Year 12 WJEC Statistics: Formulae & Theorems Quick Reference | 12年级 WJEC 统计：公式定理速查手册

📚 Year 12 WJEC Statistics: Formulae & Theorems Quick Reference | 12年级 WJEC 统计：公式定理速查手册

This handbook provides a concise compilation of essential formulae, theorems, and key concepts for the Year 12 WJEC Statistics syllabus. Use it for quick revision and as a check-list before your exams. Each section presents the core ideas in English and Chinese, with clear mathematical notation to support your understanding.

本手册简明汇编了12年级 WJEC 统计大纲中的关键公式、定理与核心概念，方便你快速复习与考前自查。每个小节均以中英文双语呈现核心内容，并提供清晰的数学符号以辅助理解。

1. Data Types & Sampling | 数据类型与抽样

Data can be qualitative (categorical, e.g. colour) or quantitative (numerical). Quantitative data is either discrete (countable, e.g. number of students) or continuous (measurable, e.g. height). Understanding the data type is essential for choosing the right statistical tools.

数据可以是定性的（分类，如颜色）或定量的（数值）。定量数据又分为离散型（可计数，如学生人数）和连续型（可测量，如身高）。理解数据类型对于选择正确的统计工具至关重要。

A random sample ensures every member of the population has an equal chance of being selected, reducing bias. Stratified sampling divides the population into distinct groups (strata) and samples proportionally from each, ensuring representation across key characteristics.

简单随机样本确保总体中每个成员被抽中的机会均等，从而减少偏差。分层抽样先将总体划分为不同的群组（层），然后按比例从每一层中抽取样本，确保关键特征的充分代表。

2. Measures of Location | 位置度量

The sample mean summarises the centre of the data and is calculated as the sum of all observations divided by the number of observations.

样本均值概括了数据的中心，其计算方法为所有观测值之和除以观测值个数。

x̄ = ∑ xᵢ / n

For grouped data, the mean is estimated by multiplying each class midpoint by its frequency, summing these products, and dividing by the total frequency.

对于分组数据，均值可通过将每个组中值乘以该组频数，再求和并除以总频数来估计。

x̄ = ∑ fᵢ mᵢ / ∑ fᵢ

The median is the middle value when data are ordered. If n is even, the median is the average of the two centre values. For grouped data, use linear interpolation within the median class.

中位数是数据排序后位于中间的值。若 n 为偶数，则中位数为中间两个值的平均数。对于分组数据，需在包含中位数的组内进行线性插值。

The mode is the most frequently occurring value. A data set can have more than one mode or no clear mode. In grouped data, the modal class is the class with the highest frequency.

众数是出现次数最多的值。一组数据可以有一个以上的众数，也可能没有明显的众数。在分组数据中，众数所在的组是频数最高的组。

3. Measures of Dispersion | 离散度量

Dispersion measures describe the spread of the data. The range is simply the difference between the largest and smallest values, but it is sensitive to outliers.

离散度量描述数据的分散程度。极差即为最大值与最小值之差，但它容易受到异常值的影响。

Range = Xₘₐₓ – Xₘᵢₙ

The interquartile range (IQR) is the difference between the upper quartile (Q3) and the lower quartile (Q1), representing the middle 50% of the data and being more robust to outliers.

四分位距（IQR）是上四分位数 (Q₃) 与下四分位数 (Q₁) 之差，代表了中间 50% 的数据，对异常值具有更强的稳健性。

IQR = Q₃ – Q₁

Variance and standard deviation are the most common measures. For a sample, we divide by (n – 1) to obtain an unbiased estimator of the population variance.

方差和标准差是最常用的度量。对于样本数据，除以 (n – 1) 可以得到总体方差的无偏估计量。

s² = ∑ (xᵢ – x̄)² / (n – 1)

The standard deviation is the square root of the variance and has the same units as the original data.

