📚 Year 12 OCR Statistics: Formula & Theorem Quick Reference Handbook | 12年级OCR统计：公式定理速查手册
This concise reference handbook pulls together all the essential formulas, notation and theorem statements required for the Year 12 OCR Statistics syllabus. Use it alongside your class notes and past papers to familiarise yourself with the toolkit you will need in the examination. Every formula is written in clean notation, ready for quick recall.
本速查手册汇集了12年级OCR统计课程中所有必需的公式、符号和定理表述。将它配合课堂笔记和历年真题使用，帮助你熟悉考试所需的工具包。每条公式都采用清晰的表示法，便于快速回忆。
1. Data Collection and Sampling Methods | 数据收集与抽样方法
A population is the whole set of items of interest. A sample is a subset of the population. Random sampling ensures each member has an equal chance of selection, reducing bias.
总体是所关注的全部对象的集合。样本是总体的一个子集。随机抽样确保每个成员被选中的概率相等，从而减少偏差。
In stratified sampling, the population is divided into mutually exclusive strata and random samples are taken from each in proportion to its size. This guarantees representation across key groups.
在分层抽样中，先将总体划分为互斥的层，然后从每层中按比例随机抽取样本。这保证了关键群体都有代表。
Systematic sampling selects every k-th item from a list after a random start. Quota sampling selects a fixed number of individuals with predetermined characteristics, relying on interviewer discretion.
系统抽样在随机起始后从名单中每隔k个抽取一个。配额抽样选取具有预设特征的固定人数，依赖调查员的判断。
Observational studies and experiments differ: experiments impose a treatment, while observational studies simply record data without intervention.
观察性研究与实验的区别在于：实验施加处理，而观察性研究仅记录数据，不进行干预。
2. Frequency Distributions and Histograms | 频数分布与直方图
When data are grouped into intervals of equal or unequal width, a histogram can display the distribution. The vertical axis represents frequency density, not frequency.
当数据被分入等宽或不等宽的区间时，用直方图展示分布。纵轴表示频率密度，而不是频率。
Frequency density = frequency ÷ class width. The area of each bar is proportional to the frequency in that interval.
频率密度 = 频率 ÷ 组距。每个长方条的面积与该区间的频率成比例。
| Key formula | Description |
| Frequency density = f / w | f = frequency, w = class width |
| Area = k × frequency | k is the scale factor (usually k = 1 on a frequency density histogram, making area = frequency) |
对于频率密度直方图，每个长方条的面积 = k × 频率，通常取 k = 1，使得面积就等于频率。
3. Measures of Central Tendency | 集中趋势度量
The three main measures are the mean, median and mode. Their properties help summarise the centre of the data.
三个主要的度量是平均数、中位数和众数。它们的性质有助于概括数据的中心。
| Statistic | Formula / Definition | Notes |
| Mean (ungrouped) | x̄ = Σxᵢ / n | Uses all data values; sensitive to outliers |
| Mean (grouped) | x̄ = Σfᵢ xᵢ / Σfᵢ | xᵢ = midpoint of i-th class |
| Median | Middle value when ordered; for n values, position = (n+1)/2 | Not affected by outliers |
| Mode | Most frequent value; for grouped data, modal class | Useful for categorical data |
For a perfectly symmetrical distribution, mean = median = mode. For skewed data, the mean is pulled in the direction of the tail.
对于完全对称的分布，平均数 = 中位数 = 众数。对于偏斜数据，平均数会被拉向尾部方向。
4. Measures of Spread | 离差与变异度量
Spread indicates how much the data vary. Key measures include range, interquartile range (IQR), variance and standard deviation.
离散程度表明数据的变异大小。主要度量有极差、四分位距（IQR）、方差和标准差。
| Measure | Formula / Definition |
| Range | max − min |
| IQR | Q₃ − Q₁ |
| Variance (population) | σ² = Σ(xᵢ − μ)² / N |
| Variance (sample) | s² = Σ(xᵢ − x̄)² / (n − 1) |
| Standard deviation | s = √s² |
| Mean squared deviation (alternative form) | s² = (Σxᵢ² − (Σxᵢ)²/n) / (n − 1) |
The sample variance uses (n − 1) as the denominator to give an unbiased estimate of the population variance. Use this formula when a sample represents a larger population.
样本方差用 (n−1) 作为分母，以得到总体方差的无偏估计。当样本代表一个更大的总体时应使用此公式。
5. Quartiles and Box Plots | 四分位数与箱线图
Quartiles split the ordered data into four equal parts. The lower quartile Q₁ separates the lowest 25%, the median Q₂ is the 50th percentile, and the upper quartile Q₃ separates the top 25%.
