📚 Year 10 AQA Statistics Formula & Theorem Quick Reference Handbook | Year 10 AQA 统计：公式定理速查手册
Welcome to your ultimate quick reference for Year 10 AQA Statistics. This handbook compiles all essential formulas, theorems, and concepts you need to master. Each section provides concise definitions with paired Chinese translations to support bilingual learners. Use this as a revision checklist and on-the-go guide.
欢迎使用 Year 10 AQA 统计终极速查手册。本手册汇集了您需要掌握的所有基本公式、定理和概念。每个部分提供简洁的定义和配对中文翻译，以支持双语学习者。将此用作复习清单和随身指南。
1. Measures of Central Tendency | 集中趋势的度量
The arithmetic mean of a dataset x₁, x₂, …, xₙ is given by x̄ = (Σxᵢ) / n, where Σxᵢ is the sum of all values.
数据集的算术平均数公式为 x̄ = (Σxᵢ) / n，其中 Σxᵢ 是所有数值的总和。
For a frequency distribution, the mean is calculated as x̄ = (Σ f × x) / Σ f, where f is the frequency of each value x.
对于频数分布，均值计算公式为 x̄ = (Σ f × x) / Σ f，其中 f 是每个数值 x 的频数。
The median is the middle value when data are ordered. For n data points, the median position is (n + 1)/2. The mode is the most frequent value.
中位数是排序后中间的值。对于 n 个数据点，中位数的位置是 (n + 1)/2。众数是出现频率最高的值。
2. Measures of Dispersion | 离散程度的度量
The range is the simplest measure: Range = Maximum value − Minimum value.
极差是最简单的度量：极差 = 最大值 − 最小值。
he difference between the upper and lower quartiles: IQR = Q₃ − Q₁, covering the middle 50% of data.
四分位距 (IQR) 是上四分位数与下四分位数之差：IQR = Q₃ − Q₁，涵盖了数据的中间 50%。
tandard deviation measure spread around the mean. For a sample, the variance s² = Σ(xᵢ − x̄)² / (n − 1) and the standard deviation s = √[ Σ(xᵢ − x̄)² / (n − 1) ].
方差和标准差衡量数据围绕均值的分散程度。样本方差 s² = Σ(xᵢ − x̄)² / (n − 1)，样本标准差 s = √[ Σ(xᵢ − x̄)² / (n − 1) ]。
3. Quartiles and Interquartile Range | 四分位数与四分位距
For ungrouped data, first find the median (Q₂). The lower quartile Q₁ is the median of the lower half, and the upper quartile Q₃ is the median of the upper half. If there is an odd number of data, exclude the median from both halves.
对于未分组数据，首先找到中位数 (Q₂)。下四分位数 Q₁ 是下半部分的中位数，上四分位数 Q₃ 是上半部分的中位数。如果数据个数为奇数，则将中位数排除在两半之外。
For grouped data, quartiles are estimated from a cumulative frequency curve using interpolation.
对于分组数据，四分位数通过累积频率曲线用插值法估算。
The IQR is a robust measure of spread, unaffected by extreme values.
四分位距是稳健的离散度量，不受极端值影响。
4. Cumulative Frequency and Box Plots | 累积频率与箱线图
A cumulative frequency diagram plots cumulative frequency against the upper class boundary. The median and quartiles can be read directly from the graph.
累积频率图将累积频率相对于上组界绘制。中位数和四分位数可直接从图中读取。
mum. Outliers are typically identified as values below Q₁ − 1.5×IQR or above Q₃ + 1.5×IQR.
箱线图（盒须图）显示最小值、Q₁、中位数、Q₃ 和最大值。异常值通常被标识为低于 Q₁ − 1.5×IQR 或高于 Q₃ + 1.5×IQR 的值。
5. Probability Basics | 概率基础
ent A is P(A) = number of favourable outcomes / total number of outcomes, so 0 ≤ P(A) ≤ 1.
事件 A 的概率为 P(A) = 有利结果数 / 总结果数，因此 0 ≤ P(A) ≤ 1。
The addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。对于互斥事件，P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。
The multiplication rule: P(A ∩ B) = P(A) × P(B | A). For independent events, P(A ∩ B) = P(A) × P(B). Conditional probability is given by P(B | A) = P(A ∩ B) / P(A).
