📚 Year 9 WJEC Statistics: Case Study Practice | 九年级 WJEC 统计：案例分析实战演练
Welcome to this real-world statistics case study. In this article, we will step through a complete statistical investigation suitable for Year 9 WJEC Statistics. You will learn how to design a survey, collect and organise data, present it using charts, calculate averages and spread, and use probability to interpret your findings. The case study focuses on the physical activity habits of Year 9 students at a school. By working through this practical example, you will see how every stage of the statistical enquiry cycle connects. You will also develop the skills needed to carry out your own investigations and critically evaluate results.
欢迎来到这个真实的统计案例研究。在本文中，我们将完整地走一遍适合九年级 WJEC 统计课程的实际调查。你将学习如何设计调查问卷、收集和整理数据、用统计图表呈现数据、计算平均值和离散程度，并利用概率来解释发现的结果。本案例研究的重点是某校九年级学生的体育活动习惯。通过这个实战演练，你会看到统计探究循环的每个环节是如何衔接的。你还将培养自己开展调查并批判性评估结果的能力。
1. Understanding the Case Study | 理解案例背景
A school's PE department wants to improve its sports programme. To do this, they need to understand the current activity levels of Year 9 students. They decide to carry out a statistical investigation to answer two main questions: 'How many hours of sport do Year 9 students do per week?' and 'What is the most popular sport among Year 9 students?' These questions will guide the whole process. Before collecting any data, the team must plan carefully, considering what data is needed, how it will be gathered, and how it will be analysed. Every good statistical investigation starts with a clear set of questions that can be answered with data.
一所学校的体育部门希望改进其运动课程。为此，他们需要了解九年级学生当前的活动水平。他们决定开展一项统计调查，回答两个核心问题：“九年级学生每周进行多少小时的运动？”以及“九年级学生中最受欢迎的运动是什么？”这些问题将指导整个调查过程。在收集任何数据之前，团队必须仔细规划，考虑需要什么数据、如何收集以及如何分析。每一项好的统计调查都始于一组清晰、可以用数据回答的问题。
2. Designing the Survey | 设计调查问卷
To collect reliable data, the team designs a short questionnaire. They decide to use closed questions because they are easier to analyse. The first question asks: 'How many hours of sport do you usually do per week?' with response options: 0–2 hours, 3–5 hours, 6–8 hours, 9 or more hours. The second question asks: 'What is your favourite sport?' with options: Football, Netball, Swimming, Tennis, Other. By providing fixed categories, the answers can be quickly turned into frequency tables. The team also decides to use simple random sampling: they will randomly select 30 Year 9 students from the school register, giving every student an equal chance of being chosen. This helps to reduce bias.
为了收集可靠的数据，团队设计了一份简短的问卷。他们决定使用封闭式问题，因为这类问题更易于分析。第一个问题问：“你通常每周进行多少小时的运动？”选项为：0–2小时、3–5小时、6–8小时、9小时或以上。第二个问题问：“你最喜欢的运动是什么？”选项为：足球、无板篮球、游泳、网球、其他。通过提供固定的分类，答案可以迅速转换成频数表。团队还决定采用简单随机抽样：他们会从学校学生名册中随机选取30名九年级学生，让每名学生都有均等被选中的机会。这有助于减少偏倚。
3. Collecting Data | 收集数据
After distributing the questionnaire, the team collects responses from the 30 selected students. The raw data for hours of sport per week (categorised) and favourite sport are recorded. A snapshot of the collected data is shown below. Each row represents one student. Notice that all answers fall into one of the pre-defined groups. This dataset will now be organised for deeper analysis.
分发问卷后，团队收集了30名被选学生的回答。每周运动小时数（分类）和最喜欢的运动原始数据被记录下来。以下是收集到的部分数据快照。每一行代表一名学生。请注意，所有回答都落在预先定义的类别中。这些数据现在将被整理以便深入分析。
| Student | Hours of sport | Favourite sport |
|---|---|---|
| 1 | 3–5 | Football |
| 2 | 6–8 | Netball |
| 3 | 0–2 | Football |
| 4 | 3–5 | Swimming |
| 5 | 6–8 | Football |
| ... | ... | ... |
| 30 | 9+ | Tennis |
The full dataset contains a mix of categories. To make sense of it, we need to count how often each category appears. This step moves us from raw data to organised summary tables.
完整的数据集包含不同类别的混合。为了理解这些数据，我们需要统计每个类别出现的次数。这一步将我们从原始数据带到了有组织的汇总表格。
4. Types of Data | 数据类型
Before organising the data, it is important to identify the types of data we have. The 'hours of sport' variable is numerical, but because responses were collected in groups, it is treated as grouped discrete (or ordinal categorical) data. The groups have a natural order: 0–2, 3–5, etc. The 'favourite sport' variable is categorical (nominal) data because the options are names of sports with no inherent order. Recognising data types helps us choose the right charts and the correct summary statistics. For example, you can calculate the mean only for numerical data, while mode can be used for both numerical and categorical data.
