📚 Year 11 AQA Statistics: Core Concepts Revision | AQA 统计：核心知识点梳理
This guide brings together the essential knowledge you need for the AQA GCSE Statistics exam. Whether you are revising data types, working with charts, or calculating averages and probability, a clear understanding of each topic will boost your confidence and performance. Use this structured revision resource to consolidate key concepts, practise with examples, and master exam techniques.
本文梳理了 AQA GCSE 统计学考试所需的核心知识点。无论你是复习数据类型、处理图表，还是计算平均数与概率，清晰地理解每个主题都能增强信心、提升成绩。利用这份结构化的复习资源来巩固关键概念、练习例题并掌握考试技巧。
1. Types of Data | 数据类型
Data is classified into qualitative (categorical) and quantitative (numerical) types. Qualitative data, such as eye colour or favourite sport, describes qualities and cannot be measured on a numerical scale. Quantitative data involves numbers and can be further divided into discrete data (countable, whole-number values like the number of students in a class) and continuous data (measurable values that can take any number within a range, such as height or time).
数据可分为定性（分类）数据和定量（数值）数据。定性数据，如眼睛颜色或最喜欢的运动，描述的是属性，不能用数值尺度测量。定量数据涉及数字，可进一步分为离散数据（可计数的整数值，如班级学生数）和连续数据（可测量的、在一定范围内可取任意值的量，如身高或时间）。
Another important distinction is between primary and secondary data. Primary data is collected first-hand by the researcher for a specific purpose, while secondary data has been gathered by someone else, such as census figures or newspaper reports. Knowing the type of data helps you choose appropriate diagrams, averages, and analysis methods.
另一个重要区分是初级数据和二手数据。初级数据由研究者为特定目的亲自收集；二手数据则是由他人收集的，如人口普查数据或新闻报道。了解数据类型有助于选择合适的图表、平均数及分析方法。
2. Data Collection & Sampling | 数据收集与抽样
Data collection methods include surveys, questionnaires, interviews, experiments, and observation. When designing a questionnaire, questions should be clear, unbiased, and avoid leading the respondent. Pilot surveys help test and refine the questions before the main data collection.
数据收集方法包括调查、问卷、访谈、实验和观察。设计问卷时，问题应清晰、无偏见，避免引导受访者。先导调查有助于在主数据收集前测试并改进问题。
Sampling is used when it is impractical to collect data from an entire population. A simple random sample ensures every member has an equal chance of selection. A stratified sample divides the population into distinct groups (strata) and takes a proportional random sample from each. Systematic sampling selects every kth individual after a random start. A representative sample reduces bias and allows valid conclusions about the population.
当难以从整个总体收集数据时，就采用抽样。简单随机样本确保每个成员被选中的机会均等。分层抽样将总体分为不同组（层），然后从每组按比例抽取随机样本。系统抽样在随机起点后每隔 k 个个体选取一个。有代表性的样本可减少偏差，使得关于总体的结论有效。
3. Presenting Data: Charts & Diagrams | 数据展示：图表
t categories. Pie charts display proportions of a whole, with each sector angle calculated as (frequency ÷ total) × 360°. Pictograms use symbols to represent a certain number of units, and a key must be included.
图形表示使数据更易于解读。条形图用于分类或离散数据，条形之间分开以显示不同类别。饼图显示整体中各部分的比例，每个扇形的角度为（频数 ÷ 总数）× 360°。象形图用符号表示一定数量的单位，必须附上图例。
quency polygon, points are plotted at the midpoints of class intervals and joined with straight lines. Stem-and-leaf diagrams keep original data values visible while showing the shape of the distribution.
对于连续数据，直方图用频率密度展示，面积与频数成正比。频率密度 = 频数 ÷ 组距。在频率多边形中，将点标在组中值上并用直线连接。茎叶图在显示分布形态的同时保留了原始数据值。
4. Measures of Central Tendency | 集中趋势度量
The three main averages are the mean, median, and mode. The mean is calculated as Σx ÷ n for raw data or Σfx ÷ Σf for a frequency table. The median is the middle value when data is ordered; for an odd number of values, it is the (n+1)/2 th term. The mode is the most frequent value. Each average has strengths: the mean uses all data but is affected by outliers, the median is robust to outliers, and the mode is useful for non‑numerical data.
