5D Paired Samples | 5D 配对样本

📚 5D Paired Samples | 5D 配对样本

When we collect two measurements from the same subject or from matched subjects, the data are paired. Analysing such data with a standard two‑sample t‑test ignores the natural link between the observations and can lead to a loss of power. The paired samples t‑test uses the differences within each pair to determine whether the population mean difference is significantly different from zero (or some other reference value).

当我们从同一个对象或匹配的对象收集两次测量值时，数据就是配对的。用普通的双样本 t 检验分析这类数据会忽略观测值之间天然的联系，可能导致检验功效降低。配对样本 t 检验利用每对数据的差值，判断总体均值差是否显著不同于零（或其他参考值）。

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1. What Are Paired Samples? | 什么是配对样本？

Paired samples arise when observations are naturally linked. Common examples include before‑and‑after studies, twin studies, left‑hand vs right‑hand measurements, or measurements taken on the same individual under two different conditions. Each pair consists of two values that are not independent, because they share a common source of variability.

配对样本出现在观测值天然相关的情况下。常见的例子包括前后对比研究、双胞胎研究、左右手测量，或同一人在两种不同条件下的测量。每对数据由两个不独立的值组成，因为它们共享相同的变异来源。

design, we focus on the differences dᵢ = x₂ᵢ − x₁ᵢ (or vice versa) for each pair i = 1, 2, …, n. These differences become the raw data for a one‑sample t‑test. The key point is that by working with the differences, we eliminate between‑subject variability and isolate the effect of interest.

在配对设计中，我们关注的是每对数据的差值 dᵢ = x₂ᵢ − x₁ᵢ（或相反），i = 1, 2, …, n。这些差值就成了单样本 t 检验的原始数据。关键在于，通过处理差值，我们排除了对象间变异，孤立出我们关注的处理效应。

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2. When to Use a Paired t‑Test | 何时使用配对 t 检验

A paired t‑ or the central limit theorem to ensure approximate normality of the mean difference. The two measurements must be made on the same unit or on genuinely matched units (e.g., same age, gender, BMI).

当两个相关样本来自差值服从正态分布的总体，或者样本量足够大（通常 n ≥ 30）以保证中心极限定理使均值差近似正态时，适合使用配对 t 检验。两次测量必须针对同一单元或真正匹配的单元（例如年龄、性别、BMI 相同）。

Using a two‑sample t‑test for independent samples on paired data is a serious mistake, because the correlation within pairs inflates the standard error estimate and makes the test insensitive to real differences. Always check whether the data have a natural pairing before choosing the test.

对配对数据使用独立样本的双样本 t 检验是一个严重错误，因为配对内的相关性会放大标准误估计，使检验对真实差异不敏感。选择检验方法之前，一定要确认数据是否存在自然配对。

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3. Hypothesis Setup | 假设设定

e μ_d is equal to zero (or some specified value μ₀). The alternative hypothesis can be two‑sided (μ_d ≠ μ₀) or one‑ μ₀). In the majority of exam board problems, μ₀ = 0, representing no change or no effect.

原假设声称总体均值差 μ_d 等于零（或某个指定值 μ₀）。备择假设可以是双侧的（μ_d ≠ μ₀），也可以是单侧的（μ_d > μ₀ 或 μ_d < μ₀）。在大多数考试题目中，μ₀ = 0，代表无变化或无效应。

Formally:
H₀: μ_d = μ₀
H₁: μ_d ≠ μ₀ (two‑tailed) or μ_d > μ₀ (one‑tailed), etc.

正式地：
H₀: μ_d = μ₀
H₁: μ_d ≠ μ₀（双尾）或 μ_d > μ₀（单尾），等等。

The choice between a one‑tailed and a two‑tailed test depends on the research question. If we suspect an increase after an intervention, a right‑tailed test is justified. If we simply want to detect any difference, use a two‑tailed test.

单尾还是双尾检验取决于研究问题。如果我们怀疑干预后有所提高，可以使用右尾检验。如果我们只是想探测任何差异，则使用双尾检验。

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4. Test Statistic for Paired Data | 配对数据的检验统计量

The paired t‑statistic is computed from the sample of differences d₁, d₂, …, dₙ. Let d̄ be the sample mean of the differences, s_d the sample standard deviation of the differences, and n the number of pairs. The test statistic is

配对 t 统计量由差值样本 d₁, d₂, …, dₙ 计算得到。设 d̄ 为差值的样本均值，s_d 为差值的样本标准差，n 为配对数。检验统计量为

t = (d̄ − μ₀) / (s_d/√n)

This formula is exactly the same as the one‑sample t‑test applied to the differences. The denominator s_d/√n is the standard error of the mean difference. Under H₀, the statistic follows a t‑distribution with n − 1 degrees of freedom, provided the differences are approximately normal.

