📚 GCSE OCR Statistics: Formula & Theorem Quick Reference Handbook | GCSE OCR 统计：公式定理速查手册
Statistics is about collecting, presenting, analysing, and interpreting data. This handbook brings together the essential formulas, definitions, and theorems you will need for the GCSE OCR Statistics specification. From measures of central tendency to probability rules, from data representation techniques to index numbers, everything is presented in a clear, exam-focused way. Keep this guide nearby when revising or practising past papers, and use it to check your understanding of key concepts.
统计学是关于收集、呈现、分析和解释数据的学科。本手册汇集了GCSE OCR统计学考试规格所需的基本公式、定义和定理。从集中趋势的度量到概率规则，从数据表示技术到指数数字，所有内容都以清晰、以考试为重点的方式呈现。在复习或练习历年试题时，将本指南放在手边，用它来检查你对关键概念的理解。
1. Measures of Central Tendency | 集中趋势的度量
Measures of central tendency identify the ‘centre’ of a data set. The mean is the arithmetic average. For a set of n values x₁, x₂, …, xₙ, the mean is x̄ = Σxᵢ / n. The median is the middle value when data are ordered. If n is odd, it 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. The mode is the most frequently occurring value.
集中趋势的度量确定数据集的“中心”。均值是算术平均数。对于包含n个值x₁，x₂，…，xₙ的集合，均值 x̄ = Σxᵢ / n。中位数是数据排序后的中间值。若n为奇数，则为第(n+1)/2个值；若n为偶数，则为第n/2个与第(n/2+1)个值的平均数。众数是出现频率最高的值。
If data are grouped in a frequency table, the estimated mean is Σ(f × midpoint) / Σf. The modal class is the class with the highest frequency.
如果数据在频数表中分组，则估计均值为 Σ(f × 组中点) / Σf。众数类是频数最高的类。
OCR often tests the choice of a suitable average. Use median when data contain outliers; use mode for categorical data; use mean when all values contribute equally.
OCR常考查选择适当的平均数。当数据含有异常值时使用中位数；对类别数据使用众数；当所有值同等重要时使用均值。
2. Measures of Dispersion | 离散程度度量
Dispersion shows how spread out the data are. The range is simply the difference between the largest and smallest values: range = max − min. The interquartile range (IQR) is the difference between the upper quartile (Q3) and lower quartile (Q1): IQR = Q3 − Q1. Quartiles are found by ordering data and locating positions: Q1 at (n+1)/4 th, Q3 at 3(n+1)/4 th. If the position is not an integer, interpolate between values.
离散程度显示数据的分散程度。极差就是最大值与最小值的差：极差 = 最大值 – 最小值。四分位距（IQR）是上四分位数（Q3）与下四分位数（Q1）之差：IQR = Q3 – Q1。四分位数通过将数据排序并确定位置来找到：Q1在第(n+1)/4个位置，Q3在第3(n+1)/4个位置。如果位置不是整数，则在两个值之间进行插值。
The standard deviation is a more sophisticated measure. For a sample, the standard deviation s is given by the square root of the variance. Variance s² = Σ(x − x̄)² / (n−1) for sample data; for a population (full data set) use division by n. A larger standard deviation means greater spread.
标准差是一种更精密的度量。对于样本，标准差 s 是方差的平方根。样本数据方差 s² = Σ(x − x̄)² / (n−1)；对于总体（完整数据集）除以n。较大的标准差意味着更大的分散程度。
Outliers can be identified using the IQR rule: a value is an outlier if it is less than Q1 − 1.5 × IQR or greater than Q3 + 1.5 × IQR.
可以使用IQR规则识别异常值：若一个值小于 Q1 − 1.5 × IQR 或大于 Q3 + 1.5 × IQR，则为异常值。
3. Cumulative Frequency & Box Plots | 累积频数与箱线图
Cumulative frequency is the running total of frequencies. To draw a cumulative frequency graph, plot the upper class boundary against the cumulative frequency, then join points with a smooth curve. Use the graph to estimate medians, quartiles, and percentiles. The median is at cumulative frequency n/2, Q1 at n/4, Q3 at 3n/4.