三种主要的平均数是平均数、中位数和众数。对于原始数据，平均数公式为 Σx ÷ n；对于频数表，则为 Σfx ÷ Σf。中位数是数据排序后的中间值；数据个数为奇数时，为第 (n+1)/2 项。众数是出现频率最高的值。每种平均数各有优点：平均数用到所有数据但受异常值影响，中位数对异常值稳健，众数则对非数值数据有用。
From a frequency table, you can find the modal class (the class with the highest frequency) and the median class interval by working out cumulative frequencies. For grouped data, the mean is estimated using the midpoints of class intervals.
通过频数表，你可以找出众数所在组（频数最高的组），并通过累积频数确定中位数所在区间。对于分组数据，利用组中值估算平均数。
5. Measures of Dispersion | 离散程度度量
Dispersion describes how spread out the data is. The range is the difference between the largest and smallest values. It is easy to calculate but heavily influenced by outliers. The interquartile range (IQR) is the difference between the upper quartile (Q₃) and lower quartile (Q₁), where Q₁ is the median of the lower half and Q₃ the median of the upper half. IQR gives the spread of the middle 50% of data and is more stable than the range.
离散程度描述数据的分散程度。极差是最大值与最小值之差，计算简单但易受异常值影响。四分位距（IQR）是上四分位数（Q₃）与下四分位数（Q₁）之差，其中 Q₁ 是下半部分数据的中位数，Q₃ 是上半部分的中位数。四分位距反映了中间 50% 数据的散布情况，比极差更稳健。
The standard deviation measures how far values typically deviate from the mean. For a sample, it is given by:
s = √[ Σ(x - x̄)² / (n - 1) ]
. A larger standard deviation indicates more variability. The variance is s². These measures are used alongside the mean in symmetric distributions.标准差衡量数据值通常偏离平均数的程度。样本标准差的公式为：
s = √[ Σ(x - x̄)² / (n - 1) ]
。标准差越大表明变异性越大。方差即 s²。这些度量与平均数一同用于对称分布。6. Grouped Data & Cumulative Frequency | 分组数据与累积频率
ss is the interval with the highest frequency. The median interval is located by finding the position (n/2) or (n+1)/2 in a cumulative frequency table.
当连续数据被分组为区间时，我们利用组中值估算平均数。估算平均数 = Σ(f × 组中值) ÷ Σf。众数所在组是频数最高的区间。中位数所在区间通过在累积频数表中找到第 (n/2) 或 (n+1)/2 个位置来确定。
A cumulative frequency graph (ogive) plots cumulative frequency against the upper class boundary. The median, quartiles, and percentiles can be read from the graph by drawing horizontal lines from the required cumulative frequencies to the curve, then dropping vertically to the horizontal axis. The interpercentile range (e.g. 10th to 90th percentile) is a measure of spread that avoids extreme values.
累积频率图（肩形图）将累积频数绘制在组上限处。从所需的累积频数画水平线与曲线相交，再垂直向下读轴，即可得中位数、四分位数和百分位数。百分位距（如第 10 至第 90 百分位数）是一种避免极端值的离散程度度量。
7. Box Plots & Comparing Distributions | 箱线图与分布比较
ers extend to the minimum and maximum (or to 1.5 × IQR from the quartiles if identifying outliers). Box plots are excellent for comparing two or more distributions side by side.
箱线图显示最小值、下四分位数（Q₁）、中位数、上四分位数（Q₃）和最大值。箱体覆盖 Q₁ 到 Q₃，一条线标示中位数；须线延伸至最小值和最大值（若识别异常值，则至距四分位数 1.5 × IQR 处）。箱线图非常适合并排比较两个或多个分布。
When comparing distributions, comment on central tendency (median), spread (IQR or range), skewness, and any notable values such as outliers. A distribution skewed to the right has a longer whisker towards larger values; skewed to the left has a longer whisker towards smaller values. The median line inside the box indicates symmetry when it is centrally placed.
比较分布时，需评论集中趋势（中位数）、离散程度（IQR 或极差）、偏态以及任何值得注意的值（如异常值）。向右偏斜的分布在较大数值方向的须线更长；向左偏斜则在较小值方向须线更长。箱体内中位线的位置居中时表明分布对称。
8. Scatter Graphs & Correlation | 散点图与相关性
A scatter graph shows the relationship between two numerical variables. Correlation describes the direction and strength of this relationship. Positive correlation means that as one variable increases, the other tends to increase; negative correlation means as one increases, the other tends to decrease. No correlation indicates no apparent linear relationship.
