📚 Year 8 AQA Statistics: Formula and Key Facts Quick Reference Handbook | Year 8 AQA 统计：公式定理速查手册
This quick reference handbook summarises all the essential formulas, concepts and key facts you need for the Year 8 AQA Statistics topic. It covers data types, averages, range, charts, probability and sampling in a clear bilingual format to support your revision.
本速查手册总结了 Year 8 AQA 统计所需的所有基本公式、概念和关键知识点。涵盖数据类型、平均数、极差、图表、概率和抽样，以清晰的中英双语形式呈现，助力你的复习。
1. Data Types | 数据类型
Data is classified as qualitative (categorical) or quantitative (numerical). Qualitative data describe qualities or categories, such as eye colour, favourite food or car type. They are non-numerical labels.
数据分为定性（分类）数据和定量（数值）数据。定性数据描述品质或类别，如眼睛颜色、最喜欢的食物或汽车类型，是非数值标签。
Quantitative data consist of numbers and can be further split into discrete and continuous. Discrete data can only take certain exact values, usually whole numbers, like number of siblings. Continuous data can take any value within a range, like height or mass.
定量数据由数字组成，可进一步分为离散数据和连续数据。离散数据只能取特定的精确值，通常是整数，如兄弟姐妹的数量。连续数据可以在一个范围内取任意值，如身高或质量。
2. Averages and Range – The Basics | 平均数与极差基础
Three averages summarise the centre of a data set: the mean, the median and the mode. The range is a simple measure of spread. Range = Largest value − Smallest value.
三种平均数概括数据集的中心：均值、中位数和众数。极差是离散程度的简单度量。极差 = 最大值 − 最小值。
Each average has strengths. The mean uses all data but is affected by outliers. The median is not affected by very large or small values. The mode shows the most typical item.
每种平均数各有优点。均值使用所有数据，但受异常值影响。中位数不受极大或极小值影响。众数显示最典型的项目。
3. Mean Calculation | 均值计算
The mean is the sum of all values divided by the number of values. Mean = (∑x) / n, where ∑x represents the sum of all data values and n is the total number of values.
均值是所有数值的总和除以数值的个数。均值 = (∑x) / n，其中 ∑x 代表所有数据值的总和，n 是数据值的总个数。
For example, the mean of 4, 9, 11 is (4 + 9 + 11) ÷ 3 = 24 ÷ 3 = 8.
例如，4、9、11 的均值为 (4 + 9 + 11) ÷ 3 = 24 ÷ 3 = 8。
4. Median and Mode | 中位数与众数
The median is the middle value when data are arranged in order. For an odd number of values, it is the central one. For an even number, it is the mean of the two middle numbers. Position of median = (n + 1) ÷ 2 in an ordered list.
中位数是将数据排序后位于中间的值。奇数个数据时取正中间的值；偶数个时取中间两个数的均值。中位数的位置 = (n + 1) ÷ 2，在有序列表中。
The mode is the value that appears most often. A set can have one mode, two modes (bimodal), more than two (multimodal) or no mode if every value occurs once.
众数是出现频率最高的值。一个数据集中可以有一个众数、两个众数（双峰）、多个众数（多峰），或者如果每个值只出现一次则没有众数。
5. Range and Spread | 极差与离散程度
Range gives the difference between the largest and smallest values: Range = Maximum − Minimum. A larger range shows greater variability in the data.
极差给出了最大值与最小值之间的差值：极差 = 最大值 − 最小值。极差越大，表示数据变异性越大。
Because the range only uses two values, it does not tell us how the rest of the data are spread. Outliers can make the range misleading.
因为极差只用到两个值，它不能说明其余数据的分布情况。异常值可能使极差产生误导。
6. Frequency Tables | 频数表
A frequency table organises data by recording how often each value or group occurs. Use tally marks to count accurately.
频数表通过记录每个值或组出现的次数来整理数据。使用画记法准确计数。
For grouped data, first find the midpoint of each class interval: Midpoint = (Lower bound + Upper bound) ÷ 2. The mean from a grouped frequency table is Mean = ∑(f × x) / ∑f, where f is frequency and x is the midpoint.
对于分组数据，首先计算每个组区间的中点：中点值 = (下限 + 上限) ÷ 2。由分组频数表计算均值的公式为 均值 = ∑(频数 × 中点值) ÷ 总频数，其中 f 为频数，x 为中点值。
To find the median from a frequency table, form the cumulative frequency column and find the value at the (n+1)/2 position (for ungrouped data). For grouped data, the median lies in the class interval containing this position.
从频数表求中位数，需建立累积频数列，并找到第 (n+1)/2 位置对应的值（未分组数据）。对于分组数据，中位数位于包含该位置的组区间内。
7. Charts: Bar Charts, Pie Charts, and Line Graphs | 图表：条形图、饼图和折线图
Bar charts represent frequencies with rectangular bars of equal width. Gaps between bars remind us that the categories are separate. The height of each bar equals the frequency.
条形图用等宽的矩形条表示频数。条形之间的空隙表示类别是独立的。每个条形的高度等于频数。
Pie charts show proportions of a whole. To calculate the angle for each sector: Sector angle = (Frequency / Total frequency) × 360°.
饼图展示整体中的比例。计算每个扇形的角度：扇形角度 = (频数 / 总频数) × 360°。
Line graphs are used to display changes over time. Points are plotted and joined by straight lines. They are excellent for showing trends.
折线图用于显示随时间变化的情况。标出数据点并用直线连接，非常适合展示趋势。
Pictograms use symbols to represent frequency. A key must state how many units each symbol stands for.
象形图使用符号表示频数。图例必须注明每个符号代表多少单位。
8. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph plots bivariate data, with each point representing a pair of values. The independent variable goes on the horizontal axis; the dependent variable on the vertical.
散点图绘制双变量数据，每个点代表一对数值。自变量放在横轴上，因变量放在纵轴上。
Correlation describes the relationship: positive correlation means as one variable increases, the other also tends to increase. Negative correlation means as one increases, the other tends to decrease. No correlation means no clear pattern.
相关性描述变量间的关系：正相关表示一个变量增加时另一个也趋于增加；负相关表示一个增加时另一个趋于减少；无相关表示没有明显的模式。
A line of best fit can be drawn through the points to model the trend. It should have roughly the same number of points above and below it. Only use the line to estimate values within the range of the data (interpolation), not far beyond (extrapolation).
可以通过数据点绘制最佳拟合线来模拟趋势。该线上下方点的数量应大致相同。仅用该线估算数据范围内的值（内插），不可用于范围外较远的估计（外推）。
9. Probability Scale and Basic Probability | 概率尺度与基本概率
Probability measures the chance of an event happening, on a scale from 0 (impossible) to 1 (certain). Probabilities can be written as fractions, decimals or percentages.
概率衡量事件发生的机会，范围从 0（不可能）到 1（必然）。概率可以写成分数、小数或百分比。
When all outcomes are equally likely, the theoretical probability is: P(Event) = Number of favourable outcomes / Total number of possible outcomes.
当所有结果等可能时，理论概率为：P(事件) = 有利结果的数量 / 所有可能结果的总数。
For example, the probability of drawing an ace from a full deck of 52 playing cards is 4/52 = 1/13.
例如，从一副完整的 52 张扑克牌中抽到一张 A 的概率是 4/52 = 1/13。
10. Relative Frequency and Expected Outcomes | 相对频率与期望结果
When an experiment is repeated, relative frequency can be used to estimate probability: Relative frequency = Number of successful trials / Total number of trials.
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