📚 KS3 AQA Statistics: Formula & Theorem Quick Reference Handbook | KS3 AQA 统计：公式定理速查手册
This quick reference handbook provides a concise summary of the essential formulas, theorems, and key concepts for the KS3 Statistics syllabus under the AQA specification. It covers data handling, averages, probability, and statistical diagrams, serving as a handy revision aid for students who need to recall core principles quickly and accurately.
本速查手册简洁总结了AQA KS3统计课程中的核心公式、定理与关键概念，涵盖数据处理、平均数、概率和统计图表，是学生快速、准确回顾核心原理的实用复习工具。
1. Mean, Median and Mode | 平均数、中位数与众数
The mean is calculated by adding all the data values together and then dividing that total by the number of values. This is the most common measure of central tendency and uses every piece of data.
平均数是将所有数据值相加，再除以数值的个数。这是最常用的集中趋势度量，且使用了每一个数据。
Symbolically, the mean of n values x₁, x₂, …, xₙ is given by: mean = (x₁ + x₂ + … + xₙ) ÷ n, which can be written compactly as mean = Σx / n.
符号表示为：n个数值x₁, x₂, …, xₙ的平均数 = (x₁ + x₂ + … + xₙ) ÷ n，可简写为平均数 = Σx / n。
The median is the middle value when the data are arranged in ascending or descending order. If there is an even number of values, the median is the mean of the two middle numbers.
中位数是将数据按升序或降序排列后处于中间位置的数值。如果数值个数为偶数，则中位数是中间两个数的平均数。
The mode is the value that appears most frequently. A data set may have one mode (unimodal), two modes (bimodal), more than two modes (multimodal), or no mode at all if all values occur with the same frequency.
众数是出现次数最多的数值。数据集可能有一个众数（单众数）、两个众数（双众数）、多个众数（多众数），或者当所有数值出现次数相同时没有众数。
2. Range | 极差
ata’s variability. The formula is straightforward: Range = largest value − smallest value.
极差是一种简单的离散程度度量，告诉我们最小值与最大值之间相差多远，能快速反映数据的变异性。公式非常直接：极差 = 最大值 − 最小值。
A larger range indicates that the data are more spread out, while a smaller range suggests the values are more clustered together. However, the range is sensitive to outliers because it depends only on the two extreme values.
极差越大表示数据越分散，极差越小则表示数值更集中。然而，极差对异常值敏感，因为它只取决于两个极端值。
When comparing two or more data sets, the range helps alongside averages to describe the consistency of the data.
在比较多组数据时，极差与平均数一起使用，有助于描述数据的一致性。
3. Probability Basics | 概率基础
Probability measures the chance or likelihood that a particular event will happen. It always lies on a scale from 0 (impossible) to 1 (certain). A probability that is exactly 0.5 means the event is equally likely to happen or not happen.
概率衡量某个特定事件发生的可能性。它总是在0（不可能）到1（必然）的尺度上。概率恰好为0.5表示事件发生与不发生的可能性相等。
mes / total number of possible outcomes. Probabilities can be expressed as fractions, decimals, or percentages.
对于等可能结果的事件A，其理论概率为：P(A) = 有利结果的数量 / 所有可能结果的总数。概率可以用分数、小数或百分数表示。
The sum of the probabilities of all possible mutually exclusive outcomes in a sample space is always 1. This ensures that all possibilities are accounted for.
样本空间中所有可能的互斥结果的概率之和始终为1，这确保所有可能性都被考虑在内。
When an event is impossible, its probability is 0; when it is certain, its probability is 1.
当事件不可能发生时，其概率为0；当事件必然发生时，其概率为1。
4. Experimental Probability | 实验概率
as: Experimental probability = number of times the event occurs / total number of trials.
实验概率（或相对频数）基于实验或试验中收集的实际数据，其计算方式为：实验概率 = 事件发生的次数 / 试验总次数。
This type of probability is not fixed and can vary from one experiment to another. As the number of trials increases, the experimental probability tends to become more stable and approach the theoretical probability — this is known as the Law of Large Numbers.
