📚 Year 8 OCR Statistics: Case Study Practical Exercises | Year 8 OCR 统计：案例分析实战演练
In this article, we will walk through a complete statistical investigation based on a real‑style case study. You will see how to collect, organise, present and interpret data, calculate averages and spread, draw scatter graphs and use probability. The scenario is a Year 8 class project about exercise habits and homework time. Follow each step to build your skills for the OCR statistics exam.
在本文中，我们将基于一个真实的案例研究，完整地走一遍统计调查流程。你将看到如何收集、整理、呈现与解读数据，计算平均数与离散程度，绘制散点图并运用概率。案例背景是一个 Year 8 班级关于运动习惯和作业时间的项目。跟随每一步，为 OCR 统计考试打好基础。
1. Understanding the Scenario | 案例场景理解
A class of 30 Year 8 students wants to find out whether there is any link between the time they spend on physical exercise and the time they spend on homework each week. The teacher also suggests collecting data on favourite sports so that the class can practise probability calculations. The investigation will be organised as a complete statistical cycle: posing a question, collecting data, analysing it and drawing conclusions.
一个由 30 名 Year 8 学生组成的班级想了解每周体育锻炼时间与每周作业时间之间是否存在关联。老师还建议收集最喜欢的运动项目数据，以便练习概率计算。本次调查将按照完整的统计循环展开：提出问题、收集数据、分析数据并得出结论。
2. Data Collection Methods | 数据收集方法
The class decides to use a questionnaire. Each student writes down the number of hours they exercise per week (to the nearest half hour) and the number of hours they spend on homework. They also tick their favourite sport from a list: football, basketball, swimming, running or other. The questionnaire is made anonymous to encourage honest answers. The response rate is 100%, so data from all 30 students is available.
班级决定采用问卷。每位学生写下自己每周运动的时数（精确到半小时）和写作业的时数，并从列表中选择最喜欢的运动：足球、篮球、游泳、跑步或其他。问卷匿名以鼓励诚实回答。回收率为 100%，因此全部 30 名学生的数据均可使用。
3. Organising the Data | 数据整理
Once the questionnaires are returned, the data is recorded in a spreadsheet. For the two numerical variables – weekly exercise hours and weekly homework hours – the class creates an ordered list. Then they construct a frequency table for exercise hours to see the distribution at a glance. A stem‑and‑leaf diagram is also drawn because it retains the original values while showing the shape.
问卷回收后，数据被录入电子表格。对于两个数值变量——每周运动小时和每周作业小时，班级建立了一个有序列表。接着他们为运动时间制作频数表，以便一眼看出分布情况。还绘制了茎叶图，因为它既保留原始数值，又展示分布形状。
The collected exercise hours (in order) are: 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 10. The frequency table looks like this:
收集到的运动小时（按顺序）为：0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 10。频数表如下：
| Weekly Exercise (hours) | Frequency |
|---|---|
| 0 | 2 |
| 1 | 2 |
| 2 | 4 |
| 3 | 4 |
| 4 | 4 |
| 5 | 4 |
| 6 | 3 |
| 7 | 3 |
| 8 | 2 |
| 9 | 1 |
| 10 | 1 |
The stem‑and‑leaf diagram uses stems of tens and leaves of units: 0 | 0 0 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 7 7 7 8 8 9 10 (key: 1|2 means 1.2? No, here all values are whole hours so 1|2 = 12? That would be wrong. For whole numbers, a key 3|2 = 3.2? We’ll present as ordered list but stamp the idea of stem‑and‑leaf. Better: with stems 0 to 1 and leaves 0-9. E.g., 0 | 0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7,7,7,8,8,9 ; 1 | 0. Key: 1|0 = 10 hours. This works.)
茎叶图以十位数为茎，个位数为叶：0 | 0 0 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 7 7 7 8 8 9 ；1 | 0。关键：1|0 表示 10 小时。这样既保留了原始数据又直观地显示出数据集中在 2-5 小时之间。
4. Graphical Representations – Bar Chart and Pie Chart | 图表呈现 – 条形图与饼图
The favourite sport data is categorical. The frequencies are: Football 10, Basketball 8, Swimming 6, Running 4, Other 2. The class draws a bar chart with sport categories on the horizontal axis and frequency on the vertical axis. They also construct a pie chart by calculating angles: Football 10/30 × 360° = 120°, Basketball 96°, Swimming 72°, Running 48°, Other 24°.
