KS3 WJEC Statistics: Formulae and Theorems Quick-Reference Handbook | KS3 WJEC 统计：公式定理速查手册

📚 KS3 WJEC Statistics: Formulae and Theorems Quick-Reference Handbook | KS3 WJEC 统计：公式定理速查手册

This quick-reference handbook collects the essential formulae, definitions, and theorems for KS3 WJEC Statistics. Use it to revise key concepts, check your understanding, and prepare confidently for assessments. Each section presents the core idea in English first, immediately followed by its Chinese translation, so you can learn bilingually and deepen your grasp of statistical language.

本速查手册汇集了 KS3 WJEC 统计课程的核心公式、定义与定理。可用于复习关键概念、检验理解，并自信地备考。每个部分的要点均先以英文呈现，紧接着提供中文译文，帮助你在双语环境中稳固掌握统计术语与思想。

1. Data Types and Collection Methods | 数据类型与收集方法

Data can be classified as qualitative (categorical) or quantitative (numerical). Qualitative data describe qualities, such as eye colour or favourite subject. Quantitative data are numbers, such as height in centimetres or test scores.

数据可分为定性（分类）数据与定量（数值）数据。定性数据描述性质，例如眼睛颜色或最喜爱的科目；定量数据是数字，例如以厘米为单位的身高或考试分数。

Quantitative data are further split into discrete and continuous types. Discrete data can only take specific values (e.g. number of students), while continuous data can take any value within a range (e.g. temperature).

定量数据进一步分为离散型与连续型。离散数据只能取特定值（如学生人数），连续数据则可在一定范围内取任意值（如温度）。

Primary data are collected directly by the researcher for a specific purpose. Secondary data are obtained from existing sources, such as government statistics or previous surveys.

原始数据由研究者为特定目的直接收集；二手数据则取自现有来源，如政府统计数据或过往调查。

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2. Frequency Tables | 频率表

A frequency table lists distinct data values (or groups) and shows how often each occurs. The tally column records marks for each observation, then the total count is written in the frequency column.

频率表列出不同的数据值（或分组）并显示每个值出现的次数。计数栏记录每个观测的记号，然后总次数写入频率栏。

The sum of all frequencies, often called total frequency and denoted by n or Σf, equals the number of data items. If grouped, the table should include class intervals.

所有频率之和，常称为总频数，记作 n 或 Σf，等于数据项的总数。若数据分组，表格需包含组距。

Total frequency: n = Σf

总频数：n = Σf

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3. Charts and Graphs | 图表

Bar charts represent discrete or categorical data. The height or length of each bar shows the frequency. The bars must have equal widths and clear gaps between them unless drawing a histogram for continuous data.

条形图表示离散或分类数据。每个条形的高度或长度显示频数。条形宽度应相等，且条形之间应有清晰的间隙（除非为连续数据绘制直方图）。

Pie charts display proportions. To find the angle for each sector:

饼图显示比例。计算各扇区角度的公式为：

Angle = (Category frequency ÷ Total frequency) × 360°

角度 = （类别频数 ÷ 总频数） × 360°

Line graphs are suited for showing changes over time. The horizontal axis usually carries time or another continuous variable, and consecutive points are joined with straight lines.

折线图适合展示随时间的变化。横轴通常表示时间或其他连续变量，相邻数据点用直线段连接。

Pictograms use symbols to represent a certain number of items. A key must show the value of one whole symbol, and parts of symbols are used for partial amounts.

象形图使用符号代表一定数量的项目。图例必须说明一个完整符号的值，部分符号用于表示不足一个单位的部分。

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4. Measures of Central Tendency | 集中趋势度量

The mean is the arithmetic average. It is calculated by summing all data values and dividing by the total number of values.

平均值是算术平均数，计算方法为将所有数据值相加再除以数据个数。

Mean = (Sum of all values) ÷ (Number of values) or x̄ = Σxᵢ ÷ n

平均值 = 总和 ÷ 数据个数，或 x̄ = Σxᵢ ÷ n

For grouped data, use the midpoints of class intervals. Multiply each midpoint by its frequency, sum these products, and divide by total frequency.

