📚 Year 12 SQA Statistics: Core Knowledge Summary | SQA 12年级统计：核心知识点梳理
SQA Year 12 Statistics equips students with essential data analysis skills, covering data collection, probability, distributions, and inferential methods. A solid grasp of these core concepts is vital for success in exams and real-world applications.
SQA 12年级统计学课程培养学生的数据分析能力，涵盖数据收集、概率、分布及推断方法。掌握这些核心概念对于考试成功及实际应用至关重要。
1. Data Types and Sampling | 数据类型与抽样
In statistics, we distinguish between qualitative (categorical) data and quantitative (numerical) data. Qualitative data include nominal (e.g., colours) and ordinal (e.g., rankings) types, while quantitative data can be discrete (countable) or continuous (measurable).
在统计学中，我们区分定性（分类）数据与定量（数值）数据。定性数据包括名义数据（如颜色）和顺序数据（如排名），定量数据可以是离散（可数）或连续（可测）的。
To collect data, various sampling techniques are used. Simple random sampling gives each member an equal chance; stratified sampling divides the population into groups and samples proportionally; systematic sampling selects every k-th item; and cluster sampling selects entire groups. Convenience sampling is non-probability and may introduce bias.
收集数据时使用多种抽样方法。简单随机抽样确保每个成员有均等机会；分层抽样将总体分组并按比例抽样；系统抽样每隔k个选取；整群抽样选取整个群体。便利抽样是非概率方法，可能引入偏差。
2. Measures of Central Tendency and Spread | 中心趋势与离散度量
Measures of central tendency include mean, median, and mode. The mean (x̄ nce between maximum and minimum. The interquartile range (IQR = Q₃ − Q₁) measures the middle 50%. Variance s² and standard deviation s quantify variability around the mean.
集中趋势度量包括均值、中位数和众数。均值（x̄）是算术平均数，中位数是排序后的中间值，众数是出现频率最高的值。离散度量中，极差是最大值与最小值之差。四分位距（IQR = Q₃ − Q₁）衡量中间50%的数据。方差 s² 和标准差 s 量化了均值周围的变异性。
s² = Σ(x − x̄)² / (n − 1)
s = √[Σ(x − x̄)² / (n − 1)]
3. Probability Theory | 概率论基础
y measures the likelihood of events. For any event A, 0 ≤ P(A) ≤ 1. The addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0. The multiplication rule: P(A ∩ B) = P(A) × P(B|A). For independent events, P(A ∩ B) = P(A) × P(B). Conditional probability is P(A|B) = P(A ∩ B) / P(B). Tree diagrams help visualise sequential events.
概率衡量事件发生的可能性。对于任何事件 A，0 ≤ P(A) ≤ 1。加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。如果 A 和 B 互斥，P(A ∩ B) = 0。乘法法则：P(A ∩ B) = P(A) × P(B|A)。对于独立事件，P(A ∩ B) = P(A) × P(B)。条件概率为 P(A|B) = P(A ∩ B) / P(B)。树形图有助于可视化序列事件。
4. Discrete Random Variables | 离散随机变量
ance Var(X) = E(X²) − [E(X)]² or Σ(x − μ)² P(X = x). These describe the centre and spread of the distribution.
离散随机变量 X 取可数值，其概率分布为 P(X = x)。期望值（均值）为 E(X) = Σ x P(X = x)。方差 Var(X) = E(X²) − [E(X)]² 或 Σ(x − μ)² P(X = x)。它们描述了分布的中心和离散程度。
5. Binomial Distribution | 二项分布
s: fixed number of trials, independence, two outcomes, constant probability. The probability mass function is P(X = r) = C(n, r) × pʳ × (1 − p)ⁿ⁻ʳ. Mean = np, variance = np(1 − p).
二项分布 B(n, p) 模拟 n 次独立试验中成功次数 r，每次成功概率 p 恒定。条件：试验次数固定、独立、两个结果、概率恒定。概率质量函数为 P(X = r) = C(n, r) × pʳ × (1 − p)ⁿ⁻ʳ。均值 = np，方差 = np(1 − p)。
P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ
6. Normal Distribution | 正态分布
t the mean μ. It is defined by parameters μ and standard deviation σ. The standard normal Z ∼ − μ) / σ. Probabilities are found using tables or technology. The empirical rule: about 68% within 1σ, 95% within 2σ, 99.7% within 3σ.
正态分布是连续的钟形曲线，关于均值 μ 对称。由参数 μ 和标准差 σ 定义。通过标准化 z = (x − μ) / σ 使用标准正态分布 Z ∼ N(0,1)。利用表或技术求概率。经验法则：约68% 在 1σ 内，95% 在 2σ 内，99.7% 在 3σ 内。
z = (x − μ) ÷ σ
7. Correlation and Linear Regression | 相关性与线性回归
easures the strength and direction of a linear relationship between two variables. Pearson' ar 0 no linear correlation. The regression line of y on x is y = a + bx, where b = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² and a = ȳ − b x̄. It is used to predict values.
相关性衡量两变量之间线性关系的强度和方向。皮尔逊相关系数 r 范围为 −1 到 +1。接近 1 表示强正相关，接近 −1 表示强负相关，接近 0 表示无线性相关。y 对 x 的回归线为 y = a + bx，其中 b = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)²，a = ȳ − b x̄。可用于预测。
8. Hypothesis Testing | 假设检验
ternative H₁ (e.g., μ ≠ μ₀, μ > μ₀). A test statistic is computed from the sample; the p-value is the probability of obtaining a result at least as extreme under H₀. If p < significance level α (commonly 0.05), we reject H₀. One-tailed and two-tailed tests correspond to directional alternatives.
假设检验是利用数据做出决策的方法。设立原假设 H₀（如 μ = μ₀）和备择假设 H₁（如 μ ≠ μ₀, μ > μ₀）。从样本计算检验统计量；p 值是在 H₀ 下获得至少如此极端结果的概率。若 p < 显著性水平 α（通常为 0.05），则拒绝 H₀。单尾和双尾检验对应方向性备择假设。
9. Confidence Intervals | 置信区间
of plausible values for a population parameter. For a population mean with known σ, the 95% confidence interval is x̄ ± 1.96 × (σ / √n he general form: estimate ± (critical value) × (standard error). Interpretation: we are 95% confident that the interval captures the true mean.
置信区间给出总体参数的可能取值范围。当总体标准差 σ 已知，均值的 95% 置信区间为 x̄ ± 1.96 × (σ / √n)。一般形式：估计值 ± (临界值) × (标准误差)。解释：我们有 95% 的把握该区间包含真实均值。
x̄ ± z* × (σ ÷ √n)
10. Statistical Diagrams and Interpretation | 统计图表与解读
Visual representations help summarise data. Bar charts display categorical data; histograms show frequency distributions of continuous data (area proportional to frequency). Box plots illustrate median, quartiles, and outliers. Cumulative frequency diagrams and stem-and-leaf plots are also useful. Interpretation involves identifying central tendency, spread, skewness, and potential outliers.
图形展示有助于汇总数据。条形图显示分类数据；直方图展示连续数据的频率分布（面积与频率成比例）。箱线图展示中位数、四分
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