📚 AS AQA Psychology: Formula & Theorem Quick Reference Handbook | AS AQA 心理学：公式定理速查手册
This quick reference handbook brings together all the statistical formulae and key principles you need to master for the AS AQA Psychology research methods component. From descriptive statistics to inferential tests, these tools allow you to analyse data, draw conclusions, and evaluate the significance of findings. Keep this guide close when tackling practice questions or revising for your exam.
本速查手册汇集了AS AQA心理学研究方法部分需要掌握的所有统计公式和关键原理。从描述统计到推断检验，这些工具能帮助你分析数据、得出结论并评估研究发现的显著性。在练习习题或准备考试时，请随时查阅本指南。
1. Measures of Central Tendency | 集中趋势的度量
The three most common measures of central tendency are the mean (x̄), median and mode. The mean is the arithmetic average: add all scores and divide by the number of scores (n). The median is the middle value when data are arranged in order, and the mode is the most frequently occurring value. These summaries help identify a typical score in a dataset.
最常见的集中趋势度量有三种：平均数(x̄)、中位数和众数。平均数是算术平均值：将所有分数相加再除以分数个数(n)。中位数是将数据按顺序排列后位于中间的值，众数则是出现频率最高的值。这些概括性指标能帮助你识别数据集中的典型分数。
x̄ = Σx / n
平均数 = 总和 / 个数
2. Measures of Dispersion | 离散程度的度量
Dispersion tells us how spread out the scores are. The range is the simplest measure: highest score minus lowest score. A more informative measure is the standard deviation (s), which represents the average distance of each score from the mean. A larger standard deviation indicates greater variability. In inferential statistics, the sample standard deviation uses n-1 in the denominator to provide an unbiased estimate.
离散程度反映的是分数的分散情况。范围是最简单的度量：最高分减去最低分。更具信息量的度量是标准差(s)，它代表每个分数与平均数的平均距离。标准差越大，表示变异性越大。在推断统计中，样本标准差的分母使用 n-1，以提供无偏估计。
s = √[ Σ (x – x̄)² / (n – 1) ]
标准差 = √[ 离差平方和 / (样本量 – 1) ]
3. Calculating Standard Deviation Step by Step | 标准差计算步骤
To compute the standard deviation: (1) Find the mean of all scores. (2) Subtract the mean from each score to get deviation scores. (3) Square each deviation. (4) Sum all squared deviations. (5) Divide by n-1 to obtain the variance. (6) Take the square root of the variance to get s. This method is essential when analysing data for research projects and interpreting published findings.
计算标准差的步骤：(1) 求出所有分数的平均数。(2) 用每个分数减去平均数，得到离差。(3) 将每个离差平方。(4) 求所有平方离差的总和。(5) 除以 n-1 得到方差。(6) 取方差的平方根即得 s。在分析研究项目数据和解读已发表结果时，这种方法至关重要。
4. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rho (ρ) measures the strength and direction of association between two ranked variables. It ranges from -1 to +1. A value of +1 indicates a perfect positive rank correlation, -1 a perfect negative correlation, and 0 no correlation. The formula uses the sum of squared rank differences (d²) and the number of pairs (n). It is a non-parametric test often used with ordinal data.
斯皮尔曼等级相关系数(ρ)衡量两个排序变量之间关联的强度和方向。其取值范围在-1到+1之间。+1 表示完全正等级相关，-1 表示完全负相关，0 表示无相关。公式利用等级差的平方和(d²)与数据对的数量(n)。这是一个常用于顺序数据的非参数检验。
ρ = 1 – (6 Σ d²) / [ n (n² – 1) ]
斯皮尔曼系数 = 1 – (6 × 等级差平方和) / [ n (n² – 1) ]
5. Chi-Squared Test of Association | 卡方独立性检验
The chi-squared (χ²) test assesses whether there is a significant association between two categorical variables. It compares observed frequencies (O) with expected frequencies (E) under the null hypothesis of no association. The expected frequency for each cell is calculated as (row total × column total) ÷ grand total. The larger the χ² value, the greater the discrepancy between observed and expected data.
卡方(χ²)检验用于评估两个类别变量之间是否存在显著关联。它比较观察频数(O)与在无关联的零假设下的期望频数(E)。每个单元格的期望频数按 (行总和 × 列总和) ÷ 总样本量 计算。χ² 值越大，观察数据与期望数据之间的差异越大。
χ² = Σ [ (O – E)² / E ]
卡方 = Σ [ (观察值 – 期望值)² / 期望值 ]
Degrees of freedom for the test: df = (number of rows – 1) × (number of columns – 1).
检验的自由度：df = (行数 – 1) × (列数 – 1)。
6. The Sign Test | 符号检验
The sign test is a simple non-parametric test used to compare two conditions in a repeated measures design. For each participant, you record whether the score in condition B is higher (+) or lower (–) than condition A, ignoring any ties (no difference). The test statistic S is the smaller of the total number of plus signs and total number of minus signs. The null hypothesis states there is no systematic difference.
符号检验是一种简单的非参数检验，用于比较重复测量设计中的两种条件。对每个被试，记录条件B的分数是高于(+)还是低于(–)条件A，忽略任何平局（无差异）。检验统计量 S 是正号总数和负号总数中较小的那个。零假设声称不存在系统性差异。
S = min (number of + signs, number of – signs)
S = 较小符号频数 (正号数, 负号数)
You then compare S against a critical value with N = total number of participants showing a difference. If S is less than or equal to the critical value, you reject the null hypothesis.
