📚 SQA Year 8 Statistics: Case Study Practice | SQA 八年级统计：案例分析实战演练
In statistics, real-life case studies help you apply concepts to actual data. This article guides you through a complete data investigation, from designing a survey to calculating probabilities, using a school library example. You will learn how to collect, organise, display and interpret data, and how to justify your conclusions - just like a real statistician.
在统计学中，现实世界的案例分析有助于你将概念应用于实际数据。本文将通过一个学校图书馆案例，引导你完成从设计调查到计算概率的完整数据探究过程。你将学习如何收集、整理、展示和解读数据，并学会为自己的结论提供依据——就像真正的统计学家一样。
1. Introduction to the Case Study | 案例引入
Our school librarian wants to order new books that students will enjoy. To make informed decisions, we need to collect data on reading preferences and habits. A statistical investigation will help us understand which types of books are most popular and how much time students spend reading each week.
学校图书馆员希望订购学生们会喜欢的新书。为了做出明智的决定，我们需要收集有关阅读偏好和习惯的数据。一项统计调查将帮助我们了解最受欢迎的书籍类型，以及学生每周花多少时间阅读。
A statistical investigation always follows the same cycle, and SQA questions often ask you to describe it. Learn these five stages in order.
统计探究总是遵循同一个循环过程，SQA 题目经常要求你描述它。请按顺序记住以下五个阶段。
- Stage 1: Plan - decide the question, the population and the sample.
- Stage 2: Collect - design and carry out the survey or experiment.
- Stage 3: Organise and display - build tables, charts and graphs.
- Stage 4: Analyse - calculate averages, range and probabilities.
- Stage 5: Interpret and conclude - answer the original question and state the limitations.
- 阶段 1：计划——确定问题、总体和样本。
- 阶段 2：收集——设计并实施问卷调查或实验。
- 阶段 3：整理与展示——制作表格、图表和图像。
- 阶段 4：分析——计算平均数、极差和概率。
- 阶段 5：解读与结论——回答最初的问题，并说明局限。
Our investigation question is: "Which book genres should the library order more of, and how much do Year 8 pupils read each week?" Every later step in this article exists to answer that question with evidence.
我们的调查问题是：“图书馆应该多订购哪些类型的书？八年级学生每周阅读多少时间？”本文后续每一步，都是为了用证据回答这个问题。
2. Data Collection: The Survey | 数据收集：问卷调查
We designed a short questionnaire for 30 Year 8 pupils. It asked two questions: (1) "What is your favourite book genre?" with options Fantasy, Mystery, Fiction, Graphic Novels; and (2) "How many hours do you read for pleasure in a typical week?" with a written numerical answer.
我们为 30 名八年级学生设计了一份简短的问卷，包含两个问题：（1）“你最喜欢的书籍类型是什么？”选项为奇幻、悬疑、小说、图像小说；（2）“在典型的一周里，你出于兴趣阅读多少小时？”要求填写具体数字。
Question 1 produces categorical (qualitative) data because the answers are labels or categories. Question 2 produces numerical (quantitative) data because the answers are measured quantities in hours.
第 1 题产生的是分类数据（定性数据），因为答案是标签或类别；第 2 题产生的是数值数据（定量数据），因为答案是以小时为单位的量。
| Question 问题 | Data type 数据类型 | Display 展示方式 |
| Favourite genre 最喜欢的类型 | Categorical 分类数据 | Bar chart, pie chart 条形图、饼图 |
| Reading hours 阅读时长 | Numerical (discrete/continuous) 数值数据 | Frequency table, histogram, dot plot 频数表、直方图、点图 |
Good survey design matters. Options should be mutually exclusive, cover all sensible answers, and avoid leading wording such as "Do you agree that reading is better than gaming?"
良好的问卷设计很重要。选项应当互斥、覆盖所有合理答案，并且要避免“你是否同意阅读比打游戏更好？”这类诱导性措辞。
3. Types of Data and Sampling | 数据类型与抽样
A population is every member of the group you want to know about; a sample is the smaller group you actually measure. Here the population is all Year 8 pupils in the school and the sample is the 30 pupils we surveyed.
总体是你想了解的全部对象；样本是你实际测量的一小部分。在此例中，总体是全校所有八年级学生，样本是我们调查的 30 名学生。
The way the sample is chosen determines how trustworthy the results are. The main sampling methods you should recognise are listed below.
样本的选取方式决定了结果的可信度。你应当能够辨认以下几种主要抽样方法。
- Random sampling - every member of the population has an equal chance (for example, names drawn from a hat).
- Systematic sampling - every 10th name on a register.
- Stratified sampling - the sample keeps the same proportions as the population, such as 50% boys and 50% girls.
- Convenience sampling - asking whoever is nearby, which is quick but biased.
- 随机抽样——总体中每个成员被选中的机会相同（例如从帽子里抽名字）。
- 系统抽样——按名册每隔 10 人抽取一人。
- 分层抽样——样本保持与总体相同的比例，例如男生 50%、女生 50%。
- 便利抽样——谁在附近就问谁，速度很快但有偏差。
For our library case study we used convenience sampling during one lunchtime, which is easy to organise. Remember this weakness, because it becomes important again in Section 10.
在我们的图书馆案例中，我们利用一个午休时间采用了便利抽样，这样便于组织。请记住这个弱点，因为它会在第 10 节再次变得重要。
4. Organising Data: Frequency Tables | 整理数据：频数表
Raw data is hard to read, so the first job is to count how often each value occurs. This count is called the frequency.
