📚 OCR Year 10 Statistics: Formula & Theorem Quick Reference Handbook | OCR 十年级统计：公式定理速查手册
This quick reference handbook covers the essential formulae, theorems and notation required for Year 10 OCR GCSE Statistics. It is designed for rapid revision and to support problem-solving in data handling, probability and bivariate analysis.
本速查手册涵盖十年级 OCR GCSE 统计所需的核心公式、定理和符号。旨在帮助快速复习，并为数据处理、概率和双变量分析中的问题求解提供支持。
1. Basic Statistical Symbols | 基本统计符号
The following table lists the symbols you will encounter most often in statistical work. Familiarity with these will make reading formulae much easier.
下表列出了统计学习中最常见的符号。熟悉这些符号将让你更容易理解和运用公式。
| Symbol | English meaning | 中文含义 |
|---|---|---|
| n | Sample size | 样本大小 |
| N | Population size | 总体大小 |
| f | Frequency | 频数 |
| ∑ | Sum of (capital sigma) | 求和（大写西格玛） |
| x̅ | Sample mean (x-bar) | 样本均值 |
| μ | Population mean (mu) | 总体均值 |
| s | Sample standard deviation | 样本标准差 |
| σ | Population standard deviation | 总体标准差 |
| Q₁, Q₂, Q₃ | Lower quartile, median, upper quartile | 下四分位数、中位数、上四分位数 |
| IQR | Interquartile range | 四分位距 |
2. Measures of Central Tendency | 中心趋势度量
The arithmetic mean of a set of data is the sum of all values divided by the number of values. For grouped data we use the midpoints of the class intervals.
一组数据的算术平均数是所有数值之和除以数据个数。对于分组数据，我们使用组中值。
Mean (ungrouped): x̅ = ∑x / n
均值（未分组）：x̅ = ∑x / n
Mean (grouped): x̅ = ∑fx / ∑f
均值（分组）：x̅ = ∑fx / ∑f
The median is the middle value when data are ordered. If n is odd, the median is the (n+1)/2 th value; if n is even it is the average of the two middle values.
中位数是排序后位于中间的数据值。若 n 为奇数，中位数为第 (n+1)/2 个值；若 n 为偶数，则为中间两个值的平均数。
The mode is the value or class interval with the highest frequency. A data set can have more than one mode.
众数是出现频数最高的数值或组距。一组数据可以有一个以上的众数。
3. Measures of Dispersion | 离散程度度量
Dispersion describes how spread out the data are. The simplest measure is the range, while variance and standard deviation give more detailed information about variability.
离散程度描述数据的分散情况。最简单的度量是极差，而方差与标准差则提供了更详尽的变异程度信息。
Range = Maximum value − Minimum value
极差 = 最大值 − 最小值
Interquartile range (IQR) = Q₃ − Q₁
四分位距 (IQR) = Q₃ − Q₁
For a sample, variance s² and standard deviation s are calculated using n−1 as the divisor (Bessel’s correction). For a population the divisor is N.
对于样本，方差 s² 和标准差 s 使用 n−1 作为除数（贝塞尔校正）。对于总体，除数为 N。
Sample variance s² = ∑(x − x̅)² / (n − 1)
样本方差 s² = ∑(x − x̅)² / (n − 1)
Sample standard deviation s = √[ ∑(x − x̅)² / (n − 1) ]
样本标准差 s = √[ ∑(x − x̅)² / (n − 1) ]
Population variance σ² = ∑(x − μ)² / N
总体方差 σ² = ∑(x − μ)² / N
4. Percentiles and Quartiles | 百分位数与四分位数
A general method for finding the position of the k-th percentile is to use (k/100)×(n+1) when the data are ordered. If the position is not an integer, interpolation is used.
当数据已排序时，第 k 百分位数位置的通用计算方法是 (k/100)×(n+1)。若位置不是整数，则使用插值法。
Position of Pk = (k/100) × (n + 1)
第 k 百分位数 Pk 的位置 = (k/100) × (n + 1)
The lower quartile Q₁ is the 25th percentile, the median Q₂ is the 50th percentile, and the upper quartile Q₃ is the 75th percentile. They divide the ordered data into four equal parts.