然后将 S 与临界值比较，此时 N = 表现出差异的被试总数。如果 S 小于或等于临界值，则拒绝零假设。
7. Probability and Statistical Significance | 概率与统计显著性
Statistical significance is expressed by the p-value, which is the probability of obtaining an observed result (or one more extreme) if the null hypothesis is true. In psychology, the conventional significance level is 0.05 (5%). If p ≤ 0.05, the result is statistically significant, and the null hypothesis is rejected. Remember that a non-significant result (p > 0.05) does not prove the null hypothesis; it simply indicates insufficient evidence to reject it.
统计显著性用 p 值表示，即如果零假设为真，获得当前观察结果（或更极端结果）的概率。在心理学中，常规的显著性水平是 0.05 (5%)。如果 p ≤ 0.05，则结果具有统计学显著性，拒绝零假设。请注意，不显著的结果 (p > 0.05) 并不能证明零假设正确，它仅仅表明没有足够证据拒绝它。
Type I error (false positive): rejecting a true null. Type II error (false negative): failing to reject a false null. Careful choice of significance level and adequate sample size help minimise these errors.
I 型错误（假阳性）：拒绝了真的零假设。II 型错误（假阴性）：未能拒绝假的零假设。谨慎选择显著性水平和充足的样本量有助于减少这些错误。
8. Critical Values and Degrees of Freedom | 临界值与自由度
Critical values are the boundaries that determine whether a test statistic is extreme enough to reject the null hypothesis. They are found in statistical tables specific to each test. Degrees of freedom (df) reflect the number of values that are free to vary in a calculation. For a chi-squared test of association, df = (r-1)(c-1); for a Spearman’s rho, N is used directly; for the sign test, N is the number of non-zero differences.
临界值是决定检验统计量是否足够极端以拒绝零假设的边界值，可在每种检验专用的统计表中查找。自由度(df)反映了计算中可自由变化的数值个数。对于卡方独立性检验，df = (r-1)(c-1)；对于斯皮尔曼等级相关，直接使用 N；对于符号检验，N 为非零差异的个数。
Always check whether your test is one-tailed (directional hypothesis) or two-tailed (non-directional hypothesis) before looking up the critical value. The same test statistic can have different critical values depending on the chosen tail and significance level.
在查临界值前，务必要明确你的检验是单尾（方向性假设）还是双尾（非方向性假设）。依据所选的尾数和显著性水平，同一检验统计量可能有不同的临界值。
9. Using Tables of Critical Values: A Step-by-Step Guide | 临界值表使用分步指南
Step 1: Identify the correct statistical test (e.g. chi-squared, sign test). Step 2: Determine whether the hypothesis is one-tailed or two-tailed. Step 3: Select the significance level, usually p ≤ 0.05. Step 4: Calculate the degrees of freedom or find N as required. Step 5: Locate the critical value in the table. Step 6: Compare your calculated test statistic with the critical value. If the calculated value equals or exceeds the critical value (for most tests, except sign test where it should be equal or smaller), the result is significant.
第一步：确定正确的统计检验（如卡方、符号检验）。第二步：判断假设是单尾还是双尾。第三步：选择显著性水平，通常为 p ≤ 0.05。第四步：计算自由度或按要求确定 N。第五步：在表中找到临界值。第六步：将你计算的检验统计量与临界值比较。如果计算值等于或超过临界值（大多数检验如此，符号检验则要求计算值等于或小于临界值），结果即显著。
For the sign test, reject H₀ if your obtained S is less than or equal to the critical S for your N. For chi-squared and Spearman’s rho, reject H₀ if your obtained value is equal to or larger than the critical value.
对于符号检验，若计算出的 S 小于或等于对应 N 的临界 S 值，则拒绝 H₀。对于卡方检验和斯皮尔曼等级相关，若计算值等于或大于临界值，则拒绝 H₀。
10. Common Research Formulae at a Glance | 常用研究公式一览
The table below summarises the key formulae, test statistics and typical uses in AS Psychology research methods. Use it as a quick memory aid when planning data analysis or checking your exam answers.
下表汇总了AS心理学研究方法中的关键公式、检验统计量及典型用途。将它作为规划数据分析或检查考试答案时的快速记忆辅助工具。
| Measure / Test | Formula | When to Use |
|---|---|---|
| Mean | x̄ = Σx / n | Summarising interval/ratio data |
| Standard Deviation | s = √[ Σ(x – x̄)² / (n – 1) ] | Measuring spread around the mean |
| Spearman’s Rho | ρ = 1 – (6 Σ d²) / [n(n² – 1)] | Correlation between two ranked variables |
| Chi-Squared (χ²) | χ² = Σ (O – E)² / E | Association between two categorical variables |
| Sign Test | S = smaller count of + / – | Difference in related conditions (repeated measures) |
Remember that knowing the formula is only the first step — you must also understand how to choose the correct test, state the null hypothesis, interpret critical values and write a conclusion in the context of the research.
请记住，仅仅记住公式只是第一步——你还必须理解如何选择正确的检验、陈述零假设、解读临界值，并结合研究背景撰写结论。
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