📚 Year 10 SQA Statistics: In-Depth Analysis of Past Exam Papers | SQA 统计：历年真题深度解析
Working through past SQA Statistics questions is one of the most effective ways to prepare for your Year 10 assessments. This article breaks down recurring question types, highlights common pitfalls, and shows you step-by-step methods to secure full marks. By analysing real exam-style problems, you will build confidence in data handling, probability, and statistical reasoning.
反复练习 SQA 历年统计真题是备考 Year 10 评估最有效的方法之一。本文将剖析高频题型、点明常见陷阱，并分步骤展示拿满分的解题方法。通过对真实考题风格的分析，你将在数据处理、概率和统计推断方面建立扎实的信心。
1. Understanding the SQA Statistics Paper Structure | 了解 SQA 统计试卷结构
SQA Statistics for Year 10 (typically part of National 5 Applications of Mathematics) is divided into two sections: one focusing on data handling, averages and spread, and the other on probability and interpretation. Past papers reveal that about 60% of marks come from calculation and diagram questions, while the remaining 40% require written interpretation. Questions often start with a real-life context such as test scores, temperature records or survey results, and then ask you to compute statistics or draw conclusions.
Year 10 的 SQA 统计（通常属于 National 5 应用数学的一部分）分为两大块：一块侧重数据处理、平均数和离散度，另一块考查概率与解读。历年真题显示，约 60% 的分数来自计算与绘图题，其余 40% 要求文字解释。题目往往以真实情境开头，如考试成绩、温度记录或调查结果，然后要求你计算统计量或得出结论。
2. Mean, Median and Mode: Core Calculations | 均值、中位数与众数：核心计算
A classic past-paper question gives a data set such as 12, 15, 14, 18, 20, 22, 14 and asks you to find the mean, median and mode. To find the mean, add all seven numbers (12+15+14+18+20+22+14 = 115) and divide by 7, giving approximately 16.43. For the median, sort the values: 12, 14, 14, 15, 18, 20, 22. The middle value is 15. The mode is 14 because it appears twice. Always show all your working, as SQA awards method marks for the sum and the division step even if the final answer is slightly off.
一道经典的真题会给出一组数据，如 12, 15, 14, 18, 20, 22, 14，要求计算均值、中位数和众数。求均值时，把七个数字相加（12+15+14+18+20+22+14 = 115），再除以 7，约得 16.43。中位数需要排序：12, 14, 14, 15, 18, 20, 22，中间的 15 便是中位数。众数是 14，因为它出现了两次。务必展示全部计算步骤，因为 SQA 会给求和与除法的过程分，即使最后答案稍有偏差。
When data is presented in a frequency table, multiply each value by its frequency, sum these products, and divide by the total frequency. For example, Score 10 (frequency 3), Score 11 (frequency 5) gives a mean of (10×3 + 11×5) ÷ 8 = (30+55) ÷ 8 = 10.625. Many students lose marks by forgetting to divide by total frequency rather than the number of rows.
如果数据以频数表呈现，将每个数值乘以对应频数，求和后再除以总频数。例如：分数 10（频数 3），分数 11（频数 5），均值 = (10×3 + 11×5) ÷ 8 = (30+55) ÷ 8 = 10.625。很多学生因忘记除以总频数而丢分，误除以表格行数。
3. Quartiles and Box Plots | 四分位数与盒须图
SQA exams frequently require you to determine the five-number summary (minimum, Q1, median, Q3, maximum) and draw a box plot. For a sorted list of 11 values: 5, 7, 8, 9, 11, 13, 15, 16, 18, 20, 25, the median is the 6th value (13). Q1 is the median of the lower half: 5, 7, 8, 9, 11 → Q1 = 8. Q3 is the median of the upper half: 15, 16, 18, 20, 25 → Q3 = 18. The interquartile range (IQR) is Q3 – Q1 = 10. When drawing the box plot, use a scale and label the axis; the box spans Q1 to Q3, with a line at the median. Whiskers extend to the minimum and maximum, unless there are outliers.
