📚 Year 7 SQA Statistics: Quick Reference Handbook of Formulas and Theorems | Year 7 SQA 统计：公式定理速查手册
Welcome to your quick reference guide for Year 7 SQA Statistics. This handbook summarises the essential formulas, definitions and theorems you need to master data handling and probability. Keep it handy for revision and homework!
欢迎使用 Year 7 SQA 统计速查手册。本手册汇总了你需要掌握的数据处理和概率的基本公式、定义和定理，方便复习和做作业时查阅。
1. Introduction to Statistics | 统计学导论
Statistics is the study of collecting, organising, analysing and interpreting data to discover patterns and make decisions.
统计学是研究如何收集、整理、分析和解读数据以发现规律并做出决策的学科。
Data can be categorical (qualitative), such as favourite colours, or numerical (quantitative), such as heights. Numerical data can be discrete (countable) or continuous (measurable).
数据可以是分类（定性）数据，如喜欢的颜色，也可以是数值（定量）数据，如身高。数值数据可分为离散型（可数）和连续型（可测量）。
A population includes all members of a group; a sample is a smaller, manageable subset used to represent the population.
总体包含某个群体的全部成员；样本是总体的一个较小的、便于研究的子集，用来代表总体。
2. Measures of Central Tendency: Mean | 集中量数：平均数
The mean (or average) is calculated by adding all the data values together and then dividing by the number of values.
平均数是将所有数据值加起来再除以数据的个数。
If there are n data values: x₁, x₂, x₃, …, xₙ, then:
如果有 n 个数据值：x₁, x₂, x₃, …, xₙ，那么：
Mean = (x₁ + x₂ + … + xₙ) ÷ n
平均数 = (x₁ + x₂ + … + xₙ) ÷ n
The mean is sensitive to extremely high or low values (outliers). It is the most commonly used measure of average.
平均数会受到极大或极小值（异常值）的影响。它是最常用的平均数指标。
3. Measures of Central Tendency: Median | 集中量数：中位数
The median is the middle value when all data values are arranged in order from smallest to largest.
中位数是将所有数据值按从小到大的顺序排列后位于中间的那个值。
If there is an odd number of values, the median is the single middle value. Position of median = (n + 1) ÷ 2.
如果有奇数个数值，中位数就是正中间的那个值。中位数的位置 =（n + 1）÷ 2。
If there is an even number of values, the median is the mean of the two middle values.
如果有偶数个数值，中位数是中间两个数值的平均数。
The median is not affected by outliers, making it useful for skewed data.
中位数不受异常值影响，因此在偏态数据中很有用。
4. Measures of Central Tendency: Mode | 集中量数：众数
The mode is the value that occurs most frequently in a data set. A set may have one mode, more than one mode (bimodal or multimodal) or no mode at all if no value repeats.
众数是数据集中出现次数最多的值。一个数据集可能有一个众数，多于一个众数（双众数或多众数），或者如果没有值重复，则没有众数。
The mode is the only measure of central tendency suitable for categorical data.
众数是唯一适用于分类数据的集中量数。
For example, in the data set 3, 7, 7, 8, 9, the mode is 7. In 2, 4, 6, 8, there is no mode.
例如，在数据集 3, 7, 7, 8, 9 中，众数是 7。在 2, 4, 6, 8 中，没有众数。
5. Measures of Spread: Range | 离散量数：极差
The range is a simple measure of spread that tells you how spread out the data are. It is the difference between the maximum and minimum values.
极差是一个简单的离散量数，表明数据的分散程度。它是最大值和最小值之间的差值。
Range = Maximum value − Minimum value
极差 = 最大值 − 最小值
A larger range indicates greater variability. However, the range only uses two values and can be heavily influenced by outliers.
极差越大表示数据变异性越大。但极差只用到了两个值，很容易受异常值影响。
6. Frequency Tables | 频数表
A frequency table lists each data category or value alongside its frequency (count) and often includes tally marks.
频数表列出每个数据类别或数值及其频数（次数），通常还包含计数符号。
From a frequency table, the mean can be estimated using the formula:
利用频数表，可以用以下公式估算平均数：
Mean ≈ Σ (Value × Frequency) ÷ Σ Frequency
平均数 ≈ Σ（数值 × 频数）÷ Σ 频数
Always add a total frequency row. A grouped frequency table is used for continuous data or many different values.
一定要加上总频数行。当处理连续数据或众多不同值时使用分组频数表。
- Cumulative frequency adds up frequencies row by row to find ‘running totals’.