最喜欢运动的数据是分类变量。频数为：足球 10、篮球 8、游泳 6、跑步 4、其他 2。班级绘制了条形图，横轴为运动类别，纵轴为频数。他们还制作了饼图，通过计算角度：足球 10/30 × 360° = 120°，篮球 96°，游泳 72°，跑步 48°，其他 24°。
Bar charts make it easy to compare categories, while a pie chart clearly shows proportions of the whole. The class discusses which visual is more useful for the school sports committee.
条形图便于比较各类别的频数，而饼图能清晰展示整体中的占比。班级讨论了哪种图形对学校体育委员会更有用。
5. Calculating the Mean | 计算平均数
To find the mean weekly exercise hours, the class adds all 30 values and divides by 30. Total exercise hours: (0×2)+(1×2)+(2×4)+(3×4)+(4×4)+(5×4)+(6×3)+(7×3)+(8×2)+(9×1)+(10×1) = 0+2+8+12+16+20+18+21+16+9+10 = 132 hours. Mean = 132 ÷ 30 = 4.4 hours.
为求每周运动小时的平均值，班级将 30 个数值相加再除以 30。运动总小时数：(0×2)+(1×2)+(2×4)+(3×4)+(4×4)+(5×4)+(6×3)+(7×3)+(8×2)+(9×1)+(10×1) = 0+2+8+12+16+20+18+21+16+9+10 = 132 小时。均值 = 132 ÷ 30 = 4.4 小时。
Using the same method, the mean weekly homework hours (data: 2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,8,8,8,8,9,9,9,10,10,10,11,12,12,14) is calculated: Sum = 222, Mean = 7.4 hours. The formula used is Mean = Σx / n.
用同样方法计算每周作业小时的平均值（数据：2,2,3,3,4,4,4,4,5,5,5,6,6,7,7,8,8,8,8,9,9,9,10,10,10,11,12,12,14）：总和为 222，均值 = 7.4 小时。所用公式为 均值 = Σx / n。
6. Finding the Median and Mode | 求中位数与众数
To find the median exercise hours, the data is already ordered. With 30 values (an even number), the median is the mean of the 15th and 16th values. The 15th value is 4, the 16th is 5, so median = (4+5)/2 = 4.5 hours. The mode is the most frequent value: 2, 3, 4 and 5 each appear 4 times, making the data set multimodal.
为了求运动小时的中位数，数据已排序。共 30 个值（偶数），中位数是第 15 个和第 16 个值的平均数。第 15 个值为 4，第 16 个值为 5，中位数 = (4+5)/2 = 4.5 小时。众数是最频繁出现的值：2、3、4 和 5 各出现 4 次，因此该数据集是多众数的。
For homework hours, the ordered list (30 values) gives the 15th and 16th values as 7 and 8, median = 7.5 hours. The modes are 4, 5, 8 and 9 (each appears 3 times). Thus both distributions are quite spread out.
对于作业小时，排序后的 30 个值中第 15 和 16 个值为 7 和 8，中位数 = 7.5 小时。众数为 4、5、8 和 9（各出现 3 次）。由此可见，两种分布都比较分散。
7. Measures of Spread – Range | 离散程度 – 极差
The range tells us how spread out the data is. For exercise hours, maximum = 10, minimum = 0, so range = 10 − 0 = 10 hours. For homework hours, maximum = 14, minimum = 2, range = 12 hours. A large range indicates high variability among students.
极差告诉我们数据的分散程度。运动小时最大值为 10，最小值为 0，极差 = 10 − 0 = 10 小时。作业小时最大值为 14，最小值为 2，极差 = 12 小时。较大的极差表明学生之间差异很大。
The class then compares the range with the mean to get a sense of relative spread. Exercise: range/mean ≈ 10/4.4 ≈ 2.27; Homework: range/mean ≈ 12/7.4 ≈ 1.62. This suggests that exercise time is relatively more varied than homework time.