SQA 考试常要求你确定五数综合（最小值、下四分位数 Q1、中位数、上四分位数 Q3、最大值）并绘制盒须图。假设有 11 个已排序数值：5, 7, 8, 9, 11, 13, 15, 16, 18, 20, 25，中位数为第 6 个值（13）。Q1 是下半部中位数：5, 7, 8, 9, 11 → Q1 = 8。Q3 是上半部中位数：15, 16, 18, 20, 25 → Q3 = 18。四分位距 IQR = Q3 – Q1 = 10。绘制盒须图时，要选择合适刻度并标注轴；盒子从 Q1 延伸到 Q3，中间用线标出中位数。须线延伸至最小值和最大值，除非存在异常值。
A typical past-paper twist gives a box plot and asks you to compare two distributions. Comment on the median (central tendency) and the IQR (spread), using phrases like ‘Class A has a higher median score, suggesting better typical performance, but Class B has a smaller IQR, indicating more consistent results.’ This comparison skill is worth several marks.
真题中常见的变式是给出一个盒须图，要求比较两组数据的分布。要针对中位数（集中趋势）和四分位距（离散度）进行评价，例如“A 班的中位数更高，说明典型表现更好，但 B 班的 IQR 更小，表明成绩更稳定”。这种比较技巧能获得好几分。
4. Standard Deviation and Spread | 标准差与数据离散度
Standard deviation measures how data is spread around the mean. The SQA formula for a sample is
s = √( ∑(x – x̄)² / (n – 1) )
To calculate, first find the mean x̄. Then subtract the mean from each data value, square the result, and sum all the squares. Divide by (n−1) and take the square root. For the set 4, 6, 7, 9 (mean = 6.5), the squares are (4−6.5)²=6.25, (6−6.5)²=0.25, (7−6.5)²=0.25, (9−6.5)²=6.25. The sum is 13, divide by (4−1)=3 gives 4.33, and √4.33 ≈ 2.08. Many candidates forget to use n−1 and lose accuracy marks.
标准差衡量数据围绕均值的分散程度。SQA 给出的样本标准差公式为
s = √( ∑(x – x̄)² / (n – 1) )
计算时先求均值 x̄；然后用每个数据值减去均值，平方后求和；除以 (n-1) 再开平方。对于数据集 4, 6, 7, 9（均值 = 6.5），平方项为 (4−6.5)²=6.25, (6−6.5)²=0.25, (7−6.5)²=0.25, (9−6.5)²=6.25。总和为 13，除以 (4−1)=3 得 4.33，√4.33 ≈ 2.08。许多考生忘记使用 n−1 而失去准确性分数。
In interpretation questions, a smaller standard deviation means the data points are closer to the mean, i.e., more consistent. Combining this with the mean lets you fully describe a distribution. For example, ‘The mean score improved, and the standard deviation decreased, showing that the intervention raised performance and reduced variability.’
在解释题中，标准差越小表示数据点越靠近均值，即越一致。将标准差与均值结合就能完整描述分布。例如：“平均分数提高了，标准差下降了，说明干预措施提升了表现并降低了变异。”
5. Cumulative Frequency Diagrams | 累积频率图
Cumulative frequency diagrams appear regularly. You are typically given a grouped frequency table and asked to add a cumulative frequency column, then plot the graph with upper class boundaries on the x-axis. After drawing the smooth S-shaped curve, you can estimate the median by going to half the total frequency on the vertical axis, drawing across to the curve and then down to the x-axis. Similarly, you find Q1 at quarter the total frequency and Q3 at three-quarters, then calculate IQR = Q3 − Q1.