对于分组数据，使用组中值。将每个组中值乘以其频数，求和后除以总频数。

Estimated mean = Σ(f × midpoint) ÷ Σf

估计平均值 = Σ（频数 × 组中值） ÷ Σf

The median is the middle value when data are arranged in order. If there are two middle numbers, the median is their mean.

中位数是将数据按大小顺序排列后位于中间的值。若中间有两个数，则中位数为这两个数的平均值。

The mode is the value that appears most frequently. A data set may have more than one mode (bimodal) or no mode at all.

众数是出现次数最多的数据值。数据集可能有一个以上众数（双众数），也可能没有众数。

The range is a simple measure of spread: largest value minus smallest value.

极差是简单的离散度量：最大值减最小值。

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5. Comparative Statistics and the Range | 比较统计量与极差

To compare two data sets, calculate the mean, median, mode, and range for each. The mean gives an overall average, while the median is less affected by extreme values. The range tells you about consistency; a smaller range often indicates less variability.

比较两个数据集时，分别计算各组的平均值、中位数、众数和极差。平均值给出整体平均数，中位数受极端值影响较小。极差反映一致性——极差较小通常意味着变异性较小。

An outlier is a data point that lies far outside the rest of the values. Outliers can significantly change the mean but have little effect on the median.

异常值是远离其他数据点的值。异常值会显著改变平均值，但对中位数影响很小。

When comparing sets, always refer back to the context (e.g. “Class A scored higher on average, but Class B had a more consistent spread”).

比较数据集时，一定要结合背景解释（例如“A 班平均分更高，但 B 班分数更集中”）。

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6. Probability Basics | 概率基础

Probability measures how likely an event is to happen. It is expressed as a number between 0 (impossible) and 1 (certain), or as a fraction, decimal, or percentage.

概率衡量事件发生的可能性。它表示为 0（不可能）到 1（必然）之间的数，也可用分数、小数或百分数表示。

Probability of an event = (Number of favourable outcomes) ÷ (Total number of possible outcomes)

事件概率 = 有利结果数 ÷ 所有可能结果总数

For equally likely outcomes, the probability of each outcome is the same. The probability of an event not happening is 1 minus the probability that it does happen.

对于等可能结果，每个结果的概率相同。事件不发生的概率等于 1 减去该事件发生的概率。

P(not A) = 1 – P(A)

P（非 A）= 1 – P（A）

All possible outcomes together form the sample space. The sum of probabilities of all outcomes in the sample space is 1.

所有可能的结果构成样本空间。样本空间中所有结果的概率之和为 1。

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7. Experimental Probability and Expected Frequency | 实验概率与期望频数

Experimental probability (relative frequency) is calculated from actual trials or observations.

实验概率（相对频率）由实际试验或观察计算得出。

Experimental probability = (Number of times event occurs) ÷ (Total number of trials)

实验概率 = 事件发生次数 ÷ 试验总次数

The more trials are conducted, the closer the experimental probability tends to get to the theoretical probability. This is the law of large numbers.

试验次数越多，实验概率越趋近于理论概率，这称为大数定律。

Expected frequency predicts how many times an event should occur over a number of trials.

期望频数预测在一定试验次数中事件应该发生的次数。

Expected frequency = Probability × Number of trials

期望频数 = 概率 × 试验次数

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8. Two-Way Tables and Venn Diagrams | 双向表与韦恩图

A two-way table displays data according to two categorical variables. Totals are found in the margins, enabling calculations of row and column probabilities.

双向表根据两个分类变量展示数据。总计位于边缘，便于计算行概率与列概率。

From a two-way table you can find simple probabilities (e.g., probability a student is both female and studies art) and conditional frequencies.

利用双向表可求简单概率（如某学生既是女生又修读艺术的概率）和条件频率。

Venn diagrams show overlaps between sets. The rectangle represents the universal set, and circles represent subsets. The intersection is where circles overlap; the union covers all elements in either set.

韦恩图展示集合间的重叠。矩形表示全集，圆表示子集。圆重叠区域为交集；并集包含任一集合中的所有元素。

Probability from a Venn diagram uses the number in a region divided by the total in the universal set.