这类概率不是固定的，会因实验不同而变化。随着试验次数的增加，实验概率趋于稳定并接近理论概率——这就是大数定律。
For example, when tossing a fair coin, you may not get exactly 50% heads in 10 tosses, but over 1000 tosses the proportion of heads is likely to be very close to 0.5.
例如，抛一枚公平硬币时，10次抛掷中正面朝上的概率可能不是恰好50%，但经过1000次抛掷，正面朝上的比例很可能非常接近0.5。
Experimental probability is particularly useful when theoretical probability cannot be easily calculated, such as in predicting the reliability of a machine or the chance of rain.
当理论概率难以计算时，实验概率尤为有用，比如预测机器的可靠性或下雨的可能性。
5. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot occur at the same time. For instance, when rolling a single die, the outcomes ‘getting a 2’ and ‘getting a 5’ are mutually exclusive.
如果两个事件不能同时发生，则称它们为互斥事件。例如，掷一个骰子时，“得到2点”与“得到5点”就是互斥事件。
e addition rule: P(A or B) = P(A) + P(B). This is valid only when A and B have no overlap.
两个互斥事件中任一事件发生的概率可用加法法则求得：P(A或B) = P(A) + P(B)。这仅在A和B没有重叠时才成立。
If events are not mutually exclusive, the addition rule must subtract the probability of the overlap, but at KS3 the focus is mainly on mutually exclusive scenarios.
如果事件不是互斥的，加法法则需要减去重叠部分的概率，但KS3阶段主要关注互斥的情况。
Examples of mutually exclusive events include passing a test and failing the same test (if no other result is possible), or choosing a red sweet and a blue sweet in a single pick from a bag.
互斥事件的例子有：一次考试中及格与不及格（如果没有其他结果），或从袋子里一次只取一颗糖果时，取出红色糖果和取出蓝色糖果。
6. Sample Space Diagrams | 样本空间图
A sample space is the set of all possible outcomes of an experiment. A sample space diagram organises these outcomes systematically, often as a list, a two-way table, or a grid, making it easier to count and calculate probabilities.
样本空间是某一实验所有可能结果的集合。样本空间图系统地将这些结果组织起来，通常表现为列表、双向表格或网格，使计数和概率计算更简便。
For a simple experiment such as flipping a coin and rolling a six-sided die together, the sample space consists of 12 outcomes: (H,1), (H,2), …, (H,6), (T,1), …, (T,6). Each outcome is equally likely.
对于如同时抛硬币和掷六面骰的简单实验，样本空间包含12种结果：(H,1), (H,2), …, (H,6), (T,1), …, (T,6)。每一种结果等可能发生。
To find the probability of an event like ‘ting a tail and an even number’, count the outcomes that satisfy both: (T,2), (T,4), (T,6) — three outcomes. So P = 3/12 = 1/4.
要找出如“得到一个反面和一个偶数”的概率，数出同时满足两个条件的结果：(T,2), (T,4), (T,6) —— 共3种结果。因此 P = 3/12 = 1/4。
A systematic approach, such as filling all cells of a grid, prevents missing outcomes and helps check that total possible outcomes are correctly identified.
系统的方法（例如填满网格的所有格子）可防止遗漏结果，并有助于检查所有可能结果是否正确识别。
7. Bar Charts and Pictograms | 条形图与象形图
Bar charts represent categorical or discrete data using rectangular bars of equal width. The height (or length, if horizontal) of each bar is proportional to the frequency of that category. Bars are separated by small gaps to show the categories are distinct.
条形图使用等宽的矩形条来表示分类数据或离散数据。每个条的高度（若为水平条则是长度）与该类别的频数成比例。条之间留有小间隙，表明类别是独立的。
A pictogram uses a series of repeated identical pictures or symbols to represent frequency. Each symbol stands for a specific number of items, and a key is essential to show what one symbol represents.
