📚 GCSE WJEC Statistics: Formula & Theorems Quick Reference Handbook | GCSE WJEC 统计：公式定理速查手册
This quick reference handbook covers the essential formulae and theorems you need to master for the GCSE WJEC Statistics examination. Every topic is presented with clear English and Chinese explanations, ensuring you can review key concepts efficiently.
本速查手册涵盖 GCSE WJEC 统计学考试必须掌握的核心公式和定理。每个主题均配有清晰的中英文解释，帮助你高效复习关键概念。
1. Measures of Central Tendency | 集中趋势的测量
Measures of central tendency give a single value that represents the centre of a data set. The three main measures are the mean, median, and mode.
集中趋势的测量用一个数值代表数据集的中心。三种主要测量是平均数、中位数和众数。
Sample Mean: x = Σx / n
The symbol x (x-bar) denotes the sample mean. Add all data values and divide by the number of values n.
符号 x 表示样本平均数。将所有数据值相加，再除以数据个数 n。
Population Mean: μ = Σx / N
For a population, the mean is denoted by the Greek letter μ (mu) and N is the total population size.
对于总体，平均数用希腊字母 μ 表示，N 是总体大小。
Median
The median is the middle value when data are ordered. If n is odd, the median is the (n+1)/2 th value. If n is even, it is the average of the n/2 th and (n/2 + 1) th values.
中位数是数据排序后位于中间的值。如果 n 为奇数，中位数为第 (n+1)/2 个值；如果 n 为偶数，则为第 n/2 个与第 (n/2 + 1) 个值的平均数。
Mode
The mode is the most frequently occurring value in a data set. A set may have one mode, more than one mode (bimodal or multimodal), or no mode at all.
众数是数据集中出现频率最高的值。一组数据可能有一个众数、多个众数（双峰或多峰）或没有众数。
2. Measures of Dispersion | 离散程度的测量
Dispersion measures describe how spread out the data values are. They include the range, interquartile range, variance, and standard deviation.
离散程度测量描述数据值的分散程度，包括极差、四分位距、方差和标准差。
Range = Maximum value – Minimum value
The range is the simplest measure of spread. It only uses the two extreme values, so it is sensitive to outliers.
极差是最简单的离散量数，仅使用两个极端值，因此对异常值敏感。
Interquartile Range (IQR) = Q3 – Q1
Q1 is the lower quartile (25th percentile) and Q3 is the upper quartile (75th percentile). The IQR is not affected by extreme values.
Q1 是下四分位数（第 25 百分位数），Q3 是上四分位数（第 75 百分位数）。IQR 不受极端值影响。
Population Variance: σ² = Σ(x – μ)² / N
The population variance is the average of the squared deviations from the mean. σ is the population standard deviation.
总体方差是各数值与总体平均数之差的平方的平均数。σ 是总体标准差。
Sample Variance: s² = Σ(x – x)² / (n – 1)
For a sample, we divide by n-1 to get an unbiased estimate of the population variance. The sample standard deviation is s = √s².
对于样本，除以 n-1 可以得到总体方差的无偏估计。样本标准差为 s = √s²。
3. Basic Probability Rules | 基本概率规则
Probability measures the chance that an event will occur. It is always a number between 0 and 1 (or 0% and 100%).
概率衡量事件发生的可能性，值总是介于 0 和 1（或 0% 和 100%）之间。
P(A) = Number of favourable outcomes / Total number of outcomes
For equally likely outcomes, the probability of event A is the ratio of favourable outcomes to total outcomes.
对于等可能结果，事件 A 的概率等于有利结果数与所有可能结果总数之比。
Complement Rule: P(A’) = 1 – P(A)
A’ (A-prime) is the event that A does not occur. The sum of the probabilities of an event and its complement is 1.
A’ 表示事件 A 不发生。一个事件与其互补事件的概率之和为 1。
Addition Rule: P(A or B) = P(A) + P(B) – P(A and B)
For any two events, add their individual probabilities and subtract the probability of both occurring to avoid double-counting. If A and B are mutually exclusive, P(A and B) = 0.
对任意两事件，将各自概率相加后减去两者同时发生的概率以免重复计算。若 A 与 B 互斥，则 P(A and B) = 0。
Multiplication Rule for Independent Events: P(A and B) = P(A) x P(B)
Two events are independent if the occurrence of one does not affect the probability of the other. Multiply their probabilities.
如果两个事件的发生互不影响，则为独立事件。将它们的概率相乘。
Conditional Probability: P(A|B) = P(A and B) / P(B)
The probability of A given B is the joint probability divided by the probability of B. This works when P(B) > 0.
给定 B 时 A 的概率等于联合概率除以 B 的概率，前提是 P(B) > 0。
4. Permutations and Combinations | 排列与组合
Combinatorics helps count the number of ways of arranging or selecting objects. The two key tools are factorial notation, permutations, and combinations.
