📚 Year 11 Edexcel Statistics: Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This handbook brings together the essential formulas, definitions, and statistical theorems required for the Year 11 Edexcel Statistics course. Use it as a quick-look reference while revising topics from data handling to probability distributions and time series analysis.
本手册汇集了 Year 11 Edexcel 统计课程所需的核心公式、定义和定理。在你复习从数据处理到概率分布以及时间序列分析等主题时，可将它作为快速查阅的参考资料。
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.
对于未分组数据，算术平均是所有数值之和除以数值的个数。
Mean μ = Σx / n
When working with a frequency table the mean becomes Σfx / Σf, where x represents each distinct value or the class midpoint and f is the frequency.
当使用频率表时，均值变为 Σfx / Σf，其中 x 代表每个不同的数值或组中值，f 为频数。
The median is the middle value when data are ordered. For a frequency distribution, find the position (n+1)/2 and then locate the value in the cumulative frequency column or diagram.
中位数是将数据排序后位于中间的数值。对于频数分布，先求出位置 (n+1)/2，再在累积频数列或图中找出对应的数值。
The mode is the value (or class interval) with the highest frequency.
众数是出现频数最高的数值（或组距）。
2. Measures of Dispersion | 离散程度的度量
Range = maximum value – minimum value. It is a crude measure affected by extreme values.
极差 = 最大值 – 最小值。它是一个易受极端值影响的粗略度量。
The interquartile range (IQR) is the difference between the upper and lower quartiles: IQR = Q₃ – Q₁. It describes the spread of the middle 50% of the data.
四分位距 (IQR) 是上四分位数与下四分位数之差：IQR = Q₃ – Q₁。它描述了中间 50% 数据的分布范围。
For a set of n values, the variance and standard deviation measure the average squared deviation from the mean.
对于一组 n 个数值，方差和标准差衡量各个数值与均值之差的平方的平均值。
Population variance σ² = Σ(x – μ)² / N
population standard deviation σ = √[Σ(x – μ)² / N]
样本方差 s² = Σ(x – x̄)² / (n – 1)
样本标准差 s = √[Σ(x – x̄)² / (n – 1)]
A convenient computing formula for the standard deviation is s = √[ Σx²/n – (Σx/n)² ] when dealing with a full population or sample divisor adjusted accordingly.
计算标准差的一个便捷公式是 s = √[ Σx²/n – (Σx/n)² ]，处理总体时用 N，样本时除数做相应调整。
3. Basic Probability Rules | 概率基本法则
The probability of an event A, written P(A), satisfies 0 ≤ P(A) ≤ 1. The probabilities of all mutually exclusive and exhaustive outcomes sum to 1.
事件 A 的概率记为 P(A)，满足 0 ≤ P(A) ≤ 1。所有互斥且完备的结局概率之和为 1。
The addition rule for mutually exclusive events: P(A ∪ B) = P(A) + P(B). For events that are not mutually exclusive the general addition rule applies.
互斥事件的加法法则：P(A ∪ B) = P(A) + P(B)。对于非互斥的事件，使用一般加法法则。
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Two events A and B are independent if P(A ∩ B) = P(A) × P(B).
若 P(A ∩ B) = P(A) × P(B)，则事件 A 与 B 相互独立。
4. Conditional Probability & Tree Diagrams | 条件概率与树形图
Conditional probability expresses the likelihood of A occurring given that B has already occurred.
条件概率表示在事件 B 已经发生的条件下事件 A 发生的可能性。
P(A|B) = P(A ∩ B) / P(B) , P(B) > 0
Tree diagrams are used to map out sequences of events. Multiply along the branches for ‘and’ probabilities; add the final outcomes for combined probabilities.
树形图用于梳理事件序列。沿分支相乘得到“且”的概率；将最终结果相加得到组合概率。
When sampling without replacement, the probabilities on successive branches change. For conditional problems, updating the denominator using the given condition is essential.
在不放回抽样中，后续分支上的概率会发生变化。解决条件概率问题时，关键是根据给定条件更新分母。
5. Binomial Distribution | 二项分布
A binomial distribution models the number of successes in n independent trials, each with the same probability of success p.
二项分布描述了在 n 次独立试验中成功的次数，每次试验的成功概率 p 相同。
P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ , r = 0,1,…,n
Mean and variance of a binomial random variable X ~ B(n, p) are:
二项随机变量 X ~ B(n, p) 的均值与方差为：
E(X) = μ = np
Var(X) = σ² = np(1 – p)
Tables or technology are often used to find cumulative probabilities P(X ≤ r). You can also use the symmetry property P(X ≥ r) = 1 – P(X ≤ r – 1).
通常使用表格或技术工具求累积概率 P(X ≤ r)。也可以利用对称性 P(X ≥ r) = 1 – P(X ≤ r – 1)。
6. 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 total area under the curve equals 1.