累积频数是频数的逐次累加。绘制累积频数图时，将组上限对累积频数描点，然后用平滑曲线连接各点。使用该图估计中位数、四分位数和百分位数。中位数位于累积频数n/2处，Q1在n/4处，Q3在3n/4处。
A box plot (box-and-whisker plot) visually summarises minimum, Q1, median, Q3, and maximum. It can also show outliers as separate points. Box plots are excellent for comparing distributions.
箱线图（盒须图）直观地汇总最小值、Q1、中位数、Q3和最大值。它还可以将异常值显示为单独的点。箱线图非常适合比较分布。
Interpreting box plots involves looking at the spread. A longer box or whisker indicates greater variability. Positions of the median inside the box indicate skewness.
解读箱线图涉及观察离散性。较长的箱子或须表示更大的变异性。中位数在箱内的位置指示偏度。
4. Probability Foundations | 概率基础
Probability is a measure of the likelihood that an event will occur, given a number between 0 (impossible) and 1 (certain). The sum of probabilities of all mutually exclusive outcomes in a sample space is 1. For an event A, P(A) = number of favourable outcomes / total number of outcomes, provided all outcomes are equally likely.
概率是事件发生可能性的度量，用介于0（不可能）与1（确定）之间的数字表示。样本空间中所有互斥结果的概率之和为1。对于事件A，如果所有结果等可能，则 P(A) = 有利结果数 / 总结果数。
The complement rule: P(not A) = 1 − P(A). Two events are mutually exclusive if they cannot occur at the same time; for mutually exclusive events A and B, P(A or B) = P(A) + P(B).
互补规则：P(非A) = 1 – P(A)。如果两个事件不能同时发生，则它们互斥；对于互斥事件A和B，P(A或B) = P(A) + P(B)。
For non-mutually exclusive events, the general addition rule: P(A or B) = P(A) + P(B) − P(A and B).
对于非互斥事件，一般加法规则：P(A或B) = P(A) + P(B) – P(A且B)。
5. Conditional Probability & Tree Diagrams | 条件概率与树形图
Conditional probability is the probability of an event happening given that another event has already occurred. It is written as P(A|B) = P(A and B) / P(B). This formula is fundamental when events are not independent.
条件概率是在另一事件已经发生的情况下，某事件发生的概率。写为 P(A|B) = P(A且B) / P(B)。当事件不独立时，此公式至关重要。
Two events A and B are independent if P(A|B) = P(A) or equivalently P(A and B) = P(A) × P(B). Tree diagrams are used to represent sequences of events. Multiply probabilities along branches to find the probability of combined outcomes. If sampling without replacement, probabilities change after each pick, indicating conditional probability.
如果满足 P(A|B) = P(A) 或等价地 P(A且B) = P(A) × P(B)，则事件A和B独立。树形图用于表示事件序列。沿分支相乘概率以找到组合结果的概率。如果进行不放回抽样，每次抽取后概率发生变化，表明存在条件概率。
Use tree diagrams for multi-stage experiments. When probabilities vary depending on previous stage, label the second set of branches with conditional probabilities. The total probability of all final outcomes adds to 1.
对于多阶段实验使用树形图。当概率依赖于前一阶段时，用条件概率标注第二组分支。所有最终结果的概率总和为1。
6. Expected Value & Relative Frequency | 期望值与相对频率
Expected value (or expectation) is the long-run average outcome of a probability experiment if repeated many times. It is calculated as E(X) = Σ [x × P(X = x)]. For example, in rolling a fair die, the expected value is 3.5. Expected value can be a non-integer even when outcomes are integers.
期望值（或期望）是如果多次重复进行某个概率实验，从长期来看的平均结果。计算公式为 E(X) = Σ [x × P(X = x)]。例如，投掷一枚公平的骰子，期望值为3.5。期望值可以是非整数，即使结果是整数。
Relative frequency is an estimate of probability based on observed data: relative frequency = number of times event occurs / total number of trials. As the number of trials increases, relative frequency tends to get closer to the theoretical probability (law of large numbers).
相对频率是基于观测数据对概率的估计：相对频率 = 事件发生次数 / 总试验次数。随着试验次数的增加，相对频率往往接近理论概率（大数定律）。
Expected frequencies in a contingency table or goodness-of-fit test are found by multiplying row and column totals and dividing by grand total. Use expected values for the chi-squared test (OCR higher tier).
