📚 KS3 OCR Statistics: Formula & Theorem Quick Reference Handbook | KS3 OCR 统计：公式定理速查手册
This quick reference handbook covers all the essential statistical formulas, definitions and theorem summaries you need for the KS3 OCR Statistics course. Keep it handy while revis ng, completing homework or preparing for assessments. Each section presents key ideas in English followed by Chinese, so you can reinforce your understanding bilingually.
这本速查手册涵盖了 KS3 OCR 统计学课程中所有必要的统计公式、定义和定理摘要。无论是复习、完成作业还是准备评估，都可以随时翻阅。每一节先用英文呈现核心概念，再用中文对应讲解，帮助你以双语方式巩固理解。
1. Measures of Central Tendency and Spread | 集中趋势与离散程度的度量
The mean is the arithmetic average of a data set. To find it, add all the values together and then divide by the total number of values.
平均数 是数据集的算术平均值。求平均数的方法是将所有数值相加，然后除以数值的总个数。
Mean = (Sum of all values) ÷ (Number of values)
平均数 = 所有数值总和 ÷ 数值个数
The median is the middle value when the data is arranged in order from smallest to largest. If there is an even number of values, the median is the mean of the two middle numbers.
中位数 是将数据从小到大排序后位于正中间的值。如果数据个数是偶数，中位数就是中间两个数的平均数。
The mode is the value that appears most often in the data set. A set of data may have one mode, more than one mode, or no mode at all.
众数 是数据集中出现次数最多的数值。一组数据可以有一个众数、多个众数，或者没有众数。
The range measures how spread out the numbers are. It is calculated by subtracting the smallest value from the largest value.
极差 用来衡量数据的分散程度。计算公式是用最大值减去最小值。
Range = Maximum value − Minimum value
极差 = 最大值 − 最小值
2. Frequency Distributions | 频数分布
A frequency table lists each distinct data value or category alongside its count, called the frequency. Summing all the frequencies gives the total number of data points.
频数表 列出每个不同的数据值或类别，并注明其出现的次数，即频数。将所有频数相加便得到数据总个数。
From a frequency table, the mode is simply the value or category with the highest frequency.
在频数表中，众数 就是频数最高的那个数值或类别。
You can also use a frequency table to see how the data is distributed and to compare the popularity of different groups.
你还可以利用频数表观察数据的分布情况，并比较不同组的常见程度。
3. Mean from a Frequency Table | 由频数表求平均数
When data is organised in a frequency table, you can still find the mean. First, multiply each data value by its frequency. Then add all these products together and divide by the total frequency.
当数据以频数表的形式整理后，仍然可以求出平均数。先用每个数据值乘以其频数，然后将所有这些乘积相加，最后除以总频数。
Mean = Σ(f × x) ÷ Σf
平均数 = Σ(频数 × 数据值) ÷ 总频数
In this formula, f stands for the frequency of each value and x stands for the data value itself. Σf means the sum of all frequencies, which is the total number of observations.
在这个公式中，f 代表每个值的频数，x 代表数据值本身。Σf 表示所有频数之和，也就是总的观测次数。
4. Grouped Data and Modal Class | 分组数据与模态组
Continuous data is often organised into class intervals or groups. You cannot see the exact original values, so you estimate the mean using the midpoint of each interval.
连续数据经常被整理成 组区间 或组。由于看不到准确的原始数值，所以需要用每个区间的 组中值 来估算平均数。
Estimated Mean = Σ(midpoint × frequency) ÷ Σf
估算平均数 = Σ(组中值 × 频数) ÷ 总频数
The modal class is the interval with the highest frequency. It tells you which group contains the most data values, even though you do not know the exact mode of the raw data.
模态组 是频数最高的组区间。即便不知道原始数据的精确众数，它也能告诉你哪一个组包含了最多的数据值。
When constructing a grouped frequency table, make sure the intervals do not overlap and cover the full range of the data.
在建立分组频数表时，应确保组区间不重叠，并且覆盖数据的整个范围。
5. Bar Charts and Composite Bar Charts | 条形图与复合条形图
A bar chart uses vertical or horizontal bars of equal width to represent frequencies. The length of each bar is proportional to the frequency it represents. Always label both axes and give the chart a clear title.
条形图 用等宽的垂直或水平条形来表示频数。每个条形的长度与其代表的频数成比例。务必在坐标轴上标注清楚，并为图表加上明确的标题。
Composite bar charts (also called stacked bar charts) show sub‑categories within each main group, making it easy to compare totals and the parts inside them.
复合条形图（也叫堆积条形图）可以展示每个主要类别中的子类别情况，便于同时比较总量及其内部的组成部分。
Bar charts are suitable for discrete or categorical data and should never have bars touching unless dealing with specific histogram‑style charts at higher levels.