四分位数将排序后的数据分成四个相等的部分。下四分位数 Q₁ 划分出最低的25%，中位数 Q₂ 是第50百分位数，上四分位数 Q₃ 划分出最高的25%。
For a discrete list, use linear interpolation for grouped data or the (n+1)/4 method for raw data. The IQR = Q₃ − Q₁ measures the spread of the middle 50%.
对于分组数据，可用线性插值法；对于原始列表，可用 (n+1)/4 定位法。IQR = Q₃ − Q₁ 度量中间50%的数据跨度。
A box plot shows the five-number summary: minimum, Q₁, median, Q₃, maximum. Outliers are typically marked as individual points beyond 1.5 × IQR from the quartiles.
箱线图展示五数概括：最小值、Q₁、中位数、Q₃、最大值。离群值通常标记为超出四分位数 1.5×IQR 以外的点。
6. Skewness | 偏度
Skewness describes the asymmetry of a distribution. Three common ways to recognise skewness are given below.
偏度描述分布的不对称性。以下是三种常见的偏度判定方法。
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Using mean and median: if mean > median, the data are positively skewed (right skewed); if mean < median, negatively skewed (left skewed).
利用平均数和中位数：若平均数 > 中位数，数据正偏（右偏）；若平均数 < 中位数，数据负偏（左偏）。
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Comparing quartiles: positive skew when (Q₂ − Q₁) < (Q₃ − Q₂); negative skew when (Q₂ − Q₁) > (Q₃ − Q₂).
比较四分位数：当 (Q₂ − Q₁) < (Q₃ − Q₂) 时为正偏；当 (Q₂ − Q₁) > (Q₃ − Q₂) 时为负偏。
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Using a box plot: the distance from the median to the upper quartile and the upper whisker is greater for positive skew.
利用箱线图：正偏时从中位数到上四分位数和上触须的距离更大。
Symmetrical distributions have mean ≈ median ≈ mode and equal whisker lengths.
对称分布具有平均数 ≈ 中位数 ≈ 众数，且两个触须长度大致相等。
7. Probability Rules and Conditional Probability | 概率法则与条件概率
Probability is the measure of the likelihood of an event, satisfying 0 ≤ P(A) ≤ 1. The sample space S contains all possible outcomes.
概率是事件发生可能性的度量，满足 0 ≤ P(A) ≤ 1。样本空间 S 包含所有可能出现的结果。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
The complement rule: P(A’) = 1 − P(A). For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
互补法则：P(A’) = 1 − P(A)。对于互斥事件，P(A ∩ B) = 0，故 P(A ∪ B) = P(A) + P(B)。
Conditional probability: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. This is the probability of A given that B has occurred.
条件概率：P(A|B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。它表示在事件B已发生的条件下事件A发生的概率。
Rearranging gives the multiplication rule: P(A ∩ B) = P(A) P(B|A) = P(B) P(A|B). For independent events, P(A|B) = P(A), so P(A ∩ B) = P(A) P(B).
由条件概率可导出乘法法则：P(A ∩ B) = P(A) P(B|A) = P(B) P(A|B)。对于独立事件，P(A|B) = P(A)，因此 P(A ∩ B) = P(A) P(B)。
Tree diagrams with branches labelled with conditional probabilities can multiply along branches to find intersection probabilities.
树状图的各分支标注条件概率，沿分支相乘可求出交事件的概率。
8. Discrete Random Variables and Expectation | 离散随机变量与期望
A discrete random variable X takes distinct values x₁, x₂, … with probabilities P(X = xᵢ) = pᵢ. The sum of all probabilities is 1.
离散随机变量 X 取互异的值 x₁, x₂, …，其概率为 P(X = xᵢ) = pᵢ。所有概率之和为 1。
E(X) = Σ xᵢ pᵢ
The expectation E(X) is the long-run average. For a function g(X), E[g(X)] = Σ g(xᵢ) pᵢ.
期望 E(X) 是长期平均值。对于函数 g(X)，E[g(X)] = Σ g(xᵢ) pᵢ。
Var(X) = E(X²) − [E(X)]² = Σ xᵢ² pᵢ − (Σ xᵢ pᵢ)²
Variance measures the spread of the distribution. The standard deviation is σ = √Var(X).
方差度量分布的离散程度。标准差 σ = √Var(X)。
If a and b are constants, E(aX + b) = a E(X) + b and Var(aX + b) = a² Var(X).
若 a 和 b 为常数，则 E(aX + b) = a E(X) + b，Var(aX + b) = a² Var(X)。
9. Binomial Distribution | 二项分布
The binomial distribution models the number of successes in a fixed number n of independent trials, each with the same probability of success p.
二项分布对固定次数 n 次独立试验中的成功次数进行建模，每次试验的成功概率均为 p。
Notation: X ~ B(n, p). The random variable X can take integer values 0, 1, 2, …, n.