原始数据很难直接阅读，因此第一步是统计每个数值出现的次数，这个次数称为频数。
For Question 1 (favourite genre) the 30 responses produced the following frequency table.
对于第 1 题（最喜欢的类型），30 份回答得到如下频数表。
| Genre 类型 | Tally 划记 | Frequency 频数 | Percentage 百分比 |
| Fantasy 奇幻 | |||| |||| || | 12 | 40% |
| Mystery 悬疑 | |||| || | 7 | ≈23.3% |
| Fiction 小说 | |||| | | 6 | 20% |
| Graphic Novels 图像小说 | |||| | 5 | ≈16.7% |
| Total 合计 | - | 30 | 100% |
Always check the total row: the frequencies must add to the sample size (30) and the percentages must add to 100%. This is the quickest way to spot a counting mistake in an exam.
一定要检查合计行：频数之和必须等于样本容量（30），百分比之和必须等于 100%。这是考试中发现自己数错的最快方法。
For Question 2 (weekly reading hours) the data is spread out, so we group it into intervals. Grouping makes the table shorter but loses the exact values.
对于第 2 题（每周阅读时长），数据比较分散，因此我们按区间分组。分组使表格更简洁，但会丢失具体数值。
| Hours 小时 | Frequency 频数 | Percentage 百分比 |
| 1 to 3 | 6 | 20% |
| 3.5 to 5.5 | 8 | ≈26.7% |
| 6 to 8 | 9 | 30% |
| 8.5 to 10 | 4 | ≈13.3% |
| 10.5 to 15 | 3 | 10% |
| Total 合计 | 30 | 100% |
5. Displaying Data: Bar Charts and Pie Charts | 展示数据：条形图与饼图
Categorical data is best shown with a bar chart or a pie chart. Bars are drawn with equal width and with gaps between them, because the categories are separate labels rather than numbers on a scale.
分类数据最适合用条形图或饼图展示。条形等宽，且条形之间留有间隔，因为各个类别是彼此独立的标签，而不是数轴上的数值。
A bar chart of favourite genres would have four bars with heights 12, 7, 6 and 5. The tallest bar for Fantasy immediately shows which genre is the most popular.
最喜欢的类型的条形图会有四根条形，高度分别为 12、7、6 和 5。奇幻这一项最高，一眼就能看出它最受欢迎。
For a pie chart we convert each frequency into an angle using the fact that a full circle is 360°.
画饼图时，我们把每个频数转换成角度，因为整个圆是 360°。
Sector angle = (frequency ÷ total) × 360°
The four angles add to 144° + 84° + 72° + 60° = 360°, which confirms the calculation is correct.
代入公式可得：奇幻 (12 ÷ 30) × 360° = 144°；悬疑 (7 ÷ 30) × 360° = 84°；小说 (6 ÷ 30) × 360° = 72°；图像小说 (5 ÷ 30) × 360° = 60°。四个角度之和为 144° + 84° + 72° + 60° = 360°，说明计算正确。
Numerical data such as reading hours is shown with a dot plot, a stem-and-leaf diagram or a histogram. A histogram has no gaps between bars because the horizontal axis is a continuous numerical scale.
像阅读时长这样的数值数据，用点图、茎叶图或直方图展示。直方图的条形之间没有间隔，因为横轴是连续的数值刻度。
Every statistical display must have a clear title, labelled axes with units, and an even scale. Marks in SQA papers are regularly lost for a missing unit such as "hours" or "pupils".
每个统计图都必须有清晰的标题、带单位的坐标轴标签，以及均匀的刻度。在 SQA 考试中，因为漏写“小时”或“人”这样的单位而失分非常常见。
6. Measures of Central Tendency | 集中趋势的度量
There are three main averages: the mean, the median and the mode. You should be able to calculate all three, and explain which one best represents the data.
平均数、中位数和众数是三种主要的“平均值”。你需要会计算这三者，并说明哪一个最能代表这组数据。
ading times and divide by 30. The total is 183 hours, so the mean is 183 ÷ 30 = 6.1 hours.
求平均数时，把 30 个阅读时长相加再除以 30。总和为 183 小时，因此平均数为 183 ÷ 30 = 6.1 小时。
Mean = Σx ÷ n = 183 ÷ 30 = 6.1 hours
etween the 15th and 16th values, which are both 6, so the median is (6 + 6) ÷ 2 = 6 hours.
求中位数时，把数据从小到大排列，找出中间的那个数。共有 30 个数据，中位数位于第 15 个和第 16 个数据之间，两者都是 6，所以中位数为 (6 + 6) ÷ 2 = 6 小时。
The mode is the value that appears most often. The value 6 appears four times, more than any other value, so the mode is 6 hours.
众数是出现次数最多的数值。数值 6 出现了四次，多于其他任何数值，所以众数为 6 小时。
- Mean 6.1 hours - uses every value, but is affected by extreme values.
- Median 6 hours - not affected by extreme values, useful when data is skewed.
- Mode 6 hours - the only average that works for categorical data such as genre.
- 平均数 6.1 小时——用到所有数据，但会受极端值影响。
- 中位数 6 小时——不受极端值影响，适合偏斜的数据。
- 众数 6 小时——唯一适用于分类数据（如书籍类型）的平均值
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