该公式与用于差值的单样本 t 检验完全相同。分母 s_d/√n 是均值差的标准误。在 H₀ 下，只要差值近似正态，该统计量服从自由度为 n − 1 的 t 分布。

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5. Degrees of Freedom | 自由度

For a paired t‑test, the degrees of freedom (df) are n − 1, where n is the number of pairs. This is because we are performing a one‑sample t‑test on n differences, so we lose one degree of freedom for estimating the mean d̄.

对于配对 t 检验，自由度（df）为 n − 1，其中 n 是配对数。这是因为我们对 n 个差值进行单样本 t 检验，在估计均值 d̄ 时失去一个自由度。

The degrees of freedom determine the shape of the t‑distribution used to find critical values and p‑values. As the sample size increases, the t‑distribution approaches the standard normal distribution, but for small n the tails are heavier.

自由度决定了用于查找临界值和 p 值的 t 分布形状。随着样本量增加，t 分布趋近于标准正态分布，但在小样本时尾部更厚。

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6. Critical Values and Rejection Region | 临界值与拒绝域

At significance level α, we compare the observed |t| with the critical value t* from the t‑distribution with n−1 df. For a two‑tailed test, reject H₀ if |t| > t*. For a right‑tailed test (H₁: μ_d > μ₀), reject H₀ if t > tₙ₋₁,α; for a left‑tailed test, reject if t < −tₙ₋₁,α.

在显著性水平 α 下，我们将观测的 |t| 与自由度为 n−1 的 t 分布临界值 t* 进行比较。对于双尾检验，若 |t| > t*，则拒绝 H₀。对于右尾检验（H₁: μ_d > μ₀），若 t > tₙ₋₁,α，则拒绝 H₀；对于左尾检验，若 t < −tₙ₋₁,α，则拒绝 H₀。

statistical tables or software. For instance, with 9 degrees of freedom and α = 0.05 (two‑ the critical value is approximately 2.262. Always check whether the problem asks for a one‑ or two‑tailed test.

临界值可以从统计表或软件中获取。例如，在自由度为 9、α = 0.05（双尾）的情况下，临界值约为 2.262。务必检查题目要求的是单尾还是双尾检验。

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7. p‑value Approach | p 值方法

The p‑ is true, of obtaining a test statistic at least as extreme as the one observed. For a two‑tailed test, p = 2 × P(T > |t|), where T follows a t‑distribution with n−1 df. If p < α, we reject H₀.

p 值是在 H₀ 成立的条件下，获得至少如观测值一样极端的检验统计量的概率。对于双尾检验，p = 2 × P(T > |t|)，其中 T 服从自由度为 n−1 的 t 分布。如果 p < α，则拒绝 H₀。

Modern data analysis often relies on p‑values because they provide a continuous measure of evidence against the null hypothesis. A very small p‑value (e.g., p < 0.001) indicates strong evidence of a real difference.

现代数据分析通常依赖 p 值，因为它提供了反对原假设的连续度量。极小的 p 值（如 p < 0.001）表明存在显著差异的有力证据。

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8. Confidence Interval for Mean Difference | 均值差的置信区间

A (1−α)×100% confidence interval for the population mean difference μ_d is given by

总体均值差 μ_d 的 (1−α)×100% 置信区间为

d̄ ± t_{n−1, α/2} × (s_d/√n)

l does not contain μ₀ (often 0), we reject H₀ at significance level α. Confidence intervals are more informative than a simple reject/fail‑to‑reject decision because they indicate the size and direction of the effect.

该区间给出了 μ_d 的合理取值范围。如果区间不包含 μ₀（通常为 0），我们就在显著性水平 α 下拒绝 H₀。置信区间比简单的拒绝/不拒绝结论更具信息量，因为它显示了效应的大小和方向。

For example, a 95% confidence interval of (1.2, 5.8) for μ_d means we are 95% .8 units, and because 0 is not in the interval, the increase is statistically significant.

