📚 Year 8 AQA Statistics: International Competition Preparation Guide | Year 8 AQA 统计：国际竞赛备战攻略
Whether you are preparing for the UKMT Junior Mathematical Challenge, the AMC 8 or an International Kangaroo Mathematics competition, statistics questions always appear on the paper. This guide links the statistics you meet in Year 8 AQA lessons - averages, charts, tables and relative frequency - with competition-style questions and the strategies that actually win marks.
无论你是在备战 UKMT 初级数学挑战（JMC）、AMC 8，还是国际袋鼠数学竞赛，统计题总会出现在试卷上。本指南把 Year 8 AQA 课堂上学到的统计知识——平均数、图表、表格与相对频率——与竞赛式题目以及真正能拿分的策略连接起来。
1. Understanding Competition Statistics Questions | 理解竞赛统计题
A competition statistics question rarely looks like a textbook exercise. The data is often squeezed into one small diagram, and the wording carries extra information: a title, axis labels, and units, a key or a footnote. Before you calculate anything, read the whole question twice and decide what is actually being asked for - a single value, a comparison, or a judgement about whether a claim is fair.
竞赛统计题很少长得像课本练习。数据常被塞进一张小图里，而题干还带着额外信息：标题、坐标轴标签、单位、图例或脚注。在动笔计算之前，先把题目完整读两遍，并确定题目真正要什么——一个数值、一个比较，还是对某个说法是否合理的判断。
Underline key words as you read: 'total', 'difference', 'more than', 'per cent of', 'estimate', 'not possible'. In a multiple-choice paper these words decide which of the five options is correct, and one misread word turns an easy question into a lost mark.
边读边划出关键词：total（总计）、difference（相差）、more than（多于）、per cent of（……的百分比）、estimate（估计）、not possible（不可能）。在选择题试卷中，正是这些词决定五个选项中哪一个是正确答案，而看错一个词就会把简单题变成丢分题。
- Read the title and the key first. A small legend can change the whole meaning of a chart. | 先读标题和图例。一个小小的图例就能改变整张图的含义。
- Check what the units are. Minutes or seconds? Pounds or pence? kg or g? | 先确认单位。是分钟还是秒？英镑还是便士？千克还是克？
- Decide the type of answer required. A number, a fraction, a percentage or a written reason. | 判断答案的形式：数值、分数、百分数，还是文字理由。
- Do not start calculating before you know the target. | 在明确目标之前不要开始计算。
2. Reading Graphs Carefully: Axis Labels, Units and Zero Lines | 细读图表：坐标轴标签、单位与零线
The single most common competition trap is the truncated vertical axis. If a bar chart's vertical axis starts at 90 instead of 0, a rise from 92 to 96 looks like a huge jump when it is only a small change. Always check where the vertical axis begins before you describe any trend.
竞赛中最常见的陷阱就是被截断的纵轴。如果条形图的纵轴从 90 开始而不是从 0 开始，那么从 92 升到 96 看上去就是巨大增长，而实际上变化很小。在描述任何趋势之前，先检查纵轴的起点。
Then check the scale step. One grid square might represent 1, 5, 10 or even 0.5. Count the number of squares between two labelled values and divide to find the value of a single square.
接着检查刻度步长。一格可能代表 1、5、10，甚至 0.5。数一数两个已标注数值之间有多少格，再相除，就能得出每一格的数值。
- Vertical axis: does it start at zero? If not, differences are exaggerated. | 纵轴：是否从零开始？若不是，差异会被夸大。
- Horizontal axis: is it categorical (types of fruit) or numerical (time in minutes)? | 横轴：是分类数据（水果种类）还是数值数据（时间／分钟）？
- Units: they may be hidden in the axis label rather than written beside each number. | 单位：单位可能只写在坐标轴标签里，而不是标在每个数字旁。
- Gaps and class widths: in a grouped bar chart, unequal widths change how areas should be compared. | 间隔与组距：在分组条形图中，宽度不等会改变面积比较的方式。
3. Estimation and Elimination in Multiple-Choice Contests | 选择题中的估算与排除法
Multiple-choice papers reward estimation. Because the mean of a set of numbers must lie between the smallest and the largest value, any option outside that interval can be crossed out immediately, often leaving only one plausible answer.