在整理数据之前，重要的是确定我们拥有的数据类型。“运动小时数”变量是数值型的，但由于回答是按组收集的，因此被视为分组离散（或有序分类）数据。这些组有自然的顺序：0–2、3–5等。“最喜欢的运动”变量是分类（名义）数据，因为选项是运动名称，没有内在顺序。识别数据类型有助于我们选择正确的图表和恰当的汇总统计量。例如，只有数值数据才能计算平均数，而众数可用于数值数据和分类数据。
5. Organising Data: Frequency Tables | 数据整理：频数表
The first step in analysis is to tally the data into frequency tables. Using the full dataset of 30 students, the totals for each category are compiled. The frequency table for hours of sport is shown below, along with the cumulative frequency. The cumulative frequency will help us later to find the median position.
分析的第一步是将数据用画记法整理成频数表。使用30名学生的完整数据集，汇总出每个类别的总数。下面是运动小时数的频数表，还包含了累计频数。累计频数将有助于我们稍后确定中位数的位置。
| Hours of sport | Frequency | Cumulative frequency |
|---|---|---|
| 0–2 | 5 | 5 |
| 3–5 | 12 | 17 |
| 6–8 | 10 | 27 |
| 9+ | 3 | 30 |
For favourite sport, a separate frequency table is created. The mode is easily identified from this table. We can see that Football has the highest frequency.
对于最喜欢的运动，我们单独制作了一个频数表。从这个表中很容易看出众数。我们可以看到足球的频数最高。
| Favourite sport | Frequency |
|---|---|
| Football | 15 |
| Netball | 6 |
| Swimming | 5 |
| Tennis | 3 |
| Other | 1 |
Organised tables allow us to spot patterns immediately and form the basis for all further calculations and graphs.
有组织的表格能让我们立即发现规律，并为所有进一步的计算和图表提供基础。
6. Representing Data: Bar Charts | 数据表示：条形图
A bar chart is an excellent way to display categorical data visually. For the favourite sport data, we draw a bar chart with the sport names on the horizontal axis and frequency on the vertical axis. The bars are drawn with equal widths and gaps between them to show that the data are separate categories. The height of each bar represents the frequency. Looking at the bar chart, the tallest bar corresponds to Football, confirming it as the most popular sport. Swiming and Tennis have shorter bars, showing lower popularity.
条形图是直观显示分类数据的绝佳方式。对于最喜欢的运动数据，我们绘制一张条形图，横轴为运动名称，纵轴为频数。条形的宽度相等，且条形之间有间隔，以表明这些数据是独立的类别。每个条形的高度代表频数。观察条形图，最高的条形对应足球，确认它是最受欢迎的运动。游泳和网球的条形较短，显示受欢迎程度较低。
We can also create a bar chart for the grouped hours of sport data. Since the groups have a natural order, we keep them in sequence on the horizontal axis. The bar chart quickly reveals that the 3–5 hours group is the most common, followed by 6–8 hours. Very few students report doing 9+ hours or as little as 0–2 hours. This gives a clear picture of the typical activity level.
我们也可以为分组后的运动小时数数据绘制条形图。由于这些组有自然顺序，我们在横轴上保持它们的次序绘制。条形图迅速揭示出 3–5 小时组是最常见的，其次是 6–8 小时组。很少有学生报告进行 9 小时以上或仅有 0–2 小时的运动。这清晰地显示了典型的活动水平。
7. Measures of Central Tendency | 集中趋势测量
To summarise the hours of sport numerically, we calculate the mean, median and mode. Since the data are grouped, we estimate the mean using the midpoint of each interval. Let the midpoints be 1, 4, 7 and 10 for the groups 0–2, 3–5, 6–8 and 9+ respectively. We multiply each midpoint by its frequency, sum these products, and divide by the total number of students (30).
为了用数字概括运动小时数，我们计算平均数、中位数和众数。由于数据是分组的，我们使用各区间的组中值来估算平均数。设 0–2、3–5、6–8 和 9+ 组的组中值分别为 1、4、7 和 10。我们将每个组中值乘以对应的频数，把这些乘积相加，再除以学生总数 (30)。
Mean = (1×5 + 4×12 + 7×10 + 10×3) ÷ 30 = (5 + 48 + 70 + 30) ÷ 30 = 153 ÷ 30 = 5.1 hours
The estimated mean is 5.1 hours per week. The median is the middle value. With 30 data points, the median lies between the 15th and 16th values. From the cumulative frequency table, the 15th and 16th students are both in the 3–5 hours group. Therefore, the median class is 3–5 hours. The mode (the most frequent group) is also 3–5 hours, with a frequency of 12. All three measures point to a typical week of sport being in the 3–5 hour range.
估算的平均数是每周 5.1 小时。中位数是位于中间的值。有 30 个数据点，中位数位于第 15 和第 16 个值之间。从累计频数表可知，第 15 和第 16 名学生都在 3–5 小时组。因此，中位数组为 3–5 小时。众数（最频组）也是 3–5 小时组，频数为 12。这三项集中趋势指标都指向典型的一周运动时间在 3–5 小时区间。
For categorical data, only the mode is meaningful. The mode for favourite sport is Football, with 15 students choosing it. This tells us that Football is most typical, but it does not make sense to calculate a mean or median for sport names.