标准差是方差的平方根，与原数据的单位相同。

s = √[∑ (xᵢ – x̄)² / (n – 1)]

When data are coded linearly as y = (x – a)/b, the mean and standard deviation are transformed accordingly: the mean is coded identically, while the standard deviation is divided by |b|.

当数据经过线性编码 y = (x – a)/b 时，均值和标准差也会相应变换：均值按相同规则编码，标准差则除以 |b|。

ȳ = (x̄ – a) / b, s_y = s_x / |b|

4. Probability | 概率

Probability is the long-run relative frequency of an event and satisfies 0 ≤ P(A) ≤ 1. For any two events A and B, the addition rule accounts for overlap.

概率是一个事件的长期相对频率，且满足 0 ≤ P(A) ≤ 1。对于任意两个事件 A 和 B，加法法则需要考虑重叠部分。

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

If A and B are mutually exclusive (cannot occur together), then P(A ∩ B) = 0 and the formula simplifies to P(A ∪ B) = P(A) + P(B).

若 A 与 B 互斥（不能同时发生），则 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。

Conditional probability describes the chance of A given that B has occurred. It is fundamental to Bayes’ theorem and probability trees.

条件概率描述了在 B 已发生的条件下 A 发生的概率。它是贝叶斯定理和概率树的基础。

P(A | B) = P(A ∩ B) / P(B)

Two events are independent if the occurrence of one does not affect the probability of the other. In that case, P(A ∩ B) = P(A) × P(B), or equivalently P(A | B) = P(A).

如果两个事件的发生互不影响，则称它们独立。此时 P(A ∩ B) = P(A) × P(B)，或等价地 P(A | B) = P(A)。

Tree diagrams multiply probabilities along branches and add probabilities of different branches for combined events. Always check that the probabilities from any node sum to 1.

在概率树中，沿一条路径将分支上的概率相乘，并将不同路径的概率相加得到组合事件的概率。永远要检查从任一节点出发的分支概率之和是否为 1。

5. Discrete Random Variables | 离散随机变量

A discrete random variable X takes values with associated probabilities P(X = x). The sum of all these probabilities must equal 1. The probability function is often displayed in a table.

离散随机变量 X 以一定的概率取值 P(X = x)。所有这些概率之和必须等于 1。概率函数通常用表格展示。

The expected value E(X), also called the mean, is a weighted average of the possible values, with the weights being their probabilities.

期望值 E(X)，又称均值，是所有可能值的加权平均数，其权重即为相应的概率。

E(X) = μ = ∑ x · P(X = x)

The variance measures the spread of the distribution and can be computed using the alternative formula, which is often easier to handle.

方差衡量分布的离散程度，通常使用以下的替代公式进行计算会更加方便。

Var(X) = E(X²) – [E(X)]² = ∑ x²P(X = x) – μ²

Linear transformations of a random variable follow simple rules. Adding a constant shifts the mean but does not affect variance; multiplying by a constant scales both the mean and the standard deviation.

随机变量的线性变换遵循简单的规则。加上一个常数会平移均值，但不影响方差；乘以一个常数则同时缩放均值和标准差。

E(aX + b) = aE(X) + b, Var(aX + b) = a² Var(X)

6. Binomial Distribution | 二项分布

The binomial distribution arises from a fixed number of independent trials, each with two possible outcomes (success/failure) and a constant probability of success, p.

二项分布源于固定次数的独立试验，每次试验只有两种可能的结果（成功/失败），且成功的概率 p 保持不变。

X ~ B(n, p)

The probability of obtaining exactly r successes in n trials is given by the binomial probability formula, which utilises the combinatorial coefficient n choose r.

在 n 次试验中恰好获得 r 次成功的概率由二项概率公式给出，其中用到了组合数 n 选 r。

P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ

The mean and variance of a binomial random variable are straightforward to compute and are often used in hypothesis testing and approximations.

二项随机变量的均值和方差计算简便，常用于假设检验和近似计算中。

μ = np, σ² = np(1 – p)

Cumulative probabilities, such as P(X ≤ r) or P(X ≥ r), can be found using published tables or calculators. Remember to read the tables correctly depending on the direction of the inequality.