累积频率图是常考题。题目通常给出一个分组频数表，要求你添加累积频率列，然后以组距上界为 x 轴描点绘图。画出平滑的 S 形曲线后，你可以从纵轴上总频数的一半处画水平线交于曲线，再垂直下到 x 轴估计中位数。同样，从总频数的四分之一处找到 Q1，从四分之三处找到 Q3，再计算 IQR = Q3 − Q1。
| Height (cm) | Frequency | Cumulative Frequency |
|---|---|---|
| 140 ≤ h < 150 | 5 | 5 |
| 150 ≤ h < 160 | 12 | 17 |
| 160 ≤ h < 170 | 8 | 25 |
| 170 ≤ h < 180 | 3 | 28 |
Always plot points exactly at upper boundaries (150, 160, 170, 180) using the cumulative frequency just reached. A common mistake is using midpoints or starting at zero on the wrong boundary. Examiners expect a smooth curve, not straight lines connecting points, unless the question says otherwise.
始终用组距上界（150, 160, 170, 180）作为横坐标，并以当前累积频率描点。常见错误是使用组中值或在错误边界上从零开始。考官预期你画出平滑曲线，而非用直线连接各点，除非题目另有说明。
6. Scatter Graphs and Correlation | 散点图与相关性
Scatter graph questions provide bivariate data, such as temperature and ice cream sales. You plot points, then describe the correlation: positive (as x increases, y tends to increase), negative, or no correlation. Strength can be strong, moderate or weak. SQA often asks you to draw a line of best fit by eye, which must have roughly equal numbers of points on each side and pass through the ‘mean point’ (x̄, ȳ) if calculated. Use this line for estimation: for a given x, find the predicted y.
散点图题会提供双变量数据，如温度与冰淇淋销量。你先描点，然后描述相关性：正相关（x 增加，y 趋于增加）、负相关或无相关。关联强度有强、中、弱之分。SQA 常要求你凭目测画出最佳拟合线，这条线大致穿过均值点 (x̄, ȳ)，并且两侧点数大致相等。利用该线可进行预测：给定 x 值，找出对应的 y 估计值。
Past papers often ask ‘Does the data support the claim that higher revision hours lead to better test scores?’ Refer to the scatter graph: if points rise from left to right, the answer is yes, with explanation that the positive correlation provides evidence, but correlation does not imply causation. A rigid rule: always mention that other factors could influence the outcome.
真题中常见的问题是：“数据是否支持‘复习时间越长，考试成绩越好’的说法？”请引用散点图：如果点从左到右升高，答案就是“是”，并解释正相关提供了证据，但相关关系不意味着因果关系。一条要牢记的准则：始终指出其他因素也可能影响结果。
7. Probability and Tree Diagrams | 概率与树形图
Probability questions in SQA Year 10 papers range from simple single events to combined events using tree diagrams. For two independent events, multiply along branches. A bag contains 3 red and 5 blue counters; one is taken and replaced, then a second is taken. Tree diagram: first draw probabilities 3/8 and 5/8, then repeat for second draw. Probability of two reds = (3/8)×(3/8) = 9/64. Without replacement, the second probabilities change based on the first outcome. Always check that probabilities on each set of branches sum to 1.
SQA Year 10 试卷中的概率题从简单的单个事件到用树形图处理组合事件。对于两个独立事件，沿分支相乘。袋中有 3 个红球和 5 个蓝球；从中取出一个并放回，再取第二个。树形图：第一次抽取概率为 3/8 和 5/8，第二次重复。两个红球的概率 = (3/8)×(3/8) = 9/64。若是不放回，第二次概率会随第一次结果变化。务必检查每组分支的概率之和为 1。
When a question asks ‘Find the probability that at least one counter is red’, it is often quicker to use the complement: 1 − P(no red). Here, no red means both blue: (5/8)×(5/8)=25/64, so answer = 1 − 25/64 = 39/64. This approach saves time and reduces errors in exams.
当题目问“求至少有一个红球的概率”时，使用补集通常更快：1 − P(无红)。这里无红意味着两个都是蓝球：(5/8)×(5/8)=25/64，所以答案为 1 − 25/64 = 39/64。这种方法能节省时间并减少考试中的错误。
8. Interpreting Statistical Results in Context | 结合情境解读统计结果
A substantial part of the SQA Statistical Reasoning mark scheme is awarded for interpretation. You must link numbers back to the scenario. If a news article claims ‘The average household income rose by 15%’, but data shows the median remained unchanged while the mean increased due to a few very high earners, you should explain that the mean is distorted by outliers and the median gives a better picture of typical income.
