📚 GCSE CCEA Statistics: Formula and Theorem Quick Reference Handbook | GCSE CCEA 统计：公式定理速查手册
This handbook provides a concise collection of the key formulae, theorems, and definitions required for the GCSE CCEA Statistics examination. Each section presents the essential concepts with clear notation, enabling quick revision and confident application in data handling, probability, and inference.
本手册简要整理了 GCSE CCEA 统计考试所需的关键公式、定理和定义。每个小节都以清晰的符号呈现核心概念，方便快速复习并在数据处理、概率和推断中自信应用。
1. Measures of Central Tendency | 集中趋势度量
The sample mean for ungrouped data is calculated as the sum of all observations divided by the number of observations.
未分组数据的样本均值通过所有观测值之和除以观测值个数来计算。
x̄ = Σx / n
For grouped data, the mean is estimated using the midpoint of each class interval.
对于分组数据，均值通过每个组区间的中点来估计。
x̄ = Σ(f × m) / Σf
The median is the middle value when data are ordered. For ungrouped data with n odd, it is the (n+1)/2 th value. For grouped data, it is found using interpolation.
中位数是数据排序后的中间值。对于奇数个未分组数据，它是第 (n+1)/2 个值。对于分组数据，使用插值法求得。
The mode is the most frequent value or the modal class in a frequency table.
众数是出现频率最高的值，或频数表中的众数组。
2. Measures of Dispersion | 离差度量
The range is the difference between the largest and smallest values: Range = max – min.
全距是最大值与最小值之差：全距 = 最大值 – 最小值。
The interquartile range (IQR) is Q₃ – Q₁ and describes the spread of the middle 50% of the data.
四分位距 (IQR) 为 Q₃ – Q₁，描述中间 50% 数据的离散程度。
The sample variance is the average of the squared deviations from the mean. For ungrouped data:
样本方差是各数据与均值之差的平方的平均值。对于未分组数据：
s² = Σ(x – x̄)² / (n – 1)
The standard deviation is the square root of the variance:
标准差是方差的平方根：
s = √[Σ(x – x̄)² / (n – 1)]
For grouped data, replace x with the midpoint m and include frequency f:
对于分组数据，用组中点 m 替代 x 并乘以频数 f：
s = √[Σf(m – x̄)² / (Σf – 1)]
3. Quartiles and Percentiles | 四分位数与百分位数
Quartiles divide ordered data into four equal parts. Q₁ (lower quartile) is at position (n+1)/4, Q₂ (median) at (n+1)/2, and Q₃ (upper quartile) at 3(n+1)/4.
四分位数将排序后的数据分成四等份。Q₁（下四分位数）位于 (n+1)/4 位置，Q₂（中位数）位于 (n+1)/2，Q₃（上四分位数）位于 3(n+1)/4。
Percentiles split the data into 100 parts. The k-th percentile is the value below which k% of observations fall. Pₖ is found at position k(n+1)/100.
百分位数将数据分成 100 等份。第 k 百分位数是 k% 的观测值低于它的值。Pₖ 的位置为 k(n+1)/100。
For grouped data, linear interpolation is used to estimate quartiles and percentiles within class intervals.
对于分组数据，使用线性插值法估计组区间内的四分位数和百分位数。
4. Probability Basics | 概率基础
The probability of an event A is P(A) = number of favourable outcomes / total number of outcomes, provided all outcomes are equally likely.
事件 A 的概率为 P(A) = 有利结果数 / 总结果数，前提是所有结果等可能。
The complement rule: P(not A) = 1 – P(A).
补集规则：P(非 A) = 1 – P(A)。
For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
对于互斥事件 A 和 B，P(A 或 B) = P(A) + P(B)。
For independent events, P(A and B) = P(A) × P(B). Conditional probability is given by:
对于独立事件，P(A 且 B) = P(A) × P(B)。条件概率公式为：
P(A|B) = P(A and B) / P(B)
Tree diagrams and Venn diagrams are powerful tools for calculating combined probabilities.
树形图和韦恩图是计算复合概率的强有力工具。
5. Binomial Distribution | 二项分布
A binomial distribution arises when there are a fixed number n of independent trials, each with the same probability of success p.
当有固定次数 n 的独立试验，且每次试验的成功概率 p 相同时，出现二项分布。
The probability of exactly x successes is given by:
恰好获得 x 次成功的概率由下式给出：
P(X = x) = ⁿCₓ pˣ (1 – p)ⁿ⁻ˣ
The mean of a binomial distribution is μ = np, and the variance is σ² = np(1-p).
二项分布的均值为 μ = np，方差为 σ² = np(1-p)。
In GCSE, you may use tables or a calculator to find binomial probabilities.
在 GCSE 中，你可以使用表格或计算器求二项概率。
6. Normal Distribution | 正态分布
The normal distribution is a continuous, symmetric, bell-shaped curve defined by its mean μ and standard deviation σ.
正态分布是以均值 μ 和标准差 σ 为参数的连续、对称的钟形曲线。
About 68% of data lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ (the empirical rule).
约 68% 的数据在均值 ±1σ 内，95% 在 ±2σ 内，99.7% 在 ±3σ 内（经验法则）。
The standard normal distribution has μ = 0 and σ = 1. Any normal variable X can be standardised to a z-score:
标准正态分布的均值 μ = 0，标准差 σ = 1。任何正态变量 X 都可以标准化为 z 分数：
z = (X – μ) / σ
Probability statements for normal variables are solved using z-tables or technology.