班级随后将极差与均值比较，以感受相对离散程度。运动：极差/均值 ≈ 10/4.4 ≈ 2.27；作业：极差/均值 ≈ 12/7.4 ≈ 1.62。这表明运动时间的相对变异比作业时间更大。
8. Drawing and Interpreting a Scatter Graph | 绘制与解读散点图
The class plots each student’s exercise hours on the horizontal axis (x) and their homework hours on the vertical axis (y). The 30 points are plotted as crosses. They then examine the pattern. No obvious upward or downward trend appears; the points are quite scattered. This suggests that there is no strong correlation between the amount of exercise a student does and the time they spend on homework.
班级将每位学生的运动小时标在横轴（x），作业小时标在纵轴（y）。30 个点以叉号表示。他们观察了点阵模式。没有出现明显的上升或下降趋势；点相当分散。这说明学生运动量与其作业时间之间没有强相关性。
To check numerical correlation, they could calculate the mean point (x̄, ȳ) = (4.4, 7.4) and draw a line of best fit, but with such a weak pattern, the line would be almost flat. They record this observation as part of their conclusion.
为了检验数值相关性，他们可以计算均值点 (4.4, 7.4) 并画出最佳拟合线，但由于模式很弱，这条线几乎水平。他们将这一观察记录在结论部分。
9. Interpreting Probability from the Survey | 根据调查解释概率
The class uses the sport preference data to work out probabilities based on the survey. If one student is chosen at random from the 30, the probability that their favourite sport is football is 10/30 = 1/3. The probability that it is basketball is 8/30 = 4/15, and the probability of choosing a student who likes either swimming or running is (6+4)/30 = 10/30 = 1/3.
班级利用运动偏好数据计算基于调查的概率。若从 30 人中随机抽取一人，其最喜欢足球的概率为 10/30 = 1/3；喜欢篮球的概率为 8/30 = 4/15；喜欢游泳或跑步的概率为 (6+4)/30 = 10/30 = 1/3。
They also discuss how probability changes with sample size. If the survey had asked 60 students, the same proportions would give equivalent probabilities, but the accuracy might improve. This links probability to the idea of relative frequency.
他们还讨论了概率如何随样本大小变化。如果调查了 60 名学生，相同比例会给出等效概率，但准确性可能会提高。这便将概率与相对频数的概念联系了起来。
10. Drawing Conclusions and Making Recommendations | 得出结论与提出建议
From the investigation, the class concludes that, for this group, the average student exercises about 4.4 hours per week, with half the students doing between 2 and 5 hours. The typical homework time is higher at 7.4 hours. There is no clear evidence that more exercise means less homework time, contrary to what some expected. The sport preference pie chart shows football and basketball together account for more than half the votes, so the school might prioritise those facilities.
通过调查，班级得出结论：对这组学生而言，平均每周运动约 4.4 小时，半数学生的运动量在 2 至 5 小时之间。典型作业时间较高，为 7.4 小时。没有明确证据表明运动越多作业时间越少，这与部分人的预期相反。运动偏好饼图显示足球和篮球合计超过半数，学校或可优先改善这些设施。
They also note that the ranges are wide, so any decision should consider individual differences. They suggest repeating the survey with a larger sample and including other variables like sleep.
他们还注意到极差较大，因此任何决策都应考虑个体差异。他们建议扩大样本量并加入睡眠等其他变量重复调查。
11. Reflecting on the Statistical Cycle | 反思统计循环
The class reviews what went well: clear questionnaire, full participation, neat graphs. They identify limitations: using exact hours might not capture mid‑week variations, and some students estimated. They discuss how to improve: asking for a daily diary instead, or using a tracking app. This reflection completes the statistical cycle and prepares them for more advanced investigations.
班级回顾了做得好的地方：问卷清晰、全员参与、图表整洁。他们也找出局限：精确的小时数可能未反映周中的变化，部分学生是估算的。他们讨论如何改进：改用每日日记记录，或使用追踪应用程序。这次反思使统计循环完整，也为更高级的调查做好准备。
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导