SQA 统计推理的评分标准中有相当一部分分数给予解读。你必须将数字与情境关联起来。如果某新闻报道称“家庭平均收入增长了 15%”，但数据显示中位数未变而均值因少数极高收入者而增长，你就需要解释均值受异常值扭曲，中位数更能反映典型收入水平。
Comparing two groups using mean and standard deviation is another staple. ‘Group X has mean 68, SD 5.2; Group Y has mean 68, SD 9.8.’ Although the means are identical, Group X’s smaller SD shows results are more tightly clustered, suggesting higher consistency. Use comparative language: higher/lower, more/less consistent, more/less spread out.
利用均值和标准差比较两组数据是另一项核心技能。“X 组均值 68，标准差 5.2；Y 组均值 68，标准差 9.8。”虽然均值相同，但 X 组较小的标准差表明数据更集中，说明一致性更高。请使用比较性语言：更高/更低、更一致/更不稳定、更分散/更集中。
9. Common Mistakes Students Make in Exams | 学生考试常见错误
Many marks are lost through preventable errors. One is confusing the median position: for n values, the median is at the (n+1)/2 th position, not n/2. Another is forgetting to order data before finding the median. In box plots, students sometimes draw whiskers to Q1−1.5×IQR instead of the actual minimum/maximum unless outliers are defined. In cumulative frequency, plotting against midpoints instead of upper boundaries is a frequent slip. In probability, failing to adjust denominators when items are not replaced causes incorrect branch probabilities.
很多分数因可避免的错误而丢失。其一是混淆中位数的位置：对于 n 个数值，中位数在第 (n+1)/2 个位置，而非 n/2。另一个是找中位数前忘记排序。在盒须图中，学生有时会把须线画到 Q1−1.5×IQR 而不是实际的最小/最大值（除非题目标明异常值）。累积频率图中，用组中值而非上界描点是常见疏忽。概率题中，不放回时没有调整分母，导致分支概率错误。
Additionally, when calculating standard deviation, using n instead of n−1 is a classic mistake. Some candidates also round intermediate values too early, leading to inaccurate final answers. Always keep full precision until the final step. Finally, neglecting to include units in final answers loses valuable communication marks.
此外，在计算标准差时，使用 n 而非 n−1 是经典错误。有些考生过早舍入中间值，导致最终答案不准确。请始终保留完整精度直到最后一步。最后，答案中忘记写单位也会失去宝贵的表达分。
10. Exam Technique and Final Revision Tips | 应试技巧与最终复习建议
Start each question by identifying the command word: ‘Calculate’ means show working; ‘Draw’ means use a ruler and labelled axes; ‘Compare’ means use both a measure of centre and spread. Allocate time according to mark weight: a 4-mark question deserves about 5–6 minutes. Show all steps, because even if your final answer is wrong, you can earn method marks. Draw diagrams in pencil first, then go over in pen if required.
开始每题时先识别指令词：“计算”意味着要展示步骤；“绘制”意味着使用直尺并标注坐标轴；“比较”意味着需要同时使用集中量和离散量。按分值分配时间：4 分的题目大约用时 5–6 分钟。展示所有步骤，因为即使最终答案错了，也能拿到方法分。先用铅笔画图，按需要用钢笔描实。
When you finish a past-paper, always self-mark using the official marking scheme. Note whether you lost marks for method, accuracy or interpretation, and target your revision accordingly. Practise at least three full papers under timed conditions, and create a formula card with key equations like mean, standard deviation and probability rules. Repeated exposure to exam-style phrasing will boost your confidence on the day.
每完成一份真题，务必对照官方评分方案自评。记下自己在方法、准确性还是解读上丢了分，并据此进行针对性复习。在限时条件下至少练习三套完整试卷，并制作一张公式卡片，涵盖均值、标准差和概率规则等关键公式。反复接触考试措辞会让你在考试当天充满信心。
Published by TutorHao | Statistics Revision Series | aleveler.com
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