下四分位数 Q₁ 是第 25 百分位数，中位数 Q₂ 是第 50 百分位数，上四分位数 Q₃ 是第 75 百分位数。它们将有序数据分成四等份。
Q₁ position = (n+1)/4, Q₂ position = (n+1)/2, Q₃ position = 3(n+1)/4
Q₁ 位置 = (n+1)/4, Q₂ 位置 = (n+1)/2, Q₃ 位置 = 3(n+1)/4
5. Probability Rules | 概率法则
The addition rule deals with the probability of the union of two events. For any events A and B it subtracts the intersection to avoid double counting.
加法法则处理两事件并集的概率。对于任意事件 A 和 B，公式减去交集部分以避免重复计算。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B are mutually exclusive they cannot happen at the same time, so P(A ∩ B) = 0, and the formula simplifies to P(A ∪ B) = P(A) + P(B).
若 A 与 B 互斥，它们不能同时发生，因此 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。
Two events are independent if the occurrence of one does not affect the probability of the other. The multiplication rule then becomes very simple.
若一个事件的发生不影响另一个事件发生的概率，则两事件独立。此时乘法法则变得非常简单。
For independent events: P(A ∩ B) = P(A) × P(B)
独立事件：P(A ∩ B) = P(A) × P(B)
Conditional probability is the probability of A given that B has already occurred. It forms the basis for working with dependent events and tree diagrams.
条件概率是在事件 B 已经发生的条件下事件 A 发生的概率。它是处理相关事件和树图的基础。
P(A | B) = P(A ∩ B) / P(B), where P(B) > 0
P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0
6. Expectation and Probability Distributions | 期望与概率分布
The expected value of a discrete random variable X is the long-run average of its outcomes, weighted by their probabilities.
离散随机变量 X 的期望值是其结果的长期加权平均值，权重为各结果发生的概率。
E(X) = ∑ [ x × P(X = x) ]
E(X) = ∑ [ x × P(X = x) ]
For a fair die the expectation is (1+2+3+4+5+6)/6 = 3.5. The concept is used to predict outcomes in games of chance and risk analysis.
对于一枚公平的骰子，期望值为 (1+2+3+4+5+6)/6 = 3.5。该概念用于预测博弈结果和进行风险分析。
7. Bivariate Data and Scatter Graphs | 双变量数据与散点图
Bivariate data consist of paired observations of two variables. A scatter graph can reveal the direction, strength and form of an association.
双变量数据由两个变量的成对观测值组成。散点图可以揭示关联的方向、强度和形状。
Correlation measures how closely two variables are linearly related. It ranges from −1 (perfect negative) to +1 (perfect positive), with 0 indicating no linear correlation.
相关度量的是两个变量之间线性关系的密切程度，范围从 −1（完全负相关）到 +1（完全正相关），0 表示无线性相关。
Note that correlation does not imply causation. A third factor may be driving both variables, or the relationship may be coincidental.
注意相关并不蕴含因果关系。可能存在第三个因素驱动两个变量，或该关系纯属巧合。
8. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s coefficient, rₛ, is used to measure the strength and direction of a monotonic relationship between two ranked variables. It is robust to outliers.
斯皮尔曼等级系数 rₛ 用于度量两个排序变量之间单调关系的强度和方向，对异常值不敏感。
rₛ = 1 − (6 ∑ d²) / [ n (n² − 1) ]
rₛ = 1 − (6 ∑ d²) / [ n (n² − 1) ]
Here d is the difference between the ranks of each pair of observations, and n is the number of data pairs. Always rank the data first, using average ranks for ties.
其中 d 为每对观测值等级之差，n 为数据对的数量。务必先对数据排序，遇到并列时使用平均等级。
9. Time Series and Moving Averages | 时间序列与移动平均
A time series records data points at successive time intervals. Smoothing techniques such as moving averages help to identify the underlying trend.
时间序列记录连续时间间隔下的数据点。移动平均等平滑技术有助于识别潜在趋势。
A simple k-point moving average replaces each value with the mean of that value and its neighbours. For example, a three-point moving average is widely used
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