记为：X ~ B(n, p)。随机变量 X 可取整数值 0, 1, 2, …, n。
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ
Where ⁿCᵣ = n! / [r! (n−r)!] is the binomial coefficient. The formula works for r = 0, 1, …, n.
其中 ⁿCᵣ = n! / [r! (n−r)!] 为二项式系数。该公式适用于 r = 0, 1, …, n。
E(X) = np
Var(X) = np(1 − p)
Conditions for a binomial model: fixed n, independent trials, constant p, and only two outcomes labelled ‘success’ and ‘failure’.
二项模型的适用条件：n 固定，试验相互独立，p 保持不变，且每次试验只有“成功”与“失败”两种结果。
10. Hypothesis Testing for the Binomial Distribution | 二项分布假设检验
A hypothesis test for a binomial proportion compares the observed number of successes with what we would expect under a stated null hypothesis H₀: p = p₀.
对二项比例进行假设检验，将观测到的成功次数与在零假设 H₀: p = p₀ 下的期望值进行比较。
The test statistic is the number of successes X. The p-value is the probability of obtaining a result at least as extreme as the observed value, assuming H₀ is true.
检验统计量为成功次数 X。p 值是在 H₀ 为真的前提下，获得与观测值至少同样极端的结果的概率。
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For a one-tailed test with alternative H₁: p > p₀, p-value = P(X ≥ observed).
对于备择假设 H₁: p > p₀ 的单尾检验，p 值 = P(X ≥ 观测值)。
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For H₁: p < p₀, p-value = P(X ≤ observed).
对于 H₁: p < p₀，p 值 = P(X ≤ 观测值)。
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For a two-tailed test with H₁: p ≠ p₀, find the sum of the probabilities in both tails that are ≤ the probability of the observed value. OCR typically uses a fixed significance level, e.g. 5%.
对于双侧检验 H₁: p ≠ p₀，找出两个尾部中概率不大于观测值概率的所有结果之和。OCR通常采用固定显著性水平，如5%。
If p-value < significance level α, reject H₀ and conclude there is sufficient evidence to support H₁. Otherwise, do not reject H₀.
若 p 值 < 显著性水平 α，则拒绝 H₀，并认为有充分证据支持 H₁。否则不拒绝 H₀。
11. Scatter Diagrams and Correlation | 散点图与相关
A scatter diagram displays pairs (xᵢ, yᵢ) of bivariate data. Visual inspection helps identify the direction, form and strength of any association.
散点图显示成对的双变量数据 (xᵢ, yᵢ)。通过目视检查可判断关联的方向、形式和强度。
The product moment correlation coefficient (PMCC), denoted r, measures the strength of linear correlation: −1 ≤ r ≤ 1.
积矩相关系数（PMCC），记为 r，度量线性相关的强度：−1 ≤ r ≤ 1。
r = Sxy / √(Sxx Syy)
| Sxx = Σ(xᵢ − x̄)² = Σxᵢ² − (Σxᵢ)²/n |
| Syy = Σ(yᵢ − ̅y)² = Σyᵢ² − (Σyᵢ)²/n |
| Sxy = Σ(xᵢ − x̄)(yᵢ − ̅y) = Σxᵢ yᵢ − (Σxᵢ)(Σyᵢ)/n |
r close to +1 indicates strong positive linear correlation; r close to −1 indicates strong negative linear correlation; r near 0 suggests weak or no linear correlation.
r 接近 +1 表示强正线性相关；r 接近 −1 表示强负线性相关；r 接近 0 表示线性相关很弱或不存在。
12. Least Squares Regression Line | 最小二乘回归线
The least squares regression line of y on x is the straight line that minimises the sum of squared vertical distances from the points to the line.
y 对 x 的最小二乘回归线是使各点到直线的垂直距离平方和最小的直线。
Equation: y = a + bx
The slope b is given by b = Sxy / Sxx, and the intercept a = ̅y − b x̄.
斜率 b 由 b = Sxy / Sxx 给出，截距 a = ̅y − b x̄。
This line can be used to predict y for given x. Interpolation (prediction within the range of the data) is usually reliable, but extrapolation (outside the data range) can be unreliable.
该直线可用于给定 x 预测 y。内插（在数据范围内预测）通常可靠，但外推（超出数据范围）可能不可靠。
The regression line always passes through the mean point (x̄, ̅y).
回归线总是通过均值点 (x̄, ̅y)。
For a regression line of x on y, the roles are swapped: x = a’ + b’ y, with b’ = Sxy / Syy.
对于 x 对 y 的回归线，角色互换：x = a’ + b’ y，其中 b’ = Sxy / Syy。
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