📚 AS Mathematics Unit 3 (Statistics 1) June 2019 Mark Scheme: Common Mistakes Summary | AS数学单元3（统计1）2019年6月评分标准易错点总结
The June 2019 AS Mathematics Unit 3 (Statistics 1) mark scheme highlights a variety of errors that consistently appeared across candidate scripts. By examining the marking points and explanatory notes, we can extract the most frequent pitfalls and understand exactly what examiners were looking for. This article summarises those common mistakes in topics including probability, discrete distributions, the normal distribution, and hypothesis testing. Each point is paired with advice on how to avoid losing marks in future exams.
2019年6月AS数学单元3（统计1）的评分方案揭示了考生答卷中反复出现的多种错误。通过分析评分要点与注释，我们可以提炼出最常见的失分陷阱，并准确理解考官的评分要求。本文汇总了概率、离散分布、正态分布及假设检验等主题中的典型易错点，并为每个点提供了避免失分的建议。
1. Probability Tree Diagrams: Skipping the Multiplication Step | 概率树图：遗漏相乘步骤
Many candidates correctly draw a probability tree and label branch probabilities, but then fail to multiply along the branches when finding the probability of a combined event. The mark scheme specifically expects to see a multiplication, e.g. P(A ∩ B) = P(A) × P(B|A). Simply writing the final probability without showing the product loses method marks.
许多考生能正确画出概率树并标注分支概率，但在求复合事件概率时却忘记将分支上的概率相乘。评分方案明确要求展示乘法步骤，例如 P(A ∩ B) = P(A) × P(B|A)。直接给出最终概率而没有写出乘积过程，会丢掉方法分。
Furthermore, a common oversight occurs when both branches lead to the same outcome; candidates add the final probabilities correctly but fail to show the individual multiplications for each path. Always set out your working as: Path 1: a × b = …; Path 2: c × d = …; then P(final) = sum. This transparent method reduces the risk of arithmetic slips.
此外，当多条分支导向同一个结果时，常见疏忽是考生会正确相加最终概率，却没有分别展示每条路径的乘法过程。始终采用这样的步骤：路径1：a × b = …；路径2：c × d = …；然后 P(最终) = 两者之和。这种透明的过程能降低计算失误的风险。
2. Conditional Probability: Misidentifying the Denominator | 条件概率：分母识别错误
A recurrent error in conditional probability questions is using the unconditional total probability as the denominator instead of the probability of the given event. The mark scheme insists on the structure P(A|B) = P(A ∩ B) / P(B). Many candidates write the correct numerator but then divide by 1 or by the total sample space size, revealing a misunderstanding of the ‘given that’ condition.
条件概率题中一个反复出现的错误是，用无条件总概率作为分母，而不是用给定事件的概率。评分方案坚持要求使用结构 P(A|B) = P(A ∩ B) / P(B)。许多考生分子写得正确，分母却除以1或样本空间总数，这表明对“已知…条件下”的理解不深。
Examiners also noted that candidates sometimes try to read the answer directly from a two-way table without applying the formula, often confusing row and column totals. It is safer to state the formula first, then substitute values, e.g. P(defective|machine A) = (number defective from A) / (total from A).
考官还注意到，有些考生试图直接从双向表中读取答案而不套用公式，常常混淆行合计与列合计。更稳妥的做法是先写出公式，再代入数值，如 P(次品|机器A) = (来自A的次品数) / (来自A的总数)。
3. Discrete Random Variables: Losing Probability Mass in Variance | 离散随机变量：方差计算丢失概率质量
Calculating the variance of a discrete random variable is a perennial source of mistakes. The mark scheme reveals that while E(X) is often found correctly, E(X²) is mishandled. A typical error is to compute Σ x² p(x) as (Σ x p(x))², which gives a completely wrong value. Others omit the probability from the x² contributions entirely, summing only x².
离散随机变量的方差计算是长期存在的失分点。评分方案显示，虽然 E(X) 往往能正确求出，E(X²) 的处理却问题频出。典型错误是把 Σ x² p(x) 算成 (Σ x p(x))²，导致结果完全错误。另一些考生在求 x² 的贡献时完全遗漏了概率，只对 x² 求和。
The correct procedure is to extend the probability distribution table to include a row for x² p(x). Sum that row to obtain E(X²), then use Var(X) = E(X²) − [E(X)]². The mark scheme often awards separate marks for E(X²) and the final variance, so even if the final answer is wrong, a correct E(X²) can secure method marks.
