📚 Year 13 AQA Psychology: Formula & Theorem Quick-Reference Handbook | AQA A-level 心理学：公式定理速查手册
This handbook compiles the essential statistical formulas, decision rules and key theorems required for Year 13 AQA Psychology. It covers descriptive statistics, the step-by-step logic of inferential tests (Sign test, Wilcoxon signed-ranks, Mann-Whitney U, Spearman’s rho and Chi-squared), standard error, test-selection criteria, and the interpretation of significance. Use it as a rapid revision tool before mocks and the final exam.
本手册汇集了 Year 13 AQA 心理学课程要求的核心统计公式、决策规则与关键定理。它覆盖描述统计、推断检验（符号检验、威尔科克森符号秩检验、曼-惠特尼 U 检验、斯皮尔曼等级相关和卡方检验）的分步逻辑、标准误、检验选择标准以及显著性的解读。可将其作为模拟考试和最终大考前的速查工具。
1. Measures of Central Tendency & Dispersion | 集中趋势与离散量数
The mean (x̄) is the arithmetic average: x̄ = Σx / n, where Σx is the sum of all scores and n is the total number of scores. It is used for interval/ratio data but is sensitive to outliers.
均值 (x̄) 是算术平均数：x̄ = Σx / n，其中 Σx 是所有得分的总和，n 是得分的总数。它用于定距/定比数据，但容易受异常值影响。
The median is the middle score when data are ordered from smallest to largest. For an odd n, median = (n+1)/2 th value; for an even n, it is the average of the two middle scores. It is robust against outliers and is appropriate for ordinal data.
中位数是将数据从小到大排序后的中间值。当 n 为奇数时，中位数 = 第 (n+1)/2 个值；当 n 为偶数时，则为中间两个值的平均。它不受异常值影响，适用于定序数据。
The mode is the most frequently occurring score. A data set can have one mode (unimodal), two (bimodal) or more. It is the only measure of central tendency suitable for nominal data.
众数是出现频率最高的得分。一个数据集可以有一个众数（单峰）、两个（双峰）或更多。它是唯一适合定类数据的集中趋势量数。
The range is the difference between the highest and lowest scores: Range = Max – Min. It gives a quick indication of spread but is affected by extreme values.
极差是最高分与最低分之间的差值：极差 = 最大值 – 最小值。它能快速反映分散程度，但受极端值影响。
Variance (s²) is the average of the squared deviations from the mean: s² = Σ(x – x̄)² / (n – 1) for a sample. Standard deviation (s) is the square root of variance: s = √[ Σ(x – x̄)² / (n – 1) ].
方差 (s²) 是离均差平方的平均数，样本方差：s² = Σ(x – x̄)² / (n – 1)。标准差 (s) 是方差的平方根：s = √[ Σ(x – x̄)² / (n – 1) ]。
2. Calculating Standard Deviation Step-by-Step | 标准差分步计算
Step 1: Calculate the mean (x̄) of the data set. Step 2: Subtract the mean from each score to obtain deviation scores (x – x̄). Step 3: Square each deviation: (x – x̄)². Step 4: Sum all squared deviations: Σ(x – x̄)². Step 5: Divide by n – 1 (for sample SD). Step 6: Take the square root to obtain s.
第一步：计算数据集的均值 (x̄)。第二步：每个得分减去均值，得到离差 (x – x̄)。第三步：将每个离差平方：(x – x̄)²。第四步：将所有平方离差相加：Σ(x – x̄)²。第五步：除以 n – 1（样本标准差）。第六步：取平方根得到 s。
The denominator n – 1 (Bessel’s correction) gives an unbiased estimate of the population standard deviation. In AQA exam work, you may be asked to use either n or n – 1; always check the question.
分母使用 n – 1（贝塞尔校正）可得到总体标准差的无偏估计。在 AQA 考试中可能要求使用 n 或 n – 1，务必看清题目要求。
3. Standard Error of the Mean & Confidence Intervals | 均值的标准误与置信区间
The standard error of the mean (SE) estimates how much sample means vary around the population mean: SE = s / √n, where s is the sample standard deviation and n is the sample size.
均值的标准误 (SE) 用于估计样本均值在总体均值周围的波动程度：SE = s / √n，其中 s 是样本标准差，n 是样本量。
A 95% confidence interval for the mean is given by: x̄ ± (1.96 × SE) when n is large and data are normally distributed. For smaller samples, the critical t-value replaces 1.96.