正态变量的概率计算可通过 z 表或技术工具求解。
7. Correlation and Regression | 相关与回归
The product-moment correlation coefficient (PMCC), denoted by r, measures the strength and direction of a linear relationship between two variables.
积矩相关系数 (PMCC) 记作 r，衡量两个变量间线性关系的强度和方向。
r = Sₓᵧ / √(Sₓₓ Sᵧᵧ)
where Sₓᵧ = Σxy – (Σx Σy)/n, Sₓₓ = Σx² – (Σx)²/n, Sᵧᵧ = Σy² – (Σy)²/n.
其中 Sₓᵧ = Σxy – (Σx Σy)/n，Sₓₓ = Σx² – (Σx)²/n，Sᵧᵧ = Σy² – (Σy)²/n。
The equation of the least-squares regression line of y on x is:
y 关于 x 的最小二乘回归直线方程为：
y = a + bx, where b = Sₓᵧ / Sₓₓ and a = ȳ – b x̄
Spearman’s rank correlation coefficient is used when data are ordinal or not normally distributed.
当数据为顺序型或非正态分布时，使用斯皮尔曼等级相关系数。
8. Hypothesis Testing | 假设检验
A hypothesis test assesses evidence about a population parameter. The null hypothesis H₀ is assumed true unless data provide sufficient evidence to reject it in favour of the alternative H₁.
假设检验评估关于总体参数的证据。零假设 H₀ 被假定为真，除非数据提供足够证据支持备择假设 H₁ 而拒绝 H₀。
The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming H₀ is true. If p-value < significance level (commonly 0.05), H₀ is rejected.
p 值是假定 H₀ 为真时，获得至少与观测值一样极端的检验统计量的概率。若 p 值 < 显著性水平（常用 0.05），则拒绝 H₀。
For a binomial test, compare the observed number of successes with the critical region or calculate the exact binomial probability. For normal tests, compare the z-statistic to critical values (e.g., ±1.96 for a two-tailed test at 5%).
对于二项检验，将观测成功次数与临界区域比较或计算精确的二项概率。对于正态检验，将 z 统计量与临界值比较（例如 5% 双侧检验的临界值为 ±1.96）。
9. Index Numbers | 指数
An index number measures the relative change in a quantity, such as price or production, compared with a base period. The base period index is usually 100.
指数衡量某一数量（如价格或产量）相对于基期的相对变化。基期指数通常为 100。
Simple price relative: (Current Price / Base Price) × 100.
简单价格比率：(现价 / 基价) × 100。
Weighted aggregate indices, such as the Laspeyres and Paasche indices, use quantities or expenditures as weights. The Laspeyres price index uses base-period quantities as weights:
加权综合指数，如拉氏指数和帕氏指数，使用数量或支出作为权重。拉氏价格指数以基期数量为权重：
L = [Σ(pₙ × q₀) / Σ(p₀ × q₀)] × 100
10. Time Series and Moving Averages | 时间序列与移动平均
A time series is a set of observations recorded over time. Its components are trend, seasonal variation, cyclical fluctuation, and random variation.
时间序列是随时间记录的一组观测值。其组成包括趋势、季节变动、循环波动和随机变动。
Moving averages smooth out short-term fluctuations to reveal the underlying trend. For quarterly data, a four-point moving average is often used; for weekly data, a seven-point average is common.
移动平均可平滑短期波动以揭示潜在趋势。对于季度数据，常用四点移动平均；对于周数据，常用七点移动平均。
The average seasonal variation can be estimated by subtracting the trend from actual values and averaging over corresponding periods.
平均季节变动可通过从实际值中减去趋势值并对相应时期求平均来估计。
11. Standardised Scores (z-scores) | 标准化分数（z 分数）
A z-score indicates how many standard deviations an observation x lies above or below the mean. It is essential for comparing values from different distributions.
z 分数表示一个观测值 x 距离均值上下多少个标准差。它对于比较不同分布的值至关重要。
z = (x – μ) / σ
In practice, when μ and σ are unknown, use the sample mean and sample standard deviation.
实际中当 μ 和 σ 未知时，使用样本均值和样本标准差。
A positive z-score means the value is above the mean; a negative z-score means below. z-scores enable the use of standard normal tables to find probabilities.
正 z 分数表示值高于均值；负 z 分数表示低于均值。z 分数使得可以使用标准正态表求概率。
12. Important Statistical Diagrams | 重要统计图表
Box-and-whisker plots display the five-number summary: minimum, Q₁, median, Q₃, and maximum. They are excellent for comparing distributions and identifying outliers.
箱线图展示五数概括：最小值、Q₁、中位数、Q₃ 和最大值。它们非常适合比较分布和识别异常值。
Cumulative frequency curves are used to estimate medians, quartiles, and percentiles. The points are plotted at the upper class boundaries.
累积频数曲线用于估计中位数、四分位数和百分位数。各点绘制在上组界处。
Histograms represent the distribution of grouped data, where the area of each bar is proportional to its frequency. Frequency density = Frequency / Class width.
直方图表示分组数据的分布，其中每个条形的面积与其频数成正比。频数密度 = 频数 / 组距。
Scatter graphs visually show the relationship between two variables and are the first step in correlation analysis.
散点图直观显示两变量间的关系，是相关分析的第一步。
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