组合数学用于计算排列或选取对象的不同方式数目。两个关键工具是阶乘、排列和组合。
Factorial: n! = n x (n-1) x (n-2) x … x 3 x 2 x 1
n! (n factorial) is the product of all positive integers up to n. By definition, 0! = 1.
n! 表示从 1 到 n 的所有正整数的乘积。定义 0! = 1。
Permutations: &nPr = n! / (n – r)!
A permutation is an ordered arrangement of r objects chosen from n distinct objects. The order matters.
排列是从 n 个不同对象中有序选取 r 个的方式数，顺序很重要。
Combinations: &nCr = n! / [r!(n – r)!]
A combination is a selection of r objects from n distinct objects without regard to order. Use the binomial coefficient &nCr.
组合是从 n 个不同对象中无序选取 r 个的方式数，使用二项式系数 &nCr。
5. Binomial Distribution | 二项分布
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p.
二项分布描述了在固定次数的独立试验中成功次数的分布，每次试验的成功概率 p 相同。
Probability Mass Function: P(X = r) = &nCr x p^r x (1-p)^(n-r)
X is the random variable representing the number of successes in n trials. r = 0,1,2,…,n. q = 1-p is the probability of failure.
X 表示 n 次试验中成功的次数，r 可取 0 至 n，q = 1-p 表示失败概率。
Mean (Expected Value): E(X) = n x p
The expected number of successes is the product of the number of trials and the success probability.
成功的期望次数是试验次数与每次成功概率的乘积。
Variance: Var(X) = n x p x (1-p) = n x p x q
The variance measures the spread of the binomial distribution. The standard deviation is √(npq).
方差衡量二项分布的离散程度，标准差为 √(npq)。
6. The Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution that is symmetrical and bell-shaped. The WJEC specification requires you to use the standard normal distribution.
正态分布是一种对称的钟形连续概率分布。WJEC 大纲要求使用标准正态分布。
Standardisation: z = (x – μ) / σ
To find probabilities for any normal distribution N(μ,σ²), convert the raw score x into a z-score. The z-score tells how many standard deviations x is from the mean.
若要计算任意正态分布 N(μ,σ²) 的概率，将原始分数 x 转换为 z 分数。z 分数表示 x 距离平均数多少个标准差。
Standard Normal Distribution: N(0,1)
After standardisation, the distribution has a mean of 0 and a standard deviation of 1. Use probability tables to find the area under the curve.
标准化后，分布的均值为 0，标准差为 1。使用概率表查找曲线下的面积。
Inverse Normal: x = μ + zσ
If you know a probability (area), you can find the corresponding z-score from tables and then calculate the original x value using the inverse formula.
如果已知概率（面积），可从表中查出对应的 z 分数，然后利用反求公式算出原始值 x。
7. Correlation Analysis | 相关分析
Correlation measures the strength and direction of a linear relationship between two variables. The most common coefficient in GCSE WJEC is Spearman’s rank correlation coefficient.
相关分析测量两个变量之间线性关系的强度和方向。GCSE WJEC 考试中最常用的系数是斯皮尔曼等级相关系数。
Spearman’s Rank Correlation Coefficient: r_s = 1 – [6Σd² / n(n² – 1)]
Rank both sets of data separately (1 for smallest, n for largest). d is the difference between the ranks for each pair. n is the number of pairs. r_s ranges from -1 (perfect negative) to +1 (perfect positive).
将两组数据分别排序（1 为最小，n 为最大）。d 是每对数据的等级差，n 是配对数量。r_s 的取值范围从 -1（完全负相关）到 +1（完全正相关）。
Product Moment Correlation (Extension): r = S_xy / √(S_xx x S_yy)
Although not always required, you may see Pearson’s r. S_xy = Σxy – (Σx&Sigmay)/n, S_xx = Σx² – (Σx)²/n, and similarly for S_yy.
尽管不一定要求，你可能见到皮尔逊相关系数 r。S_xy = Σxy – (Σx&Sigmay)/n，S_xx = Σx² – (Σx)²/n，S_yy 类似。
8. Linear Regression | 线性回归
Regression analysis finds the equation of the straight line that best fits a set of paired data. The least squares regression line minimises the sum of squared vertical distances from the points to the line.
回归分析用于找出最佳拟合一组配对数据的直线方程。最小二乘回归线使各点到直线的垂直距离的平方和最小。
Equation of Regression Line: y = a + bx
a is the y-intercept, b is the gradient (slope). The variable x is the independent (explanatory) variable, y is the dependent (response) variable.
a 是 y 轴截距，b 是斜率。变量 x 为自变量（解释变量），y 为因变量（响应变量）。
Gradient: b = S_xy / S_xx
Use S_xx = Σx² – (Σx)²/n and S_xy = Σxy – (Σx&Sigmay)/n. Then b = S_xy / S_xx.