均值的 95% 置信区间为：x̄ ± (1.96 × SE)，适用于大样本且数据呈正态分布的情况。小样本时则用临界 t 值代替 1.96。
In AQA Psychology, you are not required to compute confidence intervals in detail, but understanding SE helps you interpret how reliable the sample mean is as an estimate of the population mean.
在 AQA 心理学中无需详细计算置信区间，但理解标准误有助于解释样本均值作为总体均值估计的可靠性。
4. The Sign Test Formula | 符号检验公式
The sign test is a non-parametric test for a repeated measures or matched pairs design when the data are at least nominal (direction of difference). It examines whether there is a significant difference by counting the number of positive and negative signs.
符号检验是一种非参数检验，用于重复测量或配对设计，且数据至少为定类（差异方向）。它通过计算正号和负号的数量来检验是否存在显著差异。
Procedure: For each pair, record a ‘+’ when condition B is greater than condition A, a ‘-‘ when B is less, and ignore ties. Let S be the count of the less frequent sign. Compare S against a critical value from a binomial distribution table; if S ≤ critical value, the result is significant.
步骤：对于每一对数据，当条件 B 大于条件 A 时记 ‘+’，当 B 小于 A 时记 ‘-‘，平局忽略。设 S 为较少出现的符号的个数。将 S 与二项分布临界值表比较；若 S ≤ 临界值，结果显著。
Formula shorthand: S = min( number of ‘+’ , number of ‘-‘ ). For a two-tailed test, double the one-tailed p-value or use the critical value accordingly.
公式速记：S = min( ‘+’ 的数量 , ‘-‘ 的数量 )。对于双尾检验，将单尾 p 值翻倍或相应使用临界值。
5. Wilcoxon Signed-Ranks Test Calculation | 威尔科克森符号秩检验计算
The Wilcoxon signed-ranks test is used for a repeated measures or matched pairs design with ordinal data. It takes into account both the direction and the magnitude of the differences.
威尔科克森符号秩检验用于重复测量或配对设计的定序数据，它同时考虑差异的方向和幅度。
Steps: (1) Calculate the difference score for each pair (D = Condition 2 – Condition 1), ignoring zeros. (2) Rank the absolute differences |D| from smallest to largest, assigning the average rank for ties. (3) Attach the original sign of the difference to each rank. (4) Sum the ranks of the positive differences (R+) and the negative differences (R-). (5) The test statistic T = the smaller of R+ and R-. (6) Compare T with the critical Wilcoxon T value for the given N (number of non-zero pairs); if T ≤ critical value, the result is significant.
步骤：(1) 计算每对数据的差异分数 (D = 条件 2 – 条件 1)，忽略零差异。(2) 对差值的绝对值 |D| 从小到大排秩，遇到相同值则赋予平均秩次。(3) 将每个秩次赋予原始差值的符号。(4) 分别计算正差值秩和 (R+) 和负差值秩和 (R-)。(5) 检验统计量 T = R+ 与 R- 中的较小值。(6) 将 T 与给定 N（非零配对数）的威尔科克森 T 临界值比较；若 T ≤ 临界值，则结果显著。
Formula summary: T = min( Σ rank(+), Σ rank(-) ). Reject the null hypothesis if T ≤ critical value.
公式总结：T = min( Σ 正秩和 , Σ 负秩和 )。若 T ≤ 临界值，则拒绝零假设。
6. Mann-Whitney U Test Formula | 曼-惠特尼 U 检验公式
The Mann-Whitney U test is the independent groups equivalent of the Wilcoxon test. It is used when the design is independent measures and the dependent variable is at least ordinal.
曼-惠特尼 U 检验是独立组设计的对应检验，当研究设计为独立测量且因变量至少为定序数据时使用。
Procedure: (1) Combine all scores from both groups and rank them from lowest to highest, assigning average ranks to ties. (2) Calculate the sum of ranks for Group 1 (R₁) and for Group 2 (R₂). (3) Compute U₁ using the formula: U₁ = R₁ – [n₁(n₁ + 1)] / 2, where n₁ is the size of Group 1. (4) Compute U₂ = R₂ – [n₂(n₂ + 1)] / 2. (5) The test statistic U = min(U₁, U₂). (6) Compare U with the critical value from Mann-Whitney tables. For significance, U must be less than or equal to the critical value.
