📚 Year 10 SQA Statistics: Unit Test Mock Paper Walkthrough | 苏格兰资格认证局10年级统计：单元测试模拟卷解析
This walkthrough provides detailed solutions and exam tips for a typical SQA Year 10 Statistics unit test. Use it to sharpen your skills in data handling, probability, and statistical reasoning. Work through each question carefully, and pay close attention to the common pitfalls highlighted in the explanations.
这份模拟卷解析为典型的SQA 10年级统计单元测试提供详细解答和应试技巧。用它来提升你的数据处理、概率和统计推理能力。仔细完成每道题目，并特别关注解析中强调的常见易错点。
1. Mean, Median, Mode from Raw Data | 根据原始数据求平均数、中位数和众数
, 5, 6, 7, 8, 8, 9, 10, 12, 14. The mean is the sum of all values divided by 11, which equals 89 ÷ 11 = 8.09 (to 2 d.p.). The median is the 6th value, which is 8. The mode is 5, as it appears three times.
数据为11名学生在慈善义卖中售出的物品件数：5, 8, 12, 7, 5, 9, 10, 6, 5, 8, 14。首先将数据按升序排列：5, 5, 5, 6, 7, 8, 8, 9, 10, 12, 14。平均数是所有值之和除以11，即89 ÷ 11 = 8.09（保留两位小数）。中位数是第6个数，为8。众数是5，因为它出现了三次。
- Always write or tick off the ordered list — this avoids counting errors for median.
- 务必列出或勾选排序后的列表——这能避免中位数计数错误。
- For an even number of data values, the median is the mean of the two middle numbers.
- 当数据个数为偶数时，中位数是中间两个数的平均值。
2. Calculating Standard Deviation | 计算标准差
quare the result, sum these squares, divide by n (for population standard deviation) or n−1 (for sample). In SQA exams, unless stated otherwise, use the sample standard deviation with n−1. Here, Σ(x−x̄)² = 84.9091, so sample variance = 84.9091 ÷ 10 = 8.49091, and standard deviation = √8.49091 ≈ 2.91 (to 2 d.p.).
使用同一组数据，计算标准差作为离散程度的度量。步骤：求平均数（8.09），然后每个值减去平均数，将结果平方，求平方和，除以n（总体标准差）或n−1（样本标准差）。在SQA考试中，除非另有说明，使用n−1的样本标准差。此处，Σ(x−x̄)² = 84.9091，因此样本方差 = 84.9091 ÷ 10 = 8.49091，标准差 = √8.49091 ≈ 2.91（保留两位小数）。
- Use a table to organise x, (x−x̄), and (x−x̄)². This minimises arithmetic mistakes.
- 使用表格列出x、(x−x̄)和(x−x̄)²。这能最大限度减少计算错误。
- A smaller standard deviation indicates that the data points are closer to the mean.
- 标准差越小，表示数据点越接近平均数。
3. Interpreting a Cumulative Frequency Diagram | 解读累积频数图
go to half the total frequency (40) and read across: median ≈ 30 marks. The lower quartile (at 20) is 23, and the upper quartile (at 60) is 36. The interquartile range (IQR) is 36 − 23 = 13 marks. The IQR tells you the spread of the middle 50% of scores.
累积频数图显示满分50分的测验成绩。曲线经过点(20, 15)、(30, 40)、(40, 72)和(50, 80)。要估计中位数，从总频数的一半（40）处水平读取：中位数 ≈ 30分。下四分位数（累积频数20处）为23分，上四分位数（累积频数60处）为36分。四分位距（IQR）为36 − 23 = 13分。IQR表示中间50%成绩的分布范围。
- Always draw vertical and horizontal dotted lines on the graph to show your readings — examiners award marks for method.
- 务必在图上画出垂直和水平虚线来显示读取过程——评分人会给方法分。
- The IQR is not affected by extreme values; it is a robust measure of spread.
