📚 Year 8 Edexcel Statistics: Formula and Theorem Quick Reference Handbook | Year 8 Edexcel 统计：公式定理速查手册
This bilingual quick reference handbook presents a clear and concise summary of all the essential formulas, definitions and theorems needed for the Year 8 Edexcel Statistics curriculum. Covering topics from data types, averages and probability to charts, correlation and sampling, each key concept is explained in both English and Chinese. Use this handbook to support homework, consolidate classwork and prepare confidently for tests and examinations.
这本双语速查手册清晰、简明地总结了 Year 8 Edexcel 统计学课程所需的所有基本公式、定义和定理。内容涵盖数据类型、平均数、概率、图表、相关性以及抽样等主题，每个核心概念均配有中英文解释。利用本手册辅助作业、巩固课堂知识，并为测验和考试做好充分准备。
1. Types of Data | 数据类型
Data can be classified as qualitative or quantitative. Qualitative data describes qualities or categories (e.g. eye colour, favourite food). Quantitative data represents numerical measurements and can be further divided into discrete data and continuous data. Discrete data can only take certain separate values, often counted in whole numbers (e.g. number of siblings). Continuous data can take any value within a range and is measured rather than counted (e.g. height, mass, time).
数据可以分为定性数据或定量数据。定性数据描述的是性质或类别（如眼睛颜色、最喜欢的食物）。定量数据表示数值测量，并可进一步分为离散数据和连续数据。离散数据只能取某些独立的值，通常用整数计数（如兄弟姐妹的数量）。连续数据可以取某个范围内的任何值，是测量而非计数得到的（如身高、质量、时间）。
Identifying data types correctly helps decide which statistical measures and diagrams are appropriate. For example, you calculate a mean for quantitative data but not for qualitative data; a pie chart can show qualitative categories, while a histogram is used for continuous grouped data.
正确识别数据类型有助于确定采用何种统计度量和图表。例如，定量数据可以计算平均数，而定性数据则无法计算平均数；饼图可以展示定性类别，而直方图则用于连续的分组数据。
2. Mean, Median, Mode, and Range | 平均数、中位数、众数和极差
The three averages summarise a data set’s typical value, while the range measures spread. The mean is the sum of all data values divided by the number of values. The median is the middle value when the data are arranged in order. If there are two middle values, the median is the mean of those two. The mode is the value that appears most often. The range is the difference between the largest and smallest values.
三种平均数总结了数据集的典型值，而极差衡量数据的分散程度。平均数是所有数据值的总和除以数据的个数。中位数是将数据按顺序排列后位于中间的值；如果中间位置有两个值，则中位数是这两个值的平均数。众数是出现频率最高的值。极差是最大值与最小值之差。
Mean = (x₁ + x₂ + … + xₙ) / n
平均数 = (x₁ + x₂ + … + xₙ) / n
Range = Largest value – Smallest value
极差 = 最大值 – 最小值
For an odd number of ordered values, the median is the middle term. For an even number, locate the two middle terms, add them together, and divide by 2. The mode may not exist if no value repeats, or there can be more than one mode.
当有序数据个数为奇数时，中位数就是最中间的那一项。当为偶数时，找出中间的两项，将它们相加后除以 2。如果没有重复值，众数可能不存在；也可能存在多个众数。
3. Mean from Frequency Tables (Ungrouped and Grouped) | 频数表（未分组和分组）的平均数
When data are organised in a frequency table, the mean can be calculated using the totals of the ‘value × frequency’ products. For ungrouped data, multiply each distinct value by its frequency, sum all these products, then divide by the total frequency.
当数据以频数表形式整理时，可以利用“数值 × 频数”的乘积总和来计算平均数。对于未分组数据，将每个不同的数值乘以其频数，将所有乘积相加，然后再除以总频数。
Mean = (∑ f·x) / ∑ f
平均数 = (∑ f·x) / ∑ f
For grouped data, we do not know the exact values, so we use the midpoint (m) of each class interval as an estimate. Multiply each midpoint by its frequency, sum the products, and divide by the total frequency. The result is an estimated mean.
对于分组数据，我们不知道具体的数值，因此使用每个组距的中值 (m) 作为估算值。将每个中值乘以其频数，求和后除以总频数，所得结果即为估计的平均数。
Estimated Mean ≈ (∑ f·m) / ∑ f, where m = (lower bound + upper bound) / 2
估计平均数 ≈ (∑ f·m) / ∑ f，其中 m = (下限 + 上限) / 2
4. Probability Basics | 概率基础
Probability measures how likely an event is to happen. It is always a number between 0 and 1, where 0 means impossible and 1 means certain. Probability can be written as a fraction, decimal, or percentage.
概率衡量某个事件发生的可能性大小。概率值总是在 0 到 1 之间，0 表示不可能发生，1 表示必然发生。概率可以用分数、小数或百分数表示。
P(Event) = Number of favourable outcomes / Total number of equally likely outcomes
P(事件) = 有利结果的数量 / 所有等可能结果的总数
For a fair six-sided dice, the probability of rolling a 3 is 1/6. The sum of the probabilities of all possible mutually exclusive outcomes is 1. The probability of an event not occurring is 1 minus the probability that it does occur.