正确的做法是扩展概率分布表，加一行 x² p(x)。将这一行求和得到 E(X²)，然后使用 Var(X) = E(X²) − [E(X)]²。评分方案通常对 E(X²) 和最终方差分别给分，因此即使最终答案有误，正确的 E(X²) 也能保住一部分方法分。
4. Binomial Distribution: Miscounting ‘At Least’ and ‘At Most’ | 二项分布：错数“至少”与“至多”
Questions involving phrases such as ‘more than 5’ or ‘at most 3’ cause frequent misinterpretation. The mark scheme expects precise use of cumulative probabilities. For X ~ B(n, p), P(X > 5) must be expressed as 1 − P(X ≤ 5), and P(X ≥ 5) as 1 − P(X ≤ 4). A common mistake is to write 1 − P(X ≤ 5) for P(X ≥ 5), which over-counts.
涉及“多于5”或“至多3”之类表述的问题常被错误解读。评分方案期望精确使用累积概率。对于 X ~ B(n, p)，P(X > 5) 必须写成 1 − P(X ≤ 5)，而 P(X ≥ 5) 应为 1 − P(X ≤ 4)。常见的错误是把 P(X ≥ 5) 写成 1 − P(X ≤ 5)，从而多算了概率。
The mark scheme also penalises candidates who use calculator binomial function without writing down the correct tail expression. Even if the final numeric answer is right, the lack of a clear statement like ‘1 − binomcdf(n, p, x)’ can lose communication marks. Always define the cumulative boundary explicitly.
评分方案还会对只使用计算器二项功能却没有写出正确尾部表达式的考生扣分。即使最终得数正确，缺少像“1 − binomcdf(n, p, x)”这样清晰的表述也可能损失表达分。务必明确写出累积边界。
5. Normal Distribution: Incorrect Standardisation | 正态分布：标准化公式用错
The transformation Z = (X − μ) / σ is critical, yet numerous candidates either forget to subtract μ, divide by σ², or use σ when the variance is given. The June 2019 mark scheme repeatedly highlighted that a standardisation must show the correct denominator. For a sample mean, the standard error σ/√n is required, and many students used σ instead.
变换 Z = (X − μ) / σ 至关重要，但大量考生要么忘了减去 μ，要么除以 σ²，或者在给出方差时误用了 σ。2019年6月评分方案反复强调标准化必须展现正确的分母。对于样本均值，需使用标准误 σ/√n，许多学生却用了 σ。
Another crucial point is handling inverse normal calculations. When finding μ or σ, candidates often forget to apply the sign of the Z-value correctly. For a lower-tail probability below 0.5, Z is negative; using a positive Z leads to an absurd parameter. The mark scheme rewards a sketch of the standard normal curve to confirm the sign.
另一个关键点是处理逆正态计算。在求 μ 或 σ 时，考生常忘记正确运用 Z 值的符号。对于小于0.5的左尾概率，Z 为负；若用正 Z 值会得出荒谬的参数值。评分方案鼓励画出标准正态曲线草图以确认符号。
6. Normal Approximation to Binomial: Missing the Continuity Correction | 二项正态近似：遗漏连续性校正
When approximating a binomial distribution with a normal, the continuity correction is a must, yet it is frequently omitted. The June 2019 mark scheme insists on adjusting the boundary by ±0.5. For P(X ≥ a), use P(X > a − 0.5) in the normal approximation; for P(X ≤ b), use P(X < b + 0.5). Implementing the opposing half-unit ensures the area under the normal curve better matches the discrete bars.
用正态分布近似二项时，连续性校正是必须的，但常被遗漏。2019年6月评分方案要求必须通过 ±0.5调整边界。对于 P(X ≥ a)，在正态近似中使用 P(X > a − 0.5)；对于 P(X ≤ b)，使用 P(X < b + 0.5)。这种相反的半单位校正能使正态曲线下面积更好地匹配离散条形。
Many candidates lose marks by applying the correction in the wrong direction, e.g. using X > a + 0.5 for a right tail. Remember: to include the boundary value, widen the interval; to exclude it, narrow the interval. Writing down the adjustment step explicitly before standardising earns method marks even if the final probability is slightly off.
许多考生因校正方向弄反而丢分，例如对右尾使用 X > a + 0.5。记住：若要包含边界值，就扩展区间；若要排除边界值，就缩小区间。在标准化之前明确写出调整步骤，即使最终概率稍有偏差也能获得方法分。
7. Hypothesis Testing: Vague or Incomplete Conclusions | 假设检验：结论模糊或不完整
The mark scheme places heavy emphasis on the quality of the written conclusion. It is not enough to write ‘Reject H₀’. Examiners expect a contextualised statement: ‘There is sufficient evidence at the 5% significance level to support the claim that the proportion has increased.’ Missing the context, the significance level, or the direction of the test loses the final conclusion mark.
评分方案极其重视结论的书写质量。仅仅写“拒绝 H₀”是不够的。考官期望一个结合上下文的陈述：“在5%显著性水平下，有充分证据支持比例上升的说法。”遗漏上下文、显著性水平或检验方向都会丢掉最后的结论分。
Another frequent weakness is using ‘accept H₀’ instead of ‘do not reject H₀’. The mark scheme explicitly forbids the phrase ‘accept’ in the context of hypothesis testing. The correct wording acknowledges that insufficient evidence does not prove the null hypothesis true; it merely means we fail to disprove it.