使用 S_xx = Σx² – (Σx)²/n 和 S_xy = Σxy – (Σx&Sigmay)/n，则 b = S_xy / S_xx。
Intercept: a = y – b x
Once b is found, compute the mean of y and the mean of x, then a = (mean of y) – b x (mean of x).
求得 b 后，计算 y 的平均数和 x 的平均数，则 a = (y 的平均数) – b x (x 的平均数)。
9. Index Numbers | 指数
Index numbers measure changes in a variable over time relative to a base period. They are widely used for prices, quantities, and economic indicators.
指数用于衡量变量相对于基期的变化，广泛用于价格、数量和经济指标。
Simple Price Index: Index = (Current price / Base price) x 100
The base price is set in a reference period. An index of 110 means a 10% increase from the base period.
基期价格设定在参考期。指数为 110 表示较基期上涨 10%。
Weighted Aggregate Index: Index = [Σ(weight x price relative)] / Σweight
When different items have different importance, weights are assigned. The price relative for each item is (current price / base price) x 100. Sum the weighted relatives and divide by total weight.
当不同项目重要性不同时，需分配权重。每项的价格比价 = (当前价格/基期价格) x 100。求加权比价总和，除以总权重。
10. Time Series and Moving Averages | 时间序列与移动平均
A time series is a set of observations recorded at regular time intervals. Moving averages are used to smooth out short-term fluctuations and reveal underlying trends.
时间序列是按固定时间间隔记录的一系列观测值。移动平均用于平滑短期波动、揭示潜在趋势。
Simple Moving Average (3-point or 4-point)
For a 3-point moving average, replace each value (except the first and last) with the average of that point and its two neighbours. For a 4-point, average four consecutive values, then centre them by averaging adjacent averages.
对于三点移动平均，将每个值（首尾除外）替换为该点及其前后相邻点的平均值。四点移动平均则先平均连续四个值，再通过相邻平均值的平均进行居中处理。
Seasonal Variation / Quarterly Fluctuation
After finding the trend via moving averages, the seasonal effect is often calculated as the difference or ratio between the actual data and the trend value. These can be averaged over corresponding periods to produce typical seasonal adjustments.
经移动平均得到趋势后，季节影响常计算为实际数据与趋势值的差值或比值。可以按对应时段取平均，得到典型季节性修正量。
11. Sampling Methods | 抽样方法
Sampling is the process of selecting a subset of individuals from a population to estimate characteristics of the whole population. Understanding sampling methods is vital for data collection.
抽样是从总体中选取部分个体以估计总体特征的过程。理解抽样方法对数据收集至关重要。
Simple Random Sampling
Every member of the population has an equal chance of being selected. This can be achieved using random number tables or lottery methods.
每个总体成员被选中的机会均等，可通过随机数表或抽签方法实现。
Stratified Sampling
The population is divided into distinct groups (strata), and a random sample is taken from each, often in proportion to the stratum size. The formula for the number from a stratum is: n_stratum = (stratum size / population size) x sample size.
将总体划分为不同组别（层），然后从每层按比例随机抽样。每层抽取数量的公式为：层样本数 = (层大小/总体大小) x 总样本量。
Systematic Sampling
Select every k-th individual from a list after a random starting point. k = population size / desired sample size.
从列表中随机起点后，每隔 k 个抽取一个个体。k = 总体大小 / 所需样本量。
Quota and Convenience Sampling
Non-probability methods where interviewers fill quotas or select easily available subjects. These are quick but may lead to bias and are not truly representative.
非概率方法，访问员按配额或选取容易接触的个体。这些方法快捷但可能引入偏差，代表性不足。
12. Summary of Key Formulae Quick Look-up | 关键公式速查一览
Below is a concise table of the most important formulae you should know for the examination. Keep this page handy when revising.
以下精要表格列出了考试中必须掌握的最重要公式，复习时请将此页放在手边。
| Formula | Description |
|---|---|
| x = Σx / n | Sample mean |
| s² = Σ(x – x)² / (n-1) | Sample variance |
| IQR = Q3 – Q1 | Interquartile range |
| P(A or B) = P(A) + P(B) – P(A and B) | Addition rule |
| P(A|B) = P(A and B) / P(B) | Conditional probability |
| P(X = r) = &nCr p^r q^(n-r) | Binomial probability |
| E(X) = np, Var(X) = npq | Binomial mean and variance |
| z = (x – μ) / σ | Standard normal score |
| r_s = 1 – [6Σd² / n(n²-1)] | Spearman’s rank coefficient |
| y = a + bx, b = S_xy / S_xx | Regression line |
Remember that understanding the context and correct application of these formulae is just as important as memorising them. Practise past paper questions to become fluent in their use.
请记住，理解公式的来龙去脉以及正确应用与记忆公式同样重要。通过练习历年真题，熟练运用它们。
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