📚 AS Mathematics Unit 3 (Statistics 1) June 2019: Common Mistakes Summary | AS数学单元三（统计1）2019年6月易错点总结
The June 2019 Edexcel IAS Mathematics Unit 3 (Statistics 1) paper tested core concepts such as probability, representation of data, correlation, the normal distribution, and discrete distributions. Despite straightforward questions, many students lost marks due to common errors in notation, interpretation, and calculation. This article highlights the most frequent mistakes and provides clear guidance to help you avoid them in the exam.
2019年6月Edexcel IAS数学单元三（统计1）试卷考查了概率、数据表示、相关、正态分布和离散分布等核心概念。虽然题目直接，但很多学生因符号、解释和计算中的常见错误而失分。本文归纳最易犯的错，并提供清晰指导，帮助你在考试中避开这些陷阱。
1. Coded Data and Standard Deviation | 编码数据与标准差
back. Remember: if y = (x - a)/b, then x = b y + a. When calculating the mean of x, use x̄ = b ȳ + a, but for standard deviation, the constant a does not affect spread, so sₓ = |b| dard deviation is often given as σᵧ or sᵧ, and the transformation for variance is: Var(x) = b² Var(y).
在第1题中，题目给出编码值 y = (x - a)/b，要求计算原始数据 x 的均值和标准差。一个常见错误是在转换回原变量时颠倒了编码公式。记住：如果 y = (x - a)/b，则 x = b y + a。计算均值时，用 x̄ = b ȳ + a，但标准差中，常数 a 不影响离散程度，所以 sₓ = |b| sᵧ。很多学生错误地将 a 加到标准差上，或将编码标准差除以 b 而非乘以 b。此外，要确保在标准差公式中使用正确的 n 值；对样本使用了 n 而非 n-1 会导致结果不准确。编码数据的标准差通常记为 σᵧ 或 sᵧ，方差的转换为：Var(x) = b² Var(y)。
2. Probability Tree Diagrams and Conditional Probability | 概率树图和条件概率
Question 2 involved a tree diagram with conditional probabilities, where events were 'Pass' and 'Fail' e first test. Candidates often used the joint probability from the first branch instead of the conditional probability requested. Another mistake was failing to multiply along branches when finding an intersection probability, or omitting to consider both possible sequences that satisfy the event 'Passes on exactly one attempt'. Always label your tree with probabilities at each node and remember P(A ∩ B) = P(A) × P(B|A). For 'Passes exactly once' ised probability after additional information, use the conditional probability formula P(A|B) = P(A ∩ B) / P(B). Many forgot to divide by the total probability of the given event.
第2题涉及带条件概率的树图，事件为第一次和第二次尝试“通过”和“未通过”。最典型的错误是当条件为第一次测试未通过时，误用了条件分支的概率。考生经常使用第一条分支的联合概率，而非要求计算的条件概率。另一个错误是在求交集概率时忘记沿分支相乘，或者没有考虑“恰好通过一次”事件所包含的两种可能序列。务必在每个节点上用概率标注树图，牢记 P(A ∩ B) = P(A) × P(B|A)。对于“恰好通过一次”，应将 (通过, 未通过) 和 (未通过, 通过) 的概率相加。当题目要求在附加信息下修正概率时，要使用条件概率公式 P(A|B) = P(A ∩ B) / P(B)，许多人忘记除以给定事件的总概率。
3. Regression Line Interpretation | 回归直线解释
eted the intercept a, stating that it represents the final mark when the mock mark is zero — . Additionally, when using the regression line for prediction, many used extrapolation for an x-value far outside the range, which is unreliable. Always check the range of the given data before making predictions. Another pitfall was using the regression line of y on x when the regression line of x on y was required, leading to the wrong gradient. Note that the correlation coefficient r describes the strength of linear association, not the slope.
第3题关于双变量数据，给出了回归方程 y = a + bx，其中 y 为期末考分，x 为模拟考分。考生经常误解截距 a，称它表示当模拟分为零时的期末成绩——而零分可能不在数据范围内，因此这一解释没有意义。考官期望学生指出由于 x = 0 不在观测数据范围内，截距没有实际解释意义。此外，在使用回归线进行预测时，许多人对远超出数据范围的 x 值做外推，这是不可靠的。在预测前务必检查给定数据的范围。另一个陷阱是，当题目要求的是 x 对 y 的回归线时，却使用了 y 对 x 的回归线，导致斜率错误。注意相关系数 r 描述线性关联的强度，而非斜率。
4. Normal Distribution – Unknown Parameters | 正态分布——参数未知
.25, noting that because probability is less than 0.5, z must be negative. Students often forgot the negative sign, leading to an incorrect equation 120 = μ + |z|σ instead of 120 = μ - |z| lve systematically. Always sketch a diagram to visualise the tail probability and ensure the correct sign for z.
第4题用正态分布模拟苹果的重量，其中均值 μ 和标准差 σ 未知。一个常见的错误是在利用标准正态分布表寻找 μ 或 σ 时颠倒了步骤。例如，已知 P(X < 120) = 0.25，应找到 z 使得 Φ(z) = 0.25，注意到概率小于 0.5，因此 z 必为负数。考生经常忘记负号，导致列出错误方程 120 = μ + |z|σ，而不是 120 = μ - |z|σ。此外，当给定两个概率来建立联立方程时，解出 μ 和 σ 的代数错误也很常见。记住仔细标准化：z = (x - μ)/σ，并有条理地求解。始终画出草图，将尾部概率可视化，确保 z 的正负号正确。
5. Discrete Probability Distributions | 离散概率分布
ch gives the wrong expected value. Once k is found, verify that all probabilities are between 0 and 1. When computing E(X²) for variance, errors included forgetting to square the x-values before multiplying by probabilities. Variance formula is Var(X) = E(X²) - [E(X)]². A mistake was using E(X - μ)² directly without the shortcut, leading to arithmetic slips. Also, ensure you give probabilities as fractions or decimals as requested; unnecessary rounding early on can affect the final variance.
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