步骤：(1) 将所有得分合并并按从小到大排秩，相同值赋予平均秩次。(2) 分别计算组1的秩和 (R₁) 和组2的秩和 (R₂)。(3) 用公式计算 U₁：U₁ = R₁ – [n₁(n₁ + 1)] / 2，其中 n₁ 是组1的样本量。(4) 计算 U₂ = R₂ – [n₂(n₂ + 1)] / 2。(5) 检验统计量 U = min(U₁, U₂)。(6) 将 U 与曼-惠特尼 U 临界值表比较。要得到显著结果，U 必须小于或等于临界值。
Key equations: U₁ = R₁ – n₁(n₁+1)/2, U₂ = R₂ – n₂(n₂+1)/2, U = min(U₁, U₂). Check that U₁ + U₂ = n₁ × n₂ as a verification step.
关键公式：U₁ = R₁ – n₁(n₁+1)/2, U₂ = R₂ – n₂(n₂+1)/2, U = min(U₁, U₂)。可用 U₁ + U₂ = n₁ × n₂ 进行验证。
7. Spearman’s Rank-Order Correlation | 斯皮尔曼等级相关
Spearman’s rho (rₛ) measures the strength and direction of association between two ordinal variables or between one ordinal and one interval variable. It is used in a correlational design.
斯皮尔曼等级相关系数 (rₛ) 用于衡量两个定序变量或一个定序与一个定距变量之间关联的强度和方向，适用于相关设计。
Formula: rₛ = 1 – [ 6ΣD² ] / [ n(n² – 1) ], where D is the difference between the ranks of each pair of scores and n is the number of paired observations. Values of rₛ range from -1 (perfect negative) to +1 (perfect positive).
公式：rₛ = 1 – [ 6ΣD² ] / [ n(n² – 1) ]，其中 D 是每对得分的秩次之差，n 是配对观测的数量。rₛ 的取值从 -1（完全负相关）到 +1（完全正相关）。
Steps: Rank the scores for each variable separately, giving average ranks for ties. Compute D, the difference between the two ranks for each participant. Square each D, sum all D² values, and plug into the formula. Compare the obtained rₛ with the critical value from the Spearman table. If calculated rₛ is greater than or equal to the critical value, the correlation is significant.
步骤：分别对每个变量排秩，相同值赋予平均秩次。计算每个参与者的秩次差 D，平方后求和，代入公式。将所得 rₛ 与斯皮尔曼临界值表比较。若计算出的 rₛ 大于或等于临界值，则相关性显著。
8. Chi-Squared Test of Association | 卡方关联性检验
The chi-squared (χ²) test for independence is used when data are nominal (categories) and the design is independent groups or simply cross-tabulated frequency data. It tests whether there is a significant association between two categorical variables.
卡方 (χ²) 独立性检验用于定类数据（类别）且为独立组设计或简单的列联表频数数据。它检验两个分类变量之间是否存在显著关联。
Formula: χ² = Σ [ (O – E)² / E ], where O is the observed frequency in each cell and E is the expected frequency under the null hypothesis of no association. E is calculated as: E = (Row Total × Column Total) / Grand Total.
公式：χ² = Σ [ (O – E)² / E ]，其中 O 为每个单元格的观察频数，E 为在无关联零假设下的期望频数。E 计算公式：E = (行合计 × 列合计) / 总计。
Degrees of freedom (df) = (number of rows – 1) × (number of columns – 1). Compare the calculated χ² with the critical value from the chi-square table at the chosen significance level. A significant result means the variables are associated.
自由度 (df) = (行数 – 1) × (列数 – 1)。将计算出的 χ² 与所选显著性水平下的卡方临界值比较。结果显著表示变量之间存在关联。
Important: The test is reliable only when no more than 20% of expected frequencies are below 5 and none are below 1. If assumptions are violated, use Fisher’s exact test or combine categories.
重要提醒：只有不超过 20% 的期望频数低于 5 且没有低于 1 的期望频数时，该检验才可靠。若违反假设，需使用费雪精确检验或合并类别。
9. Choosing a Statistical Test: Decision Rules | 选择统计检验：决策规则
First, identify the design: repeated measures (or matched pairs) vs independent groups vs correlation. Then identify the level of measurement: nominal, ordinal, or interval/ratio.