累积概率，例如 P(X ≤ r) 或 P(X ≥ r)，可通过查阅标准表格或使用计算器求得。务必根据不等式的方向正确读取表格。

7. Normal Distribution | 正态分布

The normal distribution is a continuous, symmetric, bell-shaped distribution defined by its mean μ and variance σ². Its total area under the curve equals 1.

正态分布是一个连续、对称的钟形分布，由均值 μ 和方差 σ² 确定。该曲线下的总面积为 1。

X ~ N(μ, σ²)

To find probabilities for any normal variable, we standardise it by subtracting the mean and dividing by the standard deviation. This process converts X to the standard normal Z, which has mean 0 and standard deviation 1.

要计算任意正态变量的概率，可将其标准化：减去均值后再除以标准差。这一过程将 X 转化为标准正态变量 Z，其均值为 0，标准差为 1。

Z = (X – μ) / σ

Standard normal probability tables typically give P(Z < z) for positive z-values. You must apply symmetry and complement rules to obtain probabilities for negative z-values or upper tails.

标准正态概率表通常给出正 z 值下的 P(Z < z)。你需要运用对称性和补集法则来求负 z 值或上尾概率。

Sometimes you are given a probability and need to find the corresponding x-value. This inverse normal calculation involves locating the z-score from the table and then transforming back using x = μ + zσ.

有时给定概率需要求对应的 x 值。这一逆向正态运算需先从表中找出对应的 z 分位数，再通过 x = μ + zσ 变换回去。

When independent normal variables are added, the sum is also normally distributed. The mean is the sum of the means, and the variance is the sum of the variances.

当独立正态变量相加时，其和仍服从正态分布。和的均值为各自均值之和，和的方差为各自方差之和。

X₁ + X₂ ~ N(μ₁ + μ₂, σ₁² + σ₂²)

8. Correlation | 相关

Correlation measures the strength and direction of a linear relationship between two variables. The product moment correlation coefficient (PMCC), denoted by r, is the standard measure used for normally distributed data.

相关衡量两个变量之间线性关系的强弱和方向。积矩相关系数 (PMCC)，记作 r，是用于正态分布数据的标准度量。

r = Sxy / √(Sxx · Syy)

where the sums of squares and cross-products are defined as:

其中平方和和交叉乘积和定义为：

Sxx = ∑(x – x̄)² = ∑x² – (∑x)²/n

Syy = ∑(y – ȳ)² = ∑y² – (∑y)²/n

Sxy = ∑(x – x̄)(y – ȳ) = ∑xy – (∑x)(∑y)/n

The value of r always lies between –1 and 1. A value close to 1 indicates strong positive correlation, close to –1 indicates strong negative correlation, and a value around 0 suggests little or no linear correlation.

r 的取值范围始终介于 –1 与 1 之间。接近于 1 表示强正相关，接近于 –1 表示强负相关，接近 0 则表明几乎没有线性相关关系。

Spearman’s rank correlation coefficient, r_s, assesses monotonic relationships and is based on the ranks of the data rather than actual values. It is useful when data are not normally distributed or when outliers are present.

斯皮尔曼秩相关系数 r_s 评估单调关系，其基于数据的秩而非实际数值。当数据不服从正态分布或存在异常值时，它尤为有用。

r_s = 1 – 6∑d² / [n(n² – 1)]

where d is the difference between the ranks of each pair. When ties are present, a correction factor must be applied, or the PMCC formula can be used on the ranks themselves.

其中 d 为每对数据秩的差值。当存在并列秩时，必须使用校正因子，或者可以直接对秩值套用积矩相关系数公式进行计算。

9. Linear Regression | 线性回归

Regression analysis finds the best-fitting straight line through a scatter plot. The least squares regression line of y on x minimises the sum of squared vertical distances from the points to the line.