列联表或拟合优度检验中的期望频数通过行合计与列合计相乘再除以总计得到。在OCR高阶层次，使用期望值进行卡方检验。
7. Scatter Graphs & Correlation | 散点图与相关性
A scatter graph shows the relationship between two quantitative variables. Plot each pair (x, y) as a point. The pattern reveals the type of correlation: positive (as x increases, y tends to increase), negative (as x increases, y tends to decrease), or no correlation.
散点图显示两个定量变量之间的关系。将每对 (x, y) 作为一个点绘制。模式揭示相关类型：正相关（随着x增加，y倾向于增加），负相关（随着x增加，y倾向于减少），或无相关。
The strength of linear correlation is measured by the product-moment correlation coefficient, usually denoted r. In OCR GCSE Statistics, you may use technology or given formula to find r. r ranges from -1 to 1 where -1 is perfect negative, +1 perfect positive, and 0 indicates no linear correlation. r near 0 does not rule out a non-linear relationship.
线性相关强度由积差相关系数衡量，通常记为r。在OCR GCSE统计中，你可以使用技术或给定公式求出r。r的取值范围为-1到1，-1表示完全负相关，+1表示完全正相关，0表示无线性相关。r接近0并不排除非线性关系。
Spearman’s rank correlation coefficient is used when data are ordinal or when outliers make Pearson’s r inappropriate. Rank both sets of data separately, compute differences d for each pair, then use rs = 1 − (6Σd²) / (n(n²−1)).
当数据为顺序数据或异常值使得皮尔逊r不合适时，使用斯皮尔曼等级相关系数。分别对两组数据进行排秩，计算每对秩的差值d，然后使用 rs = 1 – (6Σd²) / (n(n²-1))。
8. Regression Lines & Prediction | 回归线与预测
If there is a linear correlation, we can fit a regression line to make predictions. The equation of a straight line is y = a + bx, where b is the gradient (slope) and a is the y-intercept. The least squares regression line of y on x minimises the sum of squared vertical distances of points from the line.
如果存在线性相关，我们可以拟合一条回归线进行预测。直线的方程是 y = a + bx，其中b是斜率，a是y轴截距。y对x的最小二乘回归线使各点与线的垂直距离平方和最小。
Formulas: b = Sxy / Sxx, where Sxy = Σ(x − x̄)(y − ȳ) and Sxx = Σ(x − x̄)². Then a = ȳ − b x̄. Alternatively, using raw data: b = [nΣxy − (Σx)(Σy)] / [nΣx² − (Σx)²].
公式：b = Sxy / Sxx，其中 Sxy = Σ(x – x̄)(y – ȳ) 且 Sxx = Σ(x – x̄)²。然后 a = ȳ – b x̄。或者，使用原始数据：b = [nΣxy – (Σx)(Σy)] / [nΣx² – (Σx)²]。
Use the regression line for interpolation (predicting within the range of observed x values) but be cautious about extrapolation beyond the data, as the relationship might not hold. The line gives an estimate, not an exact value.
使用回归线进行内插（在观测到的x值范围内预测），但要谨慎外推至数据范围之外，因为关系可能不成立。回归线给出的是估计值，而非精确值。
9. Index Numbers | 指数
Index numbers are used to compare changes over time. The base year index is typically set to 100. A simple price index for a single item: index number = (current price / base year price) × 100. This shows the percentage change relative to the base. For example, an index of 120 means a 20% increase from the base year.
指数用于比较随时间的变化。基年指数通常设为100。单个商品的简单价格指数：指数 = (现价 / 基年价格) × 100。这显示相对于基年的百分比变化。例如，指数为120表示比基年增长20%。
When combining multiple items, weighted index numbers are used. A weighted aggregate index combines prices and quantities. Retail Price Index (RPI) and Consumer Price Index (CPI) are real-world examples. The weighted index number formula is (Σ(weight × current price) / Σ(weight × base price)) × 100.
当组合多个商品时，使用加权指数。加权综合指数将价格和数量结合在一起。零售价格指数（RPI）和消费者价格指数（CPI）是现实世界的例子。加权指数公式为 (Σ(权重 × 现价) / Σ(权重 × 基价)) × 100。
Chain base index numbers compare each period with the immediately preceding period, rather than a fixed base. This helps monitor short-term changes. Convert between chain base and fixed base by multiplying sequential chain indices.