条形图适用于离散数据或分类数据，除在更高阶段接触的特殊直方图外，条形之间不应相互接触。
6. Pictograms | 象形图
A pictogram uses symbols or pictures to represent a certain quantity of data. It is essential to provide a key that explains what one symbol stands for, and parts of a symbol can be used to show fractions of that quantity.
象形图 用符号或图片来表示一定数量的数据。必须提供 图例 说明每个符号代表多少，并且可以使用符号的一部分来表示小数目的情况。
To read a pictogram, count the number of symbols for each category and multiply by the value each symbol represents. If a symbol is cut in half, it represents half of the key value.
解读象形图时，先数出每一类别的符号个数，再乘以每个符号代表的数量。如果符号被切成一半，则代表图例值的一半。
Pictograms are visually attractive but can be misleading if the symbols are drawn to different sizes or if the scale is not clearly labelled.
象形图直观且吸引人，但如果符号大小不一或刻度没有标注清楚，就可能产生误导。
7. Pie Charts: Sector Angles | 饼图：扇形角度
A pie chart displays proportions by dividing a circle into sectors. The angle of each sector is determined by the relative frequency of its category.
饼图 通过把圆划分为若干个扇形来展示比例关系。每个扇形的角度由该类别对应的相对频数决定。
Sector angle = (Category frequency ÷ Total frequency) × 360°
扇形角度 = (类别频数 ÷ 总频数) × 360°
Check that the sum of all sector angles adds up to 360°. It is a useful way to catch calculation errors.
检查所有扇形角度之和是否为 360°，这是一种发现计算错误的好方法。
Pie charts are best used when there are a small number of categories; too many slices can make the chart hard to read.
饼图最适合用于类别数量较少的情形；扇区太多会让图表难以阅读。
8. Line Graphs | 折线图
A line graph uses points connected by straight lines to show how a variable changes over time or another continuous quantity. Time is usually plotted on the horizontal axis.
折线图 通过点与直线连接的方式来展示某一变量随时间或其他连续量的变化趋势。时间通常绘制在横轴上。
Line graphs are particularly helpful for spotting trends, such as whether a value is increasing, decreasing or staying roughly constant.
折线图尤其有助于发现 趋势，例如数值是上升、下降还是基本保持不变。
Always read the scale carefully; sometimes the axes do not start at zero, which can make changes look more dramatic than they really are.
一定要仔细阅读刻度；有时坐标轴并不是从零开始的，这可能会让变化看起来比实际更剧烈。
If more than one set of data is shown, use different colours or line styles and include a legend.
如果图中展示了多组数据，应使用不同的颜色或线型，并附上图例。
9. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph (or scatter plot) displays values for two variables. Each point represents a pair of values from the data set. It helps you see whether there is a relationship between the two variables.
散点图 用点来展示两个变量的取值。每个点对应数据集中的一对数值，帮助你判断两个变量之间是否存在关系。
Positive correlation means that as one variable increases, the other also tends to increase. Negative correlation means that as one variable increases, the other tends to decrease. No correlation means there is no clear pattern.
正相关 表示当一个变量增加时，另一个变量也倾向于增加。负相关 表示当一个变量增加时，另一个变量倾向于减少。不相关 意味着没有明显的规律。
Correlation does not necessarily imply causation: even if two variables move together, one may not cause the change in the other.
相关性 不一定 意味着因果关系：即使两个变量一起变化，也不能说明一个引起了另一个的改变。
At KS3 you are not expected to draw a line of best fit, but you should be able to describe the correlation from the pattern of points.
在 KS3 阶段不要求绘制最佳拟合线，但你应当能根据点的分布形式描述相关类型。
10. Introduction to Probability | 概率导论
Probability is a measure of how likely an event is to happen. It can be written as a fraction, a decimal or a percentage.
概率 是对某个事件发生可能性的度量，可以用分数、小数或百分数表示。
of an event = (Number of favourable outcomes) ÷ (Total number of possible outcomes)
事件的概率 = 有利结果的数量 ÷ 所有等可能结果的总数
All probabilities lie between 0 and 1 (or 0% and 100%): a probability of 0 means the event is impossible, and a probability of 1 means it is certain.
所有概率值均在 0 到 1 之间（或 0% 到 100% 之间）：概率为 0 表示事件不可能发生，概率为 1 表示事件一定发生。
The sum of the probabilities of all possible mutually exclusive outcomes of an experiment is always 1. This is often called the probability sum rule.
一个试验中所有可能的互斥结果的概率之和 总是等于 1。这通常被称为 概率和定理。
For equally likely outcomes, you can list the sample space (the set of all possible outcomes) to help calculate probabilities.
对于等可能的结果，可以列出 样本空间（所有可能结果的集合）来帮助计算概率。
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