- IQR不受极端值影响，是一种稳健的离散度量。
4. Probability and Venn Diagrams | 概率与韦恩图
tside = 4. Find the probability that a randomly chosen student studies Art or Biology: P(A ∪ B) = P(A) + P(B) − P(A ∩ 13/15. The probability they study neither is 4/30 = 2/15. Remember to simplify fractions.
某班有30人，18人选修艺术（A），15人选修生物（B），7人两门都选。韦恩图中交集为7，只选艺术为11，只选生物为8，两门都不选为4。求随机选一名学生，其选修艺术或生物的概率：P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 18/30 + 15/30 − 7/30 = 26/30 = 13/15。两门都不选的概率为4/30 = 2/15。注意化简分数。
- Start by placing the intersection value, then subtract to find the 'only' regions.
- 先填入交集的值，然后相减得到“仅选一门”的部分。
- The total of all regions inside the rectangle must equal the sample space.
- 矩形内所有区域的总和必须等于样本空间大小。
5. Tree Diagrams for Dependent Events | 相依事件的树状图
. For the second pick, if first was R, probabilities become 4/7 (R) and 3/7 (G); if first was G, 5/7 (R) and 2/7 (G). The probability of picking two reds is (5/8) × (4/7) = 20/56 = 5/14. Probability of at least one green = 1 − P(both red) = 1 − 5/14 = 9/14. Multiply along branches and add the relevant outcomes.
袋中有5颗红色和3颗绿色糖果。随机抽取两颗，不放回。画出树状图：第一次抽取概率为5/8（红）和3/8（绿）。第二次抽取时，若第一次为红，概率变为4/7（红）和3/7（绿）；若第一次为绿，则为5/7（红）和2/7（绿）。抽到两颗红色的概率是(5/8) × (4/7) = 20/56 = 5/14。至少一颗绿色的概率 = 1 − P（两颗红）= 1 − 5/14 = 9/14。沿着分支相乘，并将相关结果相加。
- Label each branch with its probability and write the final outcomes at the end.
- 在每条分支上标记概率，并在末端写出最终结果。
- Check that the probabilities on the second-level branches sum to 1 for each cluster.
- 检查每个群组中第二级分支的概率总和是否为1。
6. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph plots hours of revision against exam scores. The points generally rise from bottom-left to top-right, indicating a positive correlation. We can draw a line of best fit by eye—it should pass through the mean point (x̄, ȳ) and balance points above and below. Use the line to estimate: a student who revises for 4.5 hours is predicted to score about 62%. Correlation does not imply causation, but it suggests a relationship. Outliers must be identified: a point (2.5, 90) is unusually high for few revision hours.
散点图以复习小时数为横轴、考试成绩为纵轴。数据点总体上从左下方向右上方延伸，表明正相关。我们可以凭目测画出最佳拟合线——它应穿过均值点（x̄, ȳ），并使线上下的点大致平衡。利用这条线进行估计：复习4.5小时的学生预计得分约62%。相关不代表因果，但它表明存在关系。必须识别异常值：点（2.5, 90）在少量复习时间下异常高分。
- Never force the line through the origin unless the context (like zero score for zero study) justifies it.
- 除非背景合理（如零学习时间得零分），切勿强行让拟合线通过原点。
- Describe correlation as 'strong positive', 'weak negative', or 'no correlation' using the spread of points.
- 根据点的分散程度，将相关性描述为“强正相关”、“弱负相关”或“无相关”。
7. Sampling Methods and Bias | 抽样方法与偏差
A researcher wants to find the average screen time of S3 students. She surveys the first 20 students entering the library. This is a convenience sample, which is likely biased because library users may study more and use screens less. A better method is stratified sampling: divide the year group into strata by gender or class, then randomly select proportional numbers from each. Random sampling gives each member an equal chance and reduces bias. The sampling frame is the list of all S3 students.
研究者想了解S3学生的平均屏幕使用时间。她调查了最先进入图书馆的20名学生。这属于便利样本，很可能有偏差，因为图书馆使用者可能学习更多、屏幕使用更少。更好的方法是分层抽样：按性别或班级将年级分组，然后从每组中按比例随机抽取。随机抽样使每个成员有均等机会，并减少偏差。抽样框是所有S3学生的名单。
- Always question the sampling method: who is included, who is left out, and why it matters.