正态分布是一种由均值 μ 和标准差 σ 确定的连续型钟形分布。曲线下的总面积等于 1。
To find probabilities, values are converted to the standard normal variable Z ~ N(0, 1) using the z-score formula.
为求概率，使用 z 分数公式将数值转化为标准正态变量 Z ~ N(0, 1)。
z = (x – μ) / σ
Approximately 68% of data lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. For inverse problems, work backwards from the probability to the z-score then to the raw score.
大约 68% 的数据落在均值 ±1σ 内，95% 在 ±2σ 内，99.7% 在 ±3σ 内。对于逆向查找问题，由概率反查 z 分数，再得到原始数值。
7. Correlation & Regression | 相关与回归
Scatter diagrams show the relationship between two variables. The strength and direction are often summarised with Pearson’s product‑moment correlation coefficient r, or Spearman’s rank coefficient when data are ranked.
散点图展示两个变量之间的关系。其强度和方向通常用皮尔逊积矩相关系数 r 加以概括，当数据以等级呈现时使用斯皮尔曼等级相关系数。
Spearman’s rank: rₛ = 1 – 6Σd² / [n(n² – 1)]
where d is the difference between ranks for each pair. The value lies between –1 and +1.
其中 d 是每对观测的秩次之差。系数取值在 –1 到 +1 之间。
The equation of a linear regression line is y = a + bx, where the slope b is given by Sxy / Sxx and the intercept a = ȳ – b x̄.
线性回归线的方程为 y = a + bx，其中斜率 b = Sxy / Sxx，截距 a = ȳ – b x̄。
b = Σ(x – x̄)(y – ȳ) / Σ(x – x̄)²
a = ȳ – b x̄
8. Data Representation & Charts | 数据表示与图表
Histograms display frequency densities for continuous grouped data. The frequency density is calculated as frequency ÷ class width, and the area of each bar is proportional to the frequency.
直方图显示连续分组数据的频率密度。频率密度 = 频数 ÷ 组距，每个矩形的面积与频数成正比。
Frequency density = frequency / class width
Cumulative frequency diagrams (ogives) are used to estimate medians, quartiles, and percentiles. A box plot (box‑and‑whisker) shows the minimum, Q₁, median, Q₃, and maximum, and can be used to identify outliers.
累积频数图（肩形图）用于估计中位数、四分位数和百分位数。箱线图（盒须图）显示最小值、Q₁、中位数、Q₃ 和最大值，可用于识别异常值。
Outlier boundaries are often defined as 1.5 × IQR below Q₁ or above Q₃. Values outside these fences are potential outliers.
异常值的界限通常定义为低于 Q₁ – 1.5×IQR 或高于 Q₃ + 1.5×IQR。超出这些界限的点可能为异常值。
9. Sampling Methods | 抽样方法
Common sampling techniques include simple random, stratified, systematic, cluster, and quota sampling. Each has its own advantages and sources of bias.
常见的抽样技术包括简单随机抽样、分层抽样、系统抽样、整群抽样和配额抽样。每种方法各有其优点和偏误来源。
In stratified sampling the population is divided into distinct groups (strata) and a random sample is taken from each stratum, often proportionally. This ensures representation of all subgroups.
在分层抽样中，总体被划分为不同组别（层），并从每层中（常按比例）随机抽取样本。这确保所有子群都有代表性。
The capture‑recapture method estimates an unknown population size N. A sample of M individuals is captured, marked, and released. Later, a second sample of size C is taken and the number of marked individuals R is recorded.
捕获－再捕获法用于估计未知的总体规模 N。首先捕获 M 个个体，做标记后放回。随后再抽取一个容量为 C 的样本，记录其中带有标记的个体数 R。
N = (M × C) / R
This assumes the marked individuals mix evenly and that no births, deaths, or migrations occur.
该方法假设有标记的个体混合均匀，且没有出生、死亡或迁入迁出。
10. Index Numbers & Time Series | 指数与时间序列
An index number compares the value of a variable at a given time with its value at a base period. The base period index is usually set to 100.
指数是将某一变量在给定时间的取值与基期取值进行比较的数值。基期指数通常设为 100。
Simple price index = (current price / base price) × 100
Weighted aggregate indices, such as a Retail Price Index, use ‘weights’ to reflect the relative importance of different items.
加权综合指数（如零售价格指数）利用“权重”来反映不同项目的相对重要性。
A time series consists of data recorded at successive points in time. It can be decomposed into trend, seasonal, cyclic, and random components. A moving average smooths short‑term fluctuations to reveal the underlying trend.
时间序列由在连续时点记录的数据组成。它可以分解为趋势、季节、循环和随机成分。移动平均能平滑短期波动，揭示潜在的趋势。
k‑point moving average = (sum of k consecutive values) / k
When the moving average is centred, it aligns with the middle time period, allowing the calculation of seasonal variation as actual value – trend value.
当移动平均居中时，它与中间的时间点对齐，从而可以计算季节变动：实际值 – 趋势值。
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