从韦恩图求概率，用某区域内的元素数量除以全集的元素总数。

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9. Tree Diagrams for Probability | 树状图与概率

Tree diagrams help list all possible outcomes of two or more events. Branches show probabilities at each stage; multiply along branches for combined outcomes.

树状图有助于列出两个或多个事件的所有可能结果。分支显示每个阶段的概率；沿分支相乘得到组合结果的概率。

The probabilities on branches from a single point must sum to 1. For independent events, the outcome of one does not affect the probability of the other.

同一点出发的各分支概率之和必须为 1。对于独立事件，一个事件的结果不影响另一个事件的概率。

The probability of two independent events both happening is found by multiplication.

两个独立事件同时发生的概率通过相乘求得。

P(A and B) = P(A) × P(B)

P（A 且 B）= P（A） × P（B）

For mutually exclusive events (they cannot happen together), use addition.

对于互斥事件（不能同时发生），使用加法求概率。

P(A or B) = P(A) + P(B)

P（A 或 B）= P（A） + P（B）

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10. Scatter Diagrams and Correlation | 散点图与相关性

A scatter diagram plots two sets of paired data on the same graph. Each point shows the values of two variables for one item.

散点图在同一图上绘制两组配对数据。每个点显示一个项目在两个变量上的数值。

Correlation describes the relationship between variables. Positive correlation: as one variable increases, the other tends to increase. Negative correlation: as one increases, the other tends to decrease. No correlation: no clear pattern.

相关性描述变量之间的关系。正相关：一个变量增大，另一个也倾向于增大。负相关：一个增大，另一个倾向于减小。无相关：无明显模式。

A line of best fit (trend line) can be drawn by eye, balancing points above and below it. The line allows predictions by interpolation (within the data range) but extrapolation (outside the range) should be treated with caution.

最佳拟合线（趋势线）可通过目测画出，使线上下的点数大致平衡。该线可用于内插预测（在数据范围内），但外推（超出范围）须谨慎对待。

Outliers are points that lie far from the general trend; they should be identified and considered separately.

远离总体趋势的点为异常点，应识别并单独考量。

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11. Sampling Methods and Bias | 抽样方法与偏差

A population is the entire group being studied. A sample is a subset selected from the population. Random sampling gives every member an equal chance of being chosen and helps avoid bias.

总体是所研究的全部对象。样本是从总体中选取的子集。随机抽样使每个成员有同等被选中的机会，有助于避免偏差。

Stratified sampling divides the population into distinct groups (strata) and a random sample is taken from each group in proportion to its size.

分层抽样先将总体分为不同组别（层），然后按比例从每组中随机抽样。

Number from stratum = (Stratum size ÷ Population size) × Total sample size

层内抽样数 = （层的大小 ÷ 总体大小） × 总样本量

Bias occurs when a sample is not representative. Convenience sampling (e.g. asking only your friends) or voluntary response sampling may produce biased results because some members are more likely to be included.

样本不具代表性即产生偏差。便利抽样（例如只调查朋友）或自愿应答抽样可能产生偏差，因为某些成员更易被纳入。

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12. Statistical Investigations and Interpreting Data | 统计调查与数据解读

A statistical investigation follows the cycle: pose a question, plan data collection, collect data, process and represent data, interpret and draw conclusions. Critically evaluate limitations, such as small sample size or biased questions.

统计调查遵循以下循环：提出问题、规划数据收集、收集数据、处理与展示数据、解释并得出结论。要批判性地评估其局限性，如样本量小或问题带有偏见。

Be aware of misleading graphs where axes do not start at zero, scales are stretched, or 3D effects distort proportions. Always check the labels and scales before trusting a visual representation.

警惕误导性图表：坐标轴不从零开始、刻度被拉伸或三维效果扭曲比例。在信任任何可视化呈现前，务必检查标签和刻度。

Whether using mean, median, or mode, relate your conclusions back to the original question and support them with numerical evidence.

无论使用平均数、中位数还是众数，都要将结论与原始问题关联，并用数值证据支撑。

Published by TutorHao | Statistics Revision Series | aleveler.com

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