- 始终对抽样方法提出质疑：谁被纳入，谁被排除，以及为何这很重要。
- Sample size also affects reliability — a larger sample generally reduces sampling error.
- 样本量也影响可靠性——较大样本通常能减少抽样误差。
8. Histograms and Frequency Density | 直方图与频数密度
ass width). For the class 10 < t ≤ 20 with frequency 8, width = 10, density = 0.8. If the next class 20 < t ≤ 25 has frequency 12, width = 5, density = 2.4. Always start classes from the exact lower boundary and label axes clearly. Unscaled paper? Use a key or calculation box to show your working.
直方图显示分组连续数据。每一条的面积代表频数，因此纵轴是频数密度（频数 ÷ 组距）。对于区间10 < t ≤ 20，频数为8，组距为10，密度为0.8。若下一个区间20 < t ≤ 25的频数为12，组距为5，密度为2.4。区间始终从精确的下限开始，并清晰标记坐标轴。未标刻度的图纸？使用图例或计算框展示你的计算过程。
- Common mistake: plotting frequency instead of frequency density — this distorts the distribution shape.
- 常见错误：直接绘制频数而非频数密度——这会扭曲分布形态。
- In an exam, the question often provides an axis for frequency density — check before plotting.
- 考试中，题目通常会提供频数密度的坐标轴——绘图前先确认。
9. Comparing Data Sets Using Statistics | 使用统计量比较数据集
Two groups of athletes record their sprint times (seconds). Group X: mean 10.8, standard deviation 0.3. Group Y: mean 11.2, standard deviation 0.6. Group X is faster on average. Group X's smaller standard deviation shows less variability — they are more consistent. The median and IQR could also be used: if Group X's median is 10.75 and IQR is 0.4, while Group Y's median is 11.1 and IQR is 0.9, it confirms the same conclusion. Always cite both a measure of central tendency and a measure of spread.
两组运动员记录短跑时间（秒）。X组：平均数10.8，标准差0.3。Y组：平均数11.2，标准差0.6。X组平均速度更快。X组较小的标准差表明变异较小——他们更稳定。也可以使用中位数和IQR：若X组中位数为10.75、IQR为0.4，而Y组中位数为11.1、IQR为0.9，则证实了相同的结论。务必同时引用中心趋势度量和变异度量。
- In commenting on the shape, mention if one set is skewed or has outliers that affect the mean.
- 在评论形态时，若某组数据偏斜或有异常值影响平均数，要予以说明。
- 'Consistent' is a key word linked to a smaller standard deviation or IQR.
- “稳定”是与较小标准差或IQR相关的关键表述。
10. Misleading Graphs and Data Representation | 误导性图表与数据呈现
A bar chart shows sales growth, but the vertical axis starts at 80 instead of 0, exaggerating the difference between 82 and 86. The chart also uses a three‑dimensional effect that distorts bar height perception. For accurate comparison, always start the axis at zero, use simple 2D bars, and label the scale evenly. Pie charts should have data labels or a key. When criticising a graph, comment on scale, labelling, area distortion, and sampling if applicable.
某条形图显示销售额增长，但纵轴从80而非0开始，夸大了82与86之间的差异。该图还使用了三维效果，扭曲了对条形高度的感知。为了准确比较，应始终将坐标轴从零开始，使用简单的二维条形，并均匀标记刻度。饼图应有数据标签或图例。在批评图表时，从刻度、标签、面积扭曲和抽样（如适用）等方面进行评论。
- Watch for missing or unclear titles and unlabelled axes — these make a graph difficult to interpret.
- 注意缺失或不清晰的标题以及未标记的坐标轴——这会使图表难以解读。
- A pictorial chart where area or volume changes in a non‑linear ratio distorts the message.
- 象形图中若面积或体积变化呈非线性比例，会扭曲信息。
Published by TutorHao | SQA Statistics Revision Series | aleveler.com
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