选择题试卷奖励估算能力。由于一组数据的平均数必定介于最小值与最大值之间，任何落在这个区间之外的选项都可以立刻划掉，往往只剩下一个合理答案。
Two quick structural checks catch many wrong options before you do any arithmetic:
在你做任何运算之前，两个快速的结构性检查就能筛掉许多错误选项：
- Range check. The mean cannot be smaller than the smallest value or larger than the largest. | 范围检查。平均数不可能小于最小值，也不可能大于最大值。
- Divisibility check. If a mean is the sum of n numbers divided by n, the total must be a multiple of n when all data are integers. | 整除检查。若平均数等于 n 个数之和除以 n，当所有数据都是整数时，总和必须是 n 的倍数。
- Magnitude check. Estimate to one significant figure: 19.8 × 5.1 is roughly 20 × 5 = 100. | 数量级检查。取一位有效数字估算：19.8 × 5.1 约为 20 × 5 = 100。
- Direction check. If the question asks for a decrease, an option larger than the original is automatically wrong. | 方向检查。若题目要求求下降量，比原数更大的选项必然错误。
÷ 6 = 13. Estimating first tells you the answer must be a little above 12, so any option such as 11 or 15 can be rejected at once.
例题：五个数的平均数是 12，所以总和是 5 × 12 = 60。加入第六个数 18 后，新总和是 60 + 18 = 78，新平均数是 78 ÷ 6 = 13。先估算就知道答案应略高于 12，因此像 11 或 15 这样的选项可以立刻排除。
4. Spotting 'Hidden' Data: Totals That Must Match | 识别"隐藏"数据：必须吻合的合计
Charts and tables carry built-in consistency checks. A pie chart showing percentages must have sectors summing to 100%, and its angles must sum to 360°. A frequency table must account for every item counted, and a two-way table must agree row by row and column by column. If the numbers do not add up, you have either misread the question or found the missing piece.
图表和表格自带一致性检查。显示百分比的扇形图，各扇区百分比之和必须是 100%，角度之和必须是 360°。频数表必须计入每一个被统计的对象，而双向表的行合计与列合计必须彼此吻合。如果数字对不上，要么是你读错了题，要么你已经找到了缺失的那一块。
| Sector angle | 扇区角度 | Fraction | 分数 | Percentage | 百分比 |
| 36° | 1/10 | 10% |
| 72° | 1/5 | 20% |
| 90° | 1/4 | 25% |
| 120° | 1/3 | 33.3% |
| 180° | 1/2 | 50% |
ludes 10 but not 20. Overlapping or missing class boundaries are a favourite way to hide an error, and so is an 'other' or 'none of these' category that quietly holds the remaining data.
在分组频数表中要记住，像 10 ≤ x < 20 这样的组包含 10 但不包含 20。组界重叠或缺失是隐藏错误的惯用手法，"其他"或"以上都不是"这类类别悄悄装着剩余数据，也是同样的套路。
5. Averages: Mean, Median and Mode Under Competition Pressure | 平均数、中位数与众数：竞赛压力下的处理
The three averages answer three different questions. The mean uses every value and is the one that changes when the total changes. The median splits the data into two equal halves and ignores how extreme the ends are. The mode records the most frequent value, and a data set may have no mode at all, one mode, or several.
三种平均数回答三个不同的问题。平均数用到每一个数值，也是随总和变化而变化的那个。中位数把数据分成相等的两半，不受两端极端值的影响。众数记录出现最频繁的数值，而一组数据可能没有众数，也可能有一个或多个众数。
Mean = (Σ of all values) ÷ (number of values)
an is the middle one, which is 7. The mode is also 7 because it appears twice. The range is 12 − 4 = 8.
例题：对于数据 4, 6, 7, 7, 9, 11, 12，总和为 56，共 7 个数值，所以平均数为 56 ÷ 7 = 8。当有序数据的个数为奇数时，中位数就是正中间的那个数，即 7。众数同样是 7，因为它出现了两次。极差为 12 − 4 = 8。
- Even number of values: the median is the mean of the two middle values. | 偶数个数值：中位数是中间两个数的平均数。
- Missing value questions: work backwards from the total. If the mean of 6 values is 9, the total is 54. | 求缺失值：从总和倒推。若 6 个数的平均数是 9，总和就是 54。
- Combined groups: total the group totals, then divide by the combined count - never average the two means directly. | 合并分组：先把各组合计相加，再除以总个数——切勿直接把两个平均数再平均一次。
- Frequency tables: multiply each value by its frequency before adding. | 频数表：先把每个数值乘以其频数，再相加。
6. Range, Spread and the Effect of Outliers | 极差、离散程度与异常值的影响
The range is the crudest measure of spread: largest minus smallest. It is easy to calculate and easy to misread, because a single extreme value can make a tightly grouped set look wildly scattered.
极差是最粗略的离散程度度量：最大值减最小值。它容易计算，也容易误导，因为一个极端值就能让本来很集中的一组数据看起来极度分散。
Consider the data 3, 5, 6, 7, 9. The mean is 6 and the median is 6. Now add the value 100. The new total is 130 over 6 values, giving a mean of about 21.7, while the median becomes 6.5. The median has barely moved; the mean has been dragged upwards by the outlier. Competition questions often exploit exactly this difference.