散点图展示两个数值变量之间的关系。相关性描述该关系的方向和强度。正相关意味着一个变量增大时另一个变量也倾向于增大；负相关则是一个增大、另一个减小。不相关表明没有明显的线性关系。
The line of best fit (regression line) is drawn by eye, passing through the mean point (x̄, ȳ). It can be used to make predictions (interpolation within the data range, which is reliable; extrapolation outside the range, which is less reliable). Spearman's rank correlation coefficient rₛ measures the strength of a monotonic relationship, especially when data is ranked.
最佳拟合线（回归线）用目测画出，经过平均值点 (x̄, ȳ)。它可用于作预测（在数据范围内内插，较为可靠；超出范围外推，不太可靠）。斯皮尔曼等级相关系数 rₛ 衡量单调关系的强度，尤其适用于排序数据。
9. Probability | 概率
e outcomes ÷ total number of possible outcomes, provided all outcomes are equally likely. Relative frequency from an experiment gives an estimate of probability when theoretical probabilities are unknown.
概率衡量事件发生的可能性，表示为 0 到 1 之间的数字。若所有结果等可能，事件 A 的概率 P(A) = 有利结果数 ÷ 可能结果总数。当理论概率未知时，通过实验得出的相对频率可用来估计概率。
tiplication rule for independent events: P(A and B) = P(A) × P(B). Tree diagrams help organise outcomes for multi-stage experiments; multiply along branches and add probabilities for combined events.
互斥事件的加法规则：P(A 或 B) = P(A) + P(B)。对于非互斥事件，P(A 或 B) = P(A) + P(B) - P(A 且 B)。独立事件的乘法规则：P(A 且 B) = P(A) × P(B)。树状图有助于组织多步实验的结果；沿分支相乘，再将组合事件的概率相加。
10. Time Series & Moving Averages | 时间序列与移动平均
A time series is a set of data collected at regular time intervals, such as monthly sales figures. A time series graph plots data against time, revealing seasonal variation and an overall trend. Trend lines can be drawn by inspection or by calculating moving averages to smooth out short-term fluctuations and highlight the long-term direction.
时间序列是按固定时间间隔收集的一组数据，例如月度销售额。时间序列图将数据按时间绘制，可揭示季节性波动和总体趋势。趋势线可通过目测画出，也可通过计算移动平均来平滑短期波动，突出长期走势。
For an odd-number of points in the moving average, each average is placed against the middle time point. For example, a 3-point moving average of values y₁, y₂, y₃ gives (y₁ + y₂ + y₃)/3 placed at time 2. Seasonal effects can be estimated by subtracting the trend value from the actual data. Forecasting future values assumes the trend and seasonal pattern continue.
若移动平均的项数为奇数，每个平均值置于时间中点上。例如，数值 y₁, y₂, y₃ 的 3 点移动平均为 (y₁ + y₂ + y₃)/3，位于时间点 2。季节性效应可通过实际数据减去趋势值来估算。预测未来数值的前提是趋势和季节模式持续不变。
11. Index Numbers | 指数
e relative is (current price ÷ base price) × 100. A weighted index, such as the Retail Prices Index, accounts for the relative importance of items by using weights.
指数通过将数量表示为相对于基期值，简化跨越时间的比较。基期通常设为 100。简单价比 = (现价 ÷ 基期价) × 100。加权指数，如零售价格指数，通过使用权重来体现各项目的重要性。
The formula for a weighted aggregate index is:
Index = Σ(weight × price relative) ÷ Σ(weights)
. Chain-based indices link together several periods by using the previous period as the base. Index numbers must be interpreted carefully; a rise from 100 to 110 indicates a 10% increase from the base period.加权综合指数的公式为：
指数 = Σ(权重 × 价比) ÷ Σ(权重)
。链式指数通过将前期作为基期，把多个时期联结起来。对指数的解读需谨慎；从 100 升至 110 表示比基期上涨 10%。Published by TutorHao | Statistics Revision Series | aleveler.com
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