首先，确认研究设计：重复测量（或配对） vs 独立组 vs 相关。然后确定测量水平：定类、定序还是定距/定比。
Decision table for non-parametric tests required in AQA:
AQA 要求的非参数检验决策表：
| Design | Level of Measurement | Test |
| Repeated measures / matched pairs | Nominal (direction) | Sign test |
| Repeated measures / matched pairs | Ordinal (or interval) | Wilcoxon signed-ranks test |
| Independent groups | Ordinal (or interval) | Mann-Whitney U test |
| Correlational | Ordinal (or interval) | Spearman’s rho |
| Independent groups, frequency data | Nominal | Chi-squared test |
If the data are normally distributed and measured on an interval/ratio scale, parametric tests (related t-test, independent t-test, Pearson’s r) could be used, but AQA focuses on non-parametric alternatives in Year 13.
如果数据呈正态分布且为定距/定比尺度，则可使用参数检验（相关 t 检验、独立 t 检验、皮尔逊 r），但 AQA Year 13 侧重于非参数替代检验。
10. Probability, Significance Levels & p-Values | 概率、显著性水平与 p 值
The null hypothesis (H₀) assumes there is no effect or no difference. The alternative hypothesis (H₁ or Ha) states there is an effect or difference. Significance is determined by comparing the calculated test statistic to a critical value from a table.
零假设 (H₀) 假定没有效应或没有差异。备择假设 (H₁ 或 Ha) 则陈述存在效应或差异。通过将计算出的检验统计量与表中的临界值进行比较，来确定显著性。
The conventional significance level in psychology is p < 0.05 (5%). If the probability of obtaining the result by chance is lower than 0.05, we reject H₀ and accept the alternative hypothesis. A result is significant when the calculated statistic is equal to or more extreme than the critical value.
心理学中惯用的显著性水平为 p < 0.05 (5%)。若偶然获得该结果的概率低于 0.05，我们就拒绝 H₀ 并接受备择假设。当计算出的统计量等于或比临界值更极端时，结果显著。
For Spearman’s rho, observed rₛ must be ≥ critical value. For Wilcoxon T, Mann-Whitney U and Sign S, the calculated statistic must be ≤ critical value. For chi-squared, observed χ² must be ≥ critical value.
对于斯皮尔曼 rₛ，观察值必须 ≥ 临界值。对于威尔科克森 T、曼-惠特尼 U 和符号 S，计算值必须 ≤ 临界值。对于卡方，观察 χ² 必须 ≥ 临界值。
11. Type I & Type II Errors | 第一类与第二类错误
A Type I error occurs when the null hypothesis is true but we incorrectly reject it. The probability of making a Type I error equals the significance level (alpha, usually 0.05).
第一类错误发生在零假设为真却被错误拒绝的情况下。犯第一类错误的概率等于显著性水平（α，通常为 0.05）。
A Type II error occurs when the null hypothesis is false but we fail to reject it. The probability of a Type II error is denoted by beta (β). Power (1 – β) is the probability of correctly rejecting a false null hypothesis.
第二类错误发生在零假设为假却没有被拒绝的情况下。犯第二类错误的概率用 β 表示。统计效力 (1 – β) 是正确拒绝错误零假设的概率。
Researchers can reduce the risk of Type I errors by setting a more stringent alpha level (e.g., p < 0.01), but this increases the risk of Type II errors. Increasing sample size improves power and reduces Type II error risk.
研究者可通过设定更严格的 α 水平（如 p < 0.01）来降低第一类错误的风险，但这会增加第二类错误的风险。增加样本量可提高统计效力并降低第二类错误风险。
12. Reporting Test Results in APA Style | 按 APA 格式报告结果
In AQA exams, you may be asked to write a short statement of results. A general template: ‘The [name of test] was used. The observed value of [statistic] was [value]. The critical value was [value] for a [one/two]-tailed test at p < 0.05, N = [number]. Since the observed value was [greater/less] than the critical value, the result is [significant/not significant]. Therefore, the null hypothesis can be [rejected/retained].'
在 AQA 考试中，你可能需要撰写简短的结果陈述。通用模板为：“采用了 [检验名称] 检验。观察到的 [统计量] 值为 [数值]。在 p < 0.05 下，[单/双] 尾检验的临界值为 [数值]，N = [数量]。由于观察值 [大于/小于
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