- 累积频数逐行累加频数以得到“累计总数”。
7. Bar Charts and Pictograms | 条形图与象形图
Bar charts use rectangular bars of equal width to represent frequencies. The height or length of each bar corresponds to the frequency of a category.
条形图使用等宽的矩形条来表示频数。每个条形的高度或长度对应相应类别的频数。
Bars in a bar chart should be separated by gaps, and the chart must have labelled axes and a title.
条形图的条形之间应有间隙，图上必须有标注的坐标轴和标题。
Pictograms use symbols or pictures to represent data. A key must indicate how many units each symbol represents (e.g., 1 picture of a book = 5 books read).
象形图用符号或图片表示数据。必须用图例说明每个符号代表多少单位（例如，一本书的图片 = 读了 5 本书）。
Both bar charts and pictograms make it easy to compare frequencies at a glance.
条形图和象形图都能让人一目了然地比较频数。
8. Pie Charts | 饼图
A pie chart shows data as sectors of a circle. The angle of each sector is proportional to the frequency it represents.
饼图将数据显示为圆的扇形。每个扇形的角度与它所代表的频数成正比。
Angle of sector = (Frequency ÷ Total frequency) × 360°
扇形角度 =（频数 ÷ 总频数）× 360°
All sector angles should add up to 360°. Pie charts are best for showing proportions or percentages, not exact frequencies.
所有扇形角度之和应为 360°。饼图最适合显示比例或百分比，而不适合显示精确频数。
9. Introduction to Probability | 概率导论
Probability measures the chance that an event will occur. It is always a number between 0 and 1, inclusive. A probability of 0 means impossible, and 1 means certain.
概率衡量某个事件发生的可能性。它始终是 0 到 1 之间的一个数，包括 0 和 1。0 表示不可能，1 表示必然。
We write the probability of an event A as P(A). The sum of the probabilities of all possible outcomes is 1.
我们将事件 A 的概率记作 P(A)。所有可能结果的概率之和为 1。
Probabilities can also be expressed as fractions, decimals or percentages.
概率也可用分数、小数或百分数表示。
10. Probability Scale and Simple Events | 概率尺度与简单事件
The probability scale helps you describe the likelihood of events using words: impossible (0), unlikely (near 0), even chance (0.5), likely (near 1), certain (1).
概率尺度有助于用词语描述事件的可能性：不可能（0）、不太可能（接近0）、机会均等（0.5）、很可能（接近1）、必然（1）。
For equally likely outcomes, the probability of an event is found by:
对于等可能的结果，可以通过以下方式求事件的概率：
P(event) = Number of favourable outcomes ÷ Total number of possible outcomes
P(事件) = 有利结果的数量 ÷ 所有可能结果的总数
Example: Rolling a fair six-sided die, P(rolling a 4) = 1 ÷ 6 ≈ 0.167.
示例：抛一个均匀的六面骰子，P(掷出4) = 1 ÷ 6 ≈ 0.167。
11. Data Collection and Sampling | 数据收集与抽样
Data can be gathered through primary methods (surveys, experiments, observations) or secondary sources (books, internet).
数据可以通过一手方法（调查、实验、观察）或二手来源（书籍、互联网）收集。
A sample is a small group selected from a population. To be useful, the sample should be random (every member has an equal chance of being chosen) and representative of the whole population.
样本是从总体中选取的一个小群体。有用的样本应该是随机的（每个成员被选中的机会均等）并能代表整个总体。
A census collects data from every member of the population. A survey typically uses a sample to save time and money.
普查从总体的每个成员收集数据。调查通常使用样本来节省时间和费用。
Bias in sampling can lead to misleading conclusions. For example, asking only your friends may not reflect the whole year group.
抽样中的偏差可能导致误导性结论。例如，只询问你的朋友可能无法反映全年级的情况。
12. Interpreting Data: Key Terms | 解读数据：关键术语
- Discrete data: values that can only take certain separate numbers (e.g., number of pets).
- 离散数据：只能取特定分隔数值的数据（如，宠物数量）。
- Continuous data: values that can take any number within a range (e.g., time taken to run 100m).
- 连续数据：可以在某个范围内取任意数值的数据（如，跑 100 米所用时间）。
- Outlier: a data value that is much larger or smaller than the other values in the set.
- 异常值：一个比数据集中其他值大得多或小得多的数值。
- Trend: a general direction in which data changes over time (increasing, decreasing, no change).
- 趋势：数据随时间变化的大致方向（上升、下降、无变化）。
- Mode class (modal class): in grouped data, the class interval with the highest frequency.
- 众数组：在分组数据中，频数最高的组距区间。
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