例如，μ_d 的 95% 置信区间为 (1.2, 5.8)，表示我们有 95% 的信心认为真实的均值差介于 1.2 和 5.8 个单位之间，并且由于 0 不在区间内，该增加在统计上是显著的。

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9. Checking Assumptions | 检验假设条件

The paired t‑ are independent and approximately normally distributed. Normality can be checked with a Q‑Q plot, a histogram, or a Shapiro– entral limit theorem relaxes the normality requirement, making the test reasonably robust.

配对 t 检验假设差值独立且近似正态分布。正态性可以通过差值的 Q‑Q 图、直方图或 Shapiro–Wilk 检验来检查。如果样本量较大（n ≥ 30），中心极限定理放宽了正态性要求，使得检验相当稳健。

If the normality assumption is severely violated and the sample size is small, consider a non‑parametric alternative such as the Wilcoxon signed‑rank test. However, for IB and Cambridge exams, the focus is usually on understanding the t‑test and its conditions.

如果正态性假设严重违背且样本量小，应考虑非参数替代方法，如 Wilcoxon 符号秩检验。不过，在 IB 和剑桥考试中，重点通常在于理解 t 检验及其条件。

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10. Worked Example | 例题演示

scores (out of 50) and differences are recorded. The mean difference (after − before) is d̄ = 4.2, and the standard deviation of the differences is s_d = 5.1. Test at the 5% significance level whether the revision tool improves performance.

一组 10 名学生在使用新的在线复习工具前后各参加了一次数学测验。成绩（满分 50）和差值已记录。差值（后 − 前）均值为 d̄ = 4.2，差值的标准差为 s_d = 5.1。在 5% 的显著性水平下检验该复习工具是否提高了成绩。

Solution:
H₀: μ_d = 0
H₁: μ_d > 0 (one‑tailed)
n = 10, df = 9
t = (4.2 − 0) / (5.1/√10) = 4.2 / 1.613 ≈ 2.604

解答：
H₀: μ_d = 0
H₁: μ_d > 0（单尾）
n = 10，df = 9
t = (4.2 − 0) / (5.1/√10) ≈ 4.2 / 1.613 ≈ 2.604

Critical value for one‑ dence to conclude that the online revision tool significantly improves test scores. The 95% 4.2 + 2.262×1.613) ≈ (0.55, 7.85). The interval does not include 0, confirming the result.

单尾 α = 0.05、自由度为 9 的临界值约为 1.833。由于 2.604 > 1.833，我们拒绝 H₀。有充分证据表明，在线复习工具显著提高了测验分数。μ_d 的 95% 置信区间为 (4.2 − 2.262×1.613, 4.2 + 2.262×1.613) ≈ (0.55, 7.85)。该区间不包含 0，进一步确认了这一结果。

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11. Common Mistakes | 常见错误

One frequent error is using a two‑sample t‑test for paired data, which leads to an incorrect standard error and reduced power. Another is forgetting to compute the differences correctly—always decide whether it is after − before or matched pair 1 − matched pair 2, and be consistent.

一个常见的错误是对配对数据使用双样本 t 检验，这会导致标准误计算错误并降低检验功效。另一个错误是忘记正确计算差值——始终要明确是后 − 前还是匹配对 1 − 匹配对 2，并保持一致。

automatically mean the two groups are equal; it means the mean of the differences is zero. Also, rounding the test statistic too early can lead to a wrong conclusion near the critical value. Always carry enough decimal places during calculation.

学生有时会误解假设：检验 μ_d = 0 并不自动意味着两组相等；它意味着差值的均值为零。此外，过早对检验统计量进行四舍五入可能导致在临界值附近得出错误结论。计算过程中应始终保留足够的小数位。

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12. Summary & Key Points | 总结与要点

• The paired t‑test analyses the differences d within n pairs. | 配对 t 检验分析 n 对数据内部的差值 d。
• Test statistic: t = (d̄ − μ₀) / (s_d/√n) with n−1 df. | 检验统计量：t = (d̄ − μ₀) / (s_d/√n)，自由度为 n−1。
• Assumptions: the differences are independent and approximately normal (or n ≥ 30). | 假设条件：差值独立且近似正态（或 n ≥ 30）。
• Confidence interval: d̄ ± t_{n−1, α/2} × (s_d/√n). | 置信区间：d̄ ± t_{n−1, α/2} × (s_d/√n)。
• Paired designs control for between‑subject variability, making them more efficient than independent designs whenever pairing is possible. | 配对设计控制了对象间变异，因此在可配对的情况下比独立设计更高效。

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