临界值（例如 95% 置信水平下 z = 1.96）。
11. Index Numbers | 指数
An index number measures the change in a variable over time relative to a base period. Simple price index:
Price relative = (Current price / Base period price) × 100
指数用于衡量变量相对于基期的变化。简单价格指数：
价比 = (当前价格 / 基期价格) × 100
A weighted aggregate index, such as the Laspeyres index, uses base-period quantities as weights:
Laspeyres Price Index = (∑ pₙ q₀ / ∑ p₀ q₀) × 100
加权综合指数，如拉氏指数，以基期数量为权重：
拉氏价格指数 = (∑ pₙ q₀ / ∑ p₀ q₀) × 100
The Paasche index uses current-period quantities:
Paasche Price Index = (∑ pₙ qₙ / ∑ p₀ qₙ) × 100
帕氏指数以当前期数量为权重：
帕氏价格指数 = (∑ pₙ qₙ / ∑ p₀ qₙ) × 100
Chain index numbers link successive period comparisons to form a continuous series.
链式指数将各期环比相连，形成连续序列。
12. Time Series & Moving Averages | 时间序列与移动平均
A time series is a set of observations recorded in chronological order. It may contain trend, seasonal variation, cyclical fluctuation, and random residual.
时间序列是按时间顺序记录的一系列观测值。它可能包含趋势、季节变动、循环波动和随机残差。
Moving averages smooth out short-term fluctuations to reveal the underlying trend. For a period of length k (odd), the centred moving average is:
k-point moving average = (xₜ₋₍ₖ₋₁₎/₂ + … + xₜ + … + xₜ₊₍ₖ₋₁₎/₂) / k
移动平均可平滑短期波动，揭示潜在趋势。对于长度为 k 的奇数期，中心化的移动平均为：
k 点移动平均 = (xₜ₋₍ₖ₋₁₎/₂ + … + xₜ + … + xₜ₊₍ₖ₋₁₎/₂) / k
If k is even, a 2-step centreing is needed. The seasonal variation can be estimated by subtracting the trend from the original data.
若 k 为偶数，则需要两步中心化。季节变动可通过从原始数据中减去趋势值来估计。
Additive model: Data = Trend + Seasonal + Residual. The seasonal components sum to zero over a complete cycle.
加法模型：数据 = 趋势 + 季节 + 残差。一个完整周期内季节成分之和为零。
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