链基指数将每个时期与紧前时期进行比较，而非固定基期。这有助于监测短期变化。通过将连续的链基指数相乘，可以在链基与固定基指数之间转换。
10. Probability Distributions: Binomial | 概率分布：二项分布
In OCR GCSE Statistics higher tier, the binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. If X ~ B(n, p), then P(X = r) = ⁿCᵣ × pʳ × (1−p)ⁿ⁻ʳ, where ⁿCᵣ = n! / (r! (n−r)!).
在OCR GCSE统计高阶层次中，二项分布模拟固定次数的独立试验中成功的次数，每次试验的成功概率p相同。若 X ~ B(n, p)，则 P(X = r) = ⁿCᵣ × pʳ × (1−p)ⁿ⁻ʳ，其中 ⁿCᵣ = n! / (r! (n−r)!)。
The mean (expected value) of the binomial distribution is μ = np. The variance is σ² = np(1−p). The standard deviation is √[np(1−p)]. For the distribution to be valid, trials must be independent and p constant.
二项分布的均值（期望值）为 μ = np。方差为 σ² = np(1−p)。标准差为 √[np(1−p)]。要使用二项分布，试验必须独立且p保持不变。
Normal approximation to the binomial (OCR may mention): when n is large and p is not too close to 0 or 1, B(n, p) can be approximated by a normal distribution with mean np and variance np(1−p), using continuity correction.
二项分布的正态近似（OCR可能提及）：当n较大且p不靠近0或1时，B(n, p)可用均值为np、方差为np(1−p)的正态分布近似，需使用连续性校正。
11. Sampling & Bias | 抽样与偏差
Sampling methods are vital for collecting representative data. A simple random sample gives each member of the population an equal chance of being selected. Systematic sampling selects every kth member after a random start. Stratified sampling divides the population into groups (strata) and samples proportionally from each. Cluster sampling is used when populations are geographically dispersed.
抽样方法对于收集代表性数据至关重要。简单随机抽样使总体中的每个成员有相等的被选中的机会。系统抽样在随机起点之后每隔k个选择一名。分层抽样将总体分成群组（层），并从每一层按比例抽样。整群抽样用于人口地理分散时。
Bias occurs when the sampling method systematically favours certain outcomes. Selection bias, non-response bias, and measurement bias are common. A questionnaire can introduce bias through leading questions or limited response options. Avoid bias by using randomisation and careful design.
当抽样方法系统性地有利于某些结果时，就会产生偏差。常见的有选择偏差、无应答偏差和测量偏差。问卷可能通过诱导性问题或有限的回答选项引入偏差。通过随机化和精心设计避免偏差。
Capture-recapture is used to estimate population size: mark a sample of size m, release, then capture a second sample of size n, count how many are marked (r). Population estimate N ≈ (m × n) / r. This is part of OCR GCSE Statistics.
捉放法用于估计种群大小：标记一个大小为m的样本，放回，然后捕获第二个大小为n的样本，统计其中有多少已标记（r）。种群估计值 N ≈ (m × n) / r。这是OCR GCSE统计的一部分。
12. Time Series & Moving Averages | 时间序列与移动平均
A time series is data recorded at regular time intervals. It often shows trend, seasonal variation, and random fluctuation. Moving averages smooth out short-term fluctuations to reveal the underlying trend. To calculate a moving average of order k, average each block of k consecutive values.
时间序列是按固定时间间隔记录的数据。它通常显示趋势、季节变动和随机波动。移动平均消除短期波动以揭示潜在趋势。计算k阶移动平均时，对每个连续的k个值求平均。
The moving average is plotted at the middle of the time interval covered. For even order moving averages (e.g., 4-point), centring is required: average two successive moving averages. Seasonal variation is then isolated by subtracting the trend (moving average) from the actual data. Average seasonal effects can be used to adjust or forecast values.
移动平均绘制在涵盖时间间隔的中间。对于偶数阶移动平均（如4点），需要中心化：求两个连续移动平均的平均。然后通过从实际数据中减去趋势（移动平均）来分离季节变动。平均季节效应可用于调整或预测数值。
Time series graphs should be labelled with time on the horizontal axis and the variable on the vertical. Plot moving averages on the same graph to show smoothing.
时间序列图应在横轴标时间，纵轴标变量。在同一图上绘制移动平均以显示平滑效果。
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