另一个常见不足是使用“接受 H₀”而非“不拒绝 H₀”。评分方案明确禁止在假设检验中使用“接受”一词。正确的措辞应承认证据不充分不代表原假设为真，只是未能将其推翻。
8. Hypothesis Testing: Getting the Critical Region Direction Wrong | 假设检验：临界区域方向弄反
Even when the test statistic is calculated correctly, candidates often place the critical region in the wrong tail. The mark scheme shows that a one-tailed test with H₁: p > 0.5 must use an upper-tail critical value, yet some students use a lower tail. This mistake stems from misreading the alternative hypothesis or confusing ‘more than’ with ‘less than’ in the context.
即便检验统计量计算正确，考生也常把临界区域放错尾部。评分方案显示，对于 H₁: p > 0.5 的单尾检验，必须使用上尾临界值，有些学生却用了下尾。这一错误源于误读备择假设，或在语境中混淆“多于”与“少于”。
To avoid this, annotate H₁ with an arrow: p > value → upper tail, p < value → lower tail. For two-tailed tests, remember to split the significance level, e.g. 5% becomes 2.5% in each tail. The mark scheme awards marks for correctly stating the critical region in terms of the test statistic before comparing with the observed value.
为避免这一点，用箭头标注 H₁：p > value → 上尾，p < value → 下尾。对于双尾检验，要记得分割显著性水平，例如5%变成每尾2.5%。评分方案对在比较观测值之前先用检验统计量正确表述临界区域的做法会给予奖励分数。
9. Sampling Distribution of the Mean: Forgetting the Standard Error | 均值的抽样分布：忘记标准误
When the test concerns a sample mean, the sampling distribution of the mean must be used: X̅ ~ N(μ, σ²/n). A very common error in the June 2019 scripts was to use the population variance σ² directly instead of σ²/n when standardising or calculating probabilities for X̅. This leads to a test statistic that is √n times too small, potentially changing the conclusion.
当检验涉及样本均值时，必须使用均值的抽样分布：X̅ ~ N(μ, σ²/n)。2019年6月答卷中一个极常见的错误是，在标准化或计算 X̅ 的概率时直接使用总体方差 σ² 而非 σ²/n。这导致检验统计量小了 √n 倍，可能改变结论。
The mark scheme explicitly separates marks for stating the correct distribution and for the correct standardisation. Always start by writing ‘If X ~ N(μ, σ²), then X̅ ~ N(μ, σ²/n)’ even if the population is not normal, as long as the Central Limit Theorem is invoked. This step earns marks and reduces later errors.
评分方案明确将表述正确分布与正确标准化作为两个独立给分点。始终先写出“若 X ~ N(μ, σ²)，则 X̅ ~ N(μ, σ²/n)”，即使总体非正态，只要引用了中心极限定理。这一步能得分，也能减少后续错误。
10. Misreading P-values: Comparing the Wrong Way | 误读P值：比较方向弄反
P-values are a standard output of hypothesis testing, yet their interpretation is often mangled. The rule is: if p-value < significance level (α), reject H₀. However, many candidates wrote 'p-value > 0.05, therefore we reject H₀’ or reversed the inequality. The mark scheme does not forgive a logically inconsistent conclusion built on a correct p-value.
P值是假设检验的标准输出，但其解读却常被曲解。规则是：若 p-value < 显著性水平 (α)，则拒绝 H₀。然而许多考生写道“p-value > 0.05，因此我们拒绝 H₀”，或将不等式方向弄反。评分方案不会宽恕基于正确p值却得出逻辑矛盾结论的情况。
A related fault is believing the p-value is the probability that H₀ is true. The mark scheme explicitly warns against this interpretation. The p-value is the probability of obtaining a result at least as extreme as the observed, assuming H₀ is true. Clarifying this distinction in revision prevents conceptual blunders.
一个相关错误是认为p值是H₀为真的概率。评分方案明确警告不要这样解读。p值是假设H₀为真的前提下，获得至少与观测结果同样极端结果的概率。复习时厘清这一区别可以避免概念性失误。
The analysis of the June 2019 Unit 3 mark scheme underlines that many lost marks come from avoidable procedural slips rather than a lack of understanding. By internalising these common mistakes—such as missing continuity corrections, misidentifying conditional denominators, and writing incomplete conclusions—students can significantly improve their accuracy and meet the examiners’ expectations.
对2019年6月单元3评分方案的分析表明，许多失分源于可以避免的程序性疏忽，而非理解不足。通过内化这些常见错误——如遗漏连续性校正、误判条件概率分母、结论表述不完整——学生能显著提升准确性，达到考官的期望。
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