对于分类数据，只有众数才有意义。最喜欢的运动的众数是足球，有 15 名学生选择了它。这告诉我们足球最具代表性，但计算运动名称的平均数或中位数是没有意义的。
8. Measures of Spread | 离散程度测量
also need to understand how spread out the data are. The simplest measure of spread is the range. For the hours of sport, the lowest recorded group is 0– is 9+ (we use the upper boundary 10, assuming 9+ means up to 10). An estimated range is 10 – siderable variation: some students do very little sport, while others do a great deal. For grouped data, the actual range might be slightly different because we do not know the exact maximum, but this estimate gives a useful sense of spread.
了解平均值很有用，但我们也需要理解数据的分散程度。最简单的离散程度测量是全距。对于运动小时数，记录的最低组是 0–2（下限为 0），最高组是 9+（假设 9+ 意味着最多 10 小时，我们使用上限 10）。估算的全距是 10 – 0 = 10 小时。这告诉我们存在相当大的变异：有些学生几乎不运动，而另一些学生运动量很大。对于分组数据，实际全距可能略有不同，因为我们不知道确切的最大值，但这个估算值提供了一个有用的离散程度概念。
Another way to describe spread is to look at the range of the middle 50% – the interquartile range – but for Year 9, the simple range is usually sufficient to show how varied the data are. A large range often suggests that a single average may not represent everyone well, so we should always report the range alongside the mean or median.
另一种描述离散程度的方法是查看中间 50% 的全距——四分位数间距——但对于九年级来说，简单全距通常就足以显示数据的变异程度。较大的全距往往表明单一的平均值可能无法很好地代表所有人，因此我们应当总是将全距与平均数或中位数一起报告。
9. Probability from Data | 从数据中计算概率
We can use the collected data to estimate probabilities. If we pick one Year 9 student at random from our sample, the probability that this student's favourite sport is Football is given by the number of students choosing Football divided by the total number of students. Using the frequency table:
我们可以用收集到的数据来估算概率。如果我们从样本中随机选取一名九年级学生，该学生最喜欢的运动是足球的概率由选择足球的学生人数除以学生总数得出。利用频数表：
P(Football) = 15 / 30 = 0.5 or 50%
tal probabilities based on observed frequencies. They can be expressed as fractions, decimals or percentages. The probabilities sum to 1, as expected. This simple probability model allows the PE department to predict the likelihood of a random student preferring a certain sport, which can help with planning equipment and sessions.
类似地，无板篮球的概率是 6/30 = 0.2，游泳的概率是 5/30 ≈ 0.167，网球是 3/30 = 0.1，其他是 1/30 ≈ 0.033。这些都是基于观察频数的实验概率。它们可以用分数、小数或百分数表示。这些概率之和为 1，符合预期。这个简单的概率模型让体育部门能够预测随机一名学生偏好某项运动的可能性，这有助于规划器材和课程。
a randomly chosen student does more than 5 hours of sport. By adding the frequencies for 6– 13/30 ≈ 0.433. This shows that nearly half the students do over 5 hours of sport per week.
对于运动小时数，我们也可以计算概率，例如随机选中的学生每周运动超过 5 小时的概率。将 6–8 小时和 9+ 小时的频数相加 (10+3=13)，得到 P(超过5小时) = 13/30 ≈ 0.433。这表明近一半的学生每周运动超过 5 小时。
10. Drawing Conclusions and Evaluation | 得出结论及评价
Returning to the original questions, the investigation yields clear answers. The most popular sport among this sample of Year 9 students is Football, chosen by 50% of respondents. The typical weekly sport participation falls in the 3–5 hour group, with an estimated mean of 5.1 hours. The range of 10 hours, however, shows that not all students are equally active. The school may consider these insights when planning its PE offering and whether to introduce more football-related sessions or encourage less active students.
回到最初的问题，调查得出了明确的答案。在这一九年级学生样本中，最受欢迎的运动是足球，有 50% 的受访者选择。典型的每周运动参与量落在 3–5 小时组，估算平均数为 5.1 小时。然而，10 小时的全距表明并非所有学生都同样活跃。学校在规划体育课程时可以考虑这些见解，并思考是否需要增加更多与足球相关的活动或鼓励活动量较少的学生。
However, the investigation has limitations. The sample size of 30 is relatively small and only from one school, so the findings may not apply to other Year 9 groups. The hours data were collected in self-reported groups, which could lead to rounding errors or over-estimation. The questionnaire only captured organised sport, possibly missing informal physical activity. To improve the investigation, a larger, more diverse sample could be taken, and the hours question could be asked as an exact number. Despite these limitations, the case study demonstrates a complete statistical cycle: posing questions, collecting data, analysing, and evaluating, which is at the heart of the WJEC Statistics curriculum.
不过，该调查也有局限性。30 人的样本量相对较小，且仅来自一所学校，因此这些发现可能不适用于其他九年级群体。运动小时数数据是以分组自报的方式收集的，这
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