回归分析寻找穿过散点图的最佳拟合直线。y 对 x 的最小二乘回归线使各点到直线的垂直距离的平方和最小。

y = a + bx

The slope b and intercept a are obtained from the summary statistics. The slope tells us how much y changes on average for a unit increase in x.

斜率 b 和截距 a 通过汇总统计量求得。斜率表示 x 每增加一个单位，y 平均变动的量。

b = Sxy / Sxx

a = ȳ – b x̄

Always confirm that the line passes through the mean point (x̄, ȳ). The regression equation should only be used to predict y for values of x within the observed range; extrapolation beyond the data is unreliable.

永远要确认该直线通过均值点 (x̄, ȳ)。回归方程只应用于预测观测范围内的 x 对应的 y 值；超出数据范围的外推是不可靠的。

When coding is applied to x or y, the regression coefficients must be interpreted in terms of the original variables. Coding can simplify arithmetic but does not alter the nature of the relationship.

当对 x 或 y 进行编码后，回归系数必须还原到原始变量的意义。编码可以简化计算，但不会改变关系的本质。

10. Hypothesis Testing for Binomial | 二项分布的假设检验

A hypothesis test examines whether sample data provide sufficient evidence to reject a null hypothesis, H₀, in favour of an alternative hypothesis, H₁. In the binomial context, the parameter under test is the population proportion (or probability of success), p.

假设检验考察样本数据是否提供足够证据来拒绝原假设 H₀ 而支持备择假设 H₁。在二项背景下，被检验的参数是总体比例（或成功概率）p。

H₀: p = p₀

The alternative hypothesis can be one-tailed (p < p₀ or p > p₀) or two-tailed (p ≠ p₀). The choice must reflect the claim being tested and should be decided before looking at the data.

备择假设可以是单尾的 (p < p₀ 或 p > p₀) 或双尾的 (p ≠ p₀)。选择必须反映所要检验的主张，且应在查看数据之前确定。

Test statistic: X ~ B(n, p₀) under H₀

We calculate the probability of obtaining the observed result, or something more extreme, assuming H₀ is true. This probability is the p-value.

我们计算在 H₀ 成立的条件下，得到观察结果或者更极端结果的概率。此概率即为 p 值。

If the p-value is less than the significance level (commonly 0.05), we reject H₀ and say the result is statistically significant. If it is greater, we do not have enough evidence to reject H₀.

若 p 值小于显著性水平（通常为 0.05），则拒绝 H₀，认为结果具有统计显著性。若 p 值大于显著性水平，则没有足够证据拒绝 H₀。

Alternatively, we can find the critical region: the set of values of the test statistic that would lead to rejection of H₀. For a one-tailed test at the 5% level, we find the largest r such that P(X ≤ r) ≤ 0.05 (for a left-tailed test) or the smallest r such that P(X ≥ r) ≤ 0.05 (for a right-tailed test).

我们也可以找出拒绝域：即会导致拒绝 H₀ 的检验统计量的取值范围。对于 5% 水平下的单尾检验，找出使 P(X ≤ r) ≤ 0.05 的最大 r（左尾检验）或使 P(X ≥ r) ≤ 0.05 的最小 r（右尾检验）。

For two-tailed tests, the significance level is split equally between the two tails (e.g. 2.5% each). We find the corresponding critical values and reject H₀ if the test statistic falls in either tail.

对于双尾检验，显著性水平被平分到两个尾部（例如各 2.5%）。求出相应的临界值；若检验统计量落入任一尾部，则拒绝 H₀。

Always state your conclusion in the context of the problem, not just a mathematical statement. For example: ‘There is sufficient evidence at the 5% level to suggest that the proportion of defective items has decreased.’

永远要在问题的实际背景下陈述结论，而不仅仅给出一个数学表述。例如：‘在 5% 的显著性水平下，有充分证据表明次品率已经下降。’

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Year 12 WJEC Statistics: Formulae & Theorems Quick Reference | 12年级 WJEC 统计：公式定理速查手册