对于一个公平的六面骰子，掷出 3 的概率是 1/6。所有互斥的可能结果概率之和为 1。某个事件不发生的概率等于 1 减去该事件发生的概率。
5. Experimental Probability and Relative Frequency | 实验概率与相对频率
When we cannot calculate theoretical probability, we can estimate it by conducting an experiment or survey. The relative frequency of an event is the number of times the event occurs divided by the total number of trials. As the number of trials increases, the relative frequency tends to settle closer to the theoretical probability.
当我们无法计算理论概率时，可以通过实验或调查来估算它。事件的相对频率等于该事件发生的次数除以试验总次数。随着试验次数的增加，相对频率会趋向稳定，更接近理论概率。
Relative Frequency = Frequency of event / Total number of trials
相对频率 = 事件发生的频数 / 试验总次数
If a coin is flipped 100 times and lands on heads 47 times, the experimental probability (relative frequency) of heads is 47/100 = 0.47. We use relative frequency to make predictions: expected number of successes = probability × number of trials.
如果一枚硬币抛掷 100 次，出现正面 47 次，那么正面的实验概率（相对频率）为 47/100 = 0.47。我们可以用相对频率做预测：期望成功次数 = 概率 × 试验次数。
6. Pie Charts and Angle Calculation | 饼图与角度计算
A pie chart is a circular diagram divided into sectors, where each sector represents a category. The angle of each sector is proportional to the frequency of the category. Since a full circle is 360°, the angle for a category is calculated using the formula below.
饼图是一种将圆形分成多个扇形的图表，每个扇形代表一个类别。每个扇形的角度与类别的频数成比例。由于整个圆为 360°，类别的角度可用以下公式计算。
Sector Angle = (Frequency / Total Frequency) × 360°
扇形角度 = (频数 / 总频数) × 360°
Always check that the sum of all sector angles equals 360° and that each angle is correctly labelled or accompanied by a key. To interpret a pie chart, compare sector sizes; the larger the angle, the greater the proportion of the whole.
务必检查所有扇形角度之和等于 360°，并且每个角度都有正确的标签或图例。解读饼图时，请比较各扇形的大小；角度越大，占总体的比例就越大。
7. Stem and Leaf Diagrams | 茎叶图
A stem and leaf diagram organises data while preserving the original values. The ‘stem’ represents the leading digit(s), and the ‘leaf’ represents the final digit. A key must always be included to show the place value. This diagram makes it easy to find the median, mode, and range.
茎叶图在整理数据的同时保留了原始数值。“茎”表示前导数字，“叶”表示最后一位数字。图中必须包含一个键来说明数位。这种图可以很方便地找出中位数、众数和极差。
A typical key: 4 | 7 means 47 or 3 | 2 means 3.2. Leaves are written in ascending order and can be repeated. A back-to-back stem and leaf diagram is used to compare two data sets sharing a common stem.
典型键如：4 | 7 表示 47 或 3 | 2 表示 3.2。叶片按升序排列并可以重复。背对背茎叶图用于比较共用同一茎的两组数据。
8. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph displays the relationship between two sets of quantitative data. Each point represents a pair of values (x, y). Correlation describes the pattern of points: positive correlation means as x increases, y tends to increase; negative correlation means as x increases, y tends to decrease; no correlation means there is no clear pattern.
散点图展示两组定量数据之间的关系。每个点代表一对数值 (x, y)。相关性描述点的分布模式：正相关意味着 x 增大时 y 也趋于增大；负相关意味着 x 增大时 y 趋于减小；无相关则没有明显的模式。
A line of best fit (trend line) can be drawn when there is clear correlation. The line should have roughly equal numbers of points on both sides and can be used to estimate values. You can estimate a missing y‑value for a given x‑value (interpolation) within the data range, but extrapolation beyond the range is less reliable.
当存在明显相关性时，可以画出最佳拟合线（趋势线）。线条两侧的点数应大致相等，并可用于估算数值。可以在数据范围内对给定的 x 值估算对应的 y 值（内插法），但超出范围的外推则不够可靠。
9. Two-way Tables and Relative Frequency | 双向表与相对频率
A two-way table (or contingency table) summarises the frequencies of two categorical variables simultaneously. It helps answer questions about joint frequencies and conditional probabilities. Marginal totals are the sums of each row and column.
双向表（或称列联表）同时汇总两个类别变量的频数。它有助于回答关于联合频数和条件概率的问题。边际总和是每行和每列的总计。
To find the relative frequency of a combination, divide the cell frequency by the total. To find a conditional probability, use the appropriate row or column total as the denominator. Always determine which total is relevant.
若要计算某个组合的相对频率，则将单元格频数除以总频数。若计算条件概率，则使用相应的行总计或列总计作为分母。务必确定哪个总计是相关的。
10. Sampling Methods | 抽样方法
When it is impractical to survey an entire population, a sample is selected. A simple random sample gives every member an equal chance of being chosen, helping to avoid bias. A systematic sample selects members at regular intervals from an ordered list. A convenience sample is based on ease of access, but it may be biased and not representative.
当调查整个总体不可行时，会选取一个样本。简单随机抽样让每个成员都有相等的被选中机会，有助于避免偏差。系统抽样从有序列表中每隔固定间隔选取成员。便利抽样基于易于获取样本，但可能存在偏差且不具代表性。
The larger the sample size, the more reliable the results tend to be. When designing a sample, it is important to specify the target population and sampling frame clearly. Avoid leading questions in surveys to maintain objectivity.
样本量越大，结果往往越可靠。设计样本时，要明确界定目标总体和抽样框。调查中应避免引导性问题，以保持客观性。
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