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)。条件概率公式为 P(B | A) = P(A ∩ B) / P(A)。
6. Probability Distributions | 概率分布
The binomial distribution applies when there are n independent trials, each with two outcomes (success/failure) and constant probability of success p. We write X ~ B(n, p).
二项分布适用于 n 次独立试验，每次试验有两种结果（成功/失败）且成功概率 p 恒定。记作 X ~ B(n, p)。
P(X = r) = nCr × pr × (1−p)n−r
其中 nCr = n! / (r!(n−r)!) 为组合数。
Mean of a binomial distribution: E(X) = np. Variance: Var(X) = np(1−p).
二项分布的均值：E(X) = np。方差：Var(X) = np(1−p)。
7. Index Numbers | 指数
ce of a single item relative to a base period: Index = (Current price / Base price) × 100.
简单价格指数显示单一物品相对于基期的价格变化：指数 = （当前价格 / 基期价格）× 100。
p₀ q₀) × 100. The Paasche index uses current-period quantities: Σ(p₁ q₁) / Σ(p₀ q₁) × 100.
加权指数考虑数量因素。拉氏指数使用基期数量：Σ(p₁ q₀) / Σ(p₀ q₀) × 100。帕氏指数使用现期数量：Σ(p₁ q₁) / Σ(p₀ q₁) × 100。
8. Time Series Analysis | 时间序列分析
A time series consists of data recorded at successive time intervals. Components include trend, seasonal variation, and irregular fluctuations.
时间序列由在连续时间间隔记录的数据组成。组成部分包括趋势、季节变动和不规则波动。
To smooth out fluctuations and reveal the trend, a moving average is calculated. A 3‑point moving average for time t is: (xₜ₋₁ + xₜ + xₜ₊₁) / 3.
为了消除波动并显示趋势，需计算移动平均数。时间 t 的三点移动平均为： (xₜ₋₁ + xₜ + xₜ₊₁) / 3。
9. Scatter Diagrams and Correlation | 散点图与相关
A scatter diagram plots paired data (x, y) to show the relationship between two variables. The pattern can reveal positive, negative, or zero correlation.
散点图绘制成对数据 (x, y)，以显示两个变量之间的关系。图案可揭示正相关、负相关或零相关。
ated as rₛ = 1 − (6 Σ d²) / (n(n² − 1)), where d is the difference in ranks for each pair.
斯皮尔曼秩相关系数衡量单调关联的强度。计算公式为 rₛ = 1 − (6 Σ d²) / (n(n² − 1))，其中 d 是每对的秩差。
10. Regression Line (Line of Best Fit) | 回归线（最佳拟合线）
equation of the least squares regression line is y = a + bx, where the slope b = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)² and the intercept a = ȳ − b x̄.
最小二乘回归线的方程为 y = a + bx，其中斜率 b = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)²，截距 a = ȳ − b x̄。
The slope b indicates the average change in y for a unit increase in x. The line can be used to make predictions within the range of observed data, but extrapolation beyond this range can be unreliable.
斜率 b 表示 x 每增加一个单位时 y 的平均变化量。该直线可用于在观测数据范围内进行预测，但超出此范围的外推可能不可靠。
11. Normal Distribution | 正态分布
The normal distribution is a continuous, bell‑shaped curve, symmetric about its mean μ. The mean, median, and mode are all equal.
正态分布是一种连续的钟形曲线，关于均值 μ 对称。均值、中位数和众数都相等。
The empirical rule: about 68% of values lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. The standardised score (z‑score) transforms any value to the standard normal distribution: z = (x − μ) / σ.
经验法则：约 68% 的值落在均值 ±1σ 范围内，95% 落在 ±2σ 内，99.7% 落在 ±3σ 内。标准化得分（z 值）将任何值转换为标准正态分布：z = (x − μ) / σ。
12. Sampling and Estimation | 抽样与估计
Good sampling methods (random, stratified, systematic) aim to give a representative sample. The sample mean x̄ is an unbiased estimator of
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