来看数据 3, 5, 6, 7, 9。平均数是 6，中位数也是 6。现在加入数值 100。新的总和为 130，共 6 个数值，平均数为约 21.7，而中位数变为 6.5。中位数几乎没动，平均数却被这个异常值拖高了。竞赛题常常正是利用这一差别出题。
- When the data are skewed, quote the median. House prices and waiting times are classic examples. | 数据偏斜时，应引用中位数。房价和等待时间就是典型例子。
- When the total matters, quote the mean. If every value contributes to a total, the mean is the right average. | 当总量重要时，应引用平均数。若每个数值都对总和有贡献，平均数才是合适的代表值。
- Always quote a measure of spread alongside an average. Two classes can share a mean of 60 yet be completely different. | 报平均数时务必同时给出离散程度的度量。两个班的平均分可以都是 60，但分布可能完全不同。
- Quartiles are a KS4 extension - useful for wider reading, but not required in Year 8. | 四分位数属于 KS4 的拓展内容——作为拓展阅读很有用，但 Year 8 并不要求。
7. Comparing Two Data Sets: Diagrams and Reasoning | 比较两组数据：图表与推理
Comparison questions almost always expect two sentences, not one. State which set has the higher average, then state which set is more consistent. A complete answer sounds like: 'Group A has the higher median, so on a typical result Group A performed better, but Group B has the smaller range, so Group B's results were more consistent.'
比较类题目几乎总是要求两句话，而不是一句。先说哪一组平均数更高，再说哪一组更稳定。完整答案的格式是："A 组的中位数更高，所以从典型表现看 A 组更好；但 B 组的极差更小，所以 B 组的成绩更稳定。"
Back-to-back stem-and-leaf diagrams are the classic competition tool here, because they put two sets of ordered data on one shared stem and make the median and range readable at a glance. If the leaves are not written in order, reorder them first - an unordered stem-and-leaf plot is a trap, not a shortcut.
背靠背茎叶图是这类题目的经典工具，因为它们把两组有序数据放在同一个"茎"上，一眼就能读出中位数和极差。如果"叶"不是按顺序写的，先重新排序——没排序的茎叶图是陷阱，不是捷径。
- Compare like with like: same units, same axis scale, same class widths. | 同类相比：单位相同、坐标轴刻度相同、组距相同。
- Use one mark per correct statement in written questions and aim for two distinct points. | 文字题中每个正确说法得一分，力争写出两个不同的要点。
- Avoid saying 'it is more spread out' without giving the range or the values. | 不要只说"分布更分散"，要给出极差或具体数值。
- Mention the outlier explicitly if one set contains an extreme value. | 若某组数据含极端值，要明确提到这个异常值。
8. Probability Meets Statistics: Relative Frequency | 概率与统计交汇：相对频率
In competitions, probability and statistics are frequently mixed in one question. Experimental probability, also called relative frequency, is found by dividing the number of successful trials by the total number of trials. As the number of trials increases, this estimate usually moves closer to the true theoretical probability.
在竞赛中，概率与统计经常出现在同一道题里。实验概率（也叫相对频率）等于成功试验次数除以试验总次数。随着试验次数增加，这个估计值通常会越来越接近真实的理论概率。
Relative frequency = (number of successful trials) ÷ (total number of trials)
ectors, the theoretical probability of red is 1 ÷ 5 = 0.2, so 0.23 is a reasonable estimate from only 200 spins. In 500 spins you would expect about 0.2 × 500 = 100 reds.
例题：一个转盘被转动 200 次，其中 46 次落在红色，相对频率为 46 ÷ 200 = 0.23。若转盘是公平的、有 5 个等大扇区，理论上出现红色的概率是 1 ÷ 5 = 0.2，所以只用 200 次试验得到 0.23 是合理的估计。若转 500 次，预期出现红色约 0.2 × 500 = 100 次。
- Expected number = probability × number of trials. | 期望次数 = 概率 × 试验次数。
- Sample size matters. A result from 20 trials is far less reliable than one from 2,000. | 样本量很重要。20 次试验的结果远不如 2,000 次可靠。
- Fairness claims need evidence. A small difference from the expected value is usually just chance. | 判断是否公平需要证据。与期望值的小偏差通常只是偶然。
- Probabilities always lie between 0 and 1, and the probabilities of all mutually exclusive outcomes sum to 1. | 概率总在 0 与 1 之间，所有互斥结果的概率之和等于 1。
9. Charts You Must Recognise Instantly | 必须一眼认出的图表
Competition papers rarely name the chart they use; you are expected to recognise it from the picture alone. Knowing what each chart is designed to show tells you what kind of question is likely to follow.
竞赛试卷很少点明用的是哪种图表，你要能仅凭图形就认出来。知道每种图表"天生"用来展示什么，就能预判接下来会问什么类型的问题。
| Chart | 图表 | What it shows | 展示内容 | Typical question | 典型问题 |