象形图使用一系列重复的相同图片或符号来表示频数。每个符号代表特定数量的项目，必须配有图例来说明一个符号代表多少项目。
When interpreting pictograms, you may see half or quarter symbols to show fractions of the symbol’s value. Always read the key carefully before interpreting.
解读象形图时，你可能会看到半个或四分之一个符号，表示该符号数值的一部分。解读前务必仔细阅读图例。
Both bar charts and pictograms are useful for comparing frequencies across different categories at a glance.
条形图和象形图均适合快速比较不同类别间的频数。
8. Pie Charts | 饼图
A pie chart displays data as sectors of a circle, where the size of each sector’s central angle is directly proportional to the frequency it represents. This makes it ideal for showing how a whole is divided into parts.
饼图将数据表示为圆的扇区，每个扇区圆心角的大小与其代表的频数成正比。这使其非常适合展示整体如何被划分成各个部分。
(frequency of category / total frequency) × 360°. The sum of all angles should equal 360°.
绘制饼图时，使用公式计算每个类别的角度：角度 = （该类别的频数 / 总频数）× 360°。所有角度之和应等于360°。
12/30) × 360° = 144°. A protractor is then used to measure and draw the angles accurately.
例如，一项30人的调查显示12人选择了曲棍球，那么曲棍球扇区的角度 = (12/30) × 360° = 144°。然后用量角器精确测量并画出角度。
Pie charts are not suitable when there are too many categories, as sectors become too small to interpret easily.
当类别过多时，饼图就不太适合了，因为扇区会变得太小而不易解读。
9. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph (or scatter plot) is used to display the relationship between two sets of numerical data. Each point represents a pair of values, plotted on a coordinate grid.
散点图（散点图）用于展示两组数值数据之间的关系。每一个点代表一对数值，绘制在坐标网格上。
Correlation describes the type and strength of the association between the two variables. Positive correlation means that as one variable increases, the other tends to increase. Negative correlation means that as one variable increases, the other tends to decrease. No correlation indicates no discernible pattern.
相关性描述了两个变量之间关联的类型和强度。正相关意味着一个变量增加时，另一个也倾向于增加。负相关意味着一个变量增加时，另一个倾向于减少。无相关则意味着没有可辨识的规律。
Correlation does not imply causation. Two variables can show a strong correlation simply due to coincidence or a third hidden factor.
相关性不意味着因果关系。两个变量可能仅仅因为巧合或某个隐藏的第三因素而表现出强相关。
Scatter graphs can also reveal outliers — points that lie far away from the general pattern of the data.
散点图还能揭示异常值——即那些远离数据总体模式的点。
10. Line of Best Fit | 最佳拟合线
When a scatter graph shows a linear correlation, we can draw a straight line to model the trend. This is called a line of best fit (or trend line). It does not have to pass through all points, but should balance them, with roughly equal numbers of points above and below the line.
当散点图显示出线性相关时，我们可以画一条直线来模拟这一趋势，这条线称为最佳拟合线（或趋势线）。它不需经过所有点，但应平衡地穿过点群，线上方和线下方的点数大致相等。
The line of best fit is used to estimate unknown values: reading within the range of the given data is interpolation, which is reliable; reading beyond the data range is extrapolation and is less reliable because the trend may change.
最佳拟合线用于估算未知数值：在给定数据范围内读取称为内插，较为可靠；超出数据范围读取称为外推，可靠性较低，因为趋势可能改变。
Do not simply join the dots. A line of best fit must be a single straight line unless the data clearly suggest a curve, which is not typically covered at KS3.
不要简单地连接各点。最佳拟合线必须是一条单一的直线，除非数据明显呈曲线关系——这在KS3阶段通常不涉及。
The equation of the line of best fit can be found using the gradient and y-intercept, but for KS3, drawing and estimating from the graph is the key skill.
最佳拟合线的方程可以通过斜率和y
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