📚 Year 12 AQA Psychology: Formula & Theorem Quick Reference Handbook | AQA 心理学公式定理速查手册
This handbook brings together every essential formula, statistical principle and decision rule you need for the Year 12 AQA Psychology specification. It covers descriptive statistics, inferential tests, distribution shapes, and the logic of hypothesis testing, with clear step-by-step explanations for each test. Use it to revise research methods, check your calculations, and feel confident in choosing the right test under exam conditions.
本手册汇集了 Year 12 AQA 心理学课程所必需的所有核心公式、统计原理和决策规则。内容涵盖描述统计、推断检验、分布形态以及假设检验的逻辑，并对每种检验提供了清晰的步骤说明。你可以用它来复习研究方法、核对计算过程，并在考试中自信地选择正确的统计检验。
1. Measures of Central Tendency | 集中趋势测量
The mean is the arithmetic average and is the most sensitive measure of central tendency because it uses all scores in the data set. It is calculated as the sum of all scores divided by the number of scores.
均值是算术平均数，是集中趋势最敏感的度量，因为它使用了数据集中的所有分数。计算方法为所有分数之和除以分数的个数。
Mean (x̄) = Σx / n
where Σx = sum of individual data values, n = sample size.
其中 Σx = 各数据值之和，n = 样本容量。
The median is the middle value when data are arranged in rank order. It is less affected by extreme scores and is the appropriate measure for ordinal data or skewed distributions.
中位数是将数据按大小排序后处于中间位置的值。它较少受极端分数影响，适用于顺序数据或偏态分布。
The mode is the most frequently occurring value. It can be used with nominal data and is the only measure of central tendency that identifies the most common category.
众数是出现次数最多的值。它可用于称名数据，并且是唯一能够识别最常见类别的集中趋势测量。
2. Measures of Dispersion | 离散程度测量
The range is the simplest measure of dispersion, showing the spread between the highest and the lowest scores. It is quick to calculate but is heavily influenced by outliers.
全距是最简单的离散程度指标，表示最高分与最低分之间的跨度。计算快捷，但受异常值的严重影响。
Range = Highest value – Lowest value
The interquartile range (IQR) is the range of the middle 50% of scores, from the 25th percentile (Q1) to the 75th percentile (Q3). It provides a more stable picture because it ignores extreme high and low values.
四分位距是中间50% 分数的范围，从第25百分位数 (Q1) 到第75百分位数 (Q3)。由于它忽略了极高和极低的值，因此能提供更稳定的描述。
IQR = Q3 – Q1
3. Standard Deviation and Variance | 标准差与方差
Variance and standard deviation measure how spread out scores are around the mean. A small standard deviation indicates that most data points are close to the mean; a large standard deviation reflects greater variability.
方差和标准差衡量分数围绕均值的分散程度。标准差小，表明大多数数据点接近均值；标准差大，反映变异程度更高。
Sample variance: s² = Σ(x – x̄)² / (n – 1)
Sample standard deviation: s = √[ Σ(x – x̄)² / (n – 1) ]
The denominator (n – 1) is used for a sample to give an unbiased estimate of the population variance. In exams, you may be given the sum of squared deviations and expected to substitute values correctly.
式中分母使用 (n – 1) 是为了得到总体方差的无偏估计值。考试中，你可能会被提供离差平方和，要求正确代入公式进行计算。
4. Normal Distribution and Skewness | 正态分布与偏态
A normal distribution is bell-shaped and symmetrical, with the mean, median and mode all coinciding at the centre. In a perfectly normal distribution, exactly 50% of scores lie below the mean and 50% above.
正态分布为钟形且对称，均值、中位数和众数在中心重合。在完全正态分布中，恰好50% 的分数低于均值，50% 高于均值。
ined solely by the mean and standard deviation. In a normal distribution, approximately 68% of data fall within ±1 SD, and 95% within ±2 SD.
正态曲线的两端永远不接触横轴，形状完全由均值和标准差决定。在正态分布中，大约68% 的数据落在均值±1 个标准差内，95% 落在±2 个标准差内。
In a positively skewed distribution, most scores cluster at the low end and the tail extends to the right (mean > median > mode). In a negatively skewed distribution, most scores are at the high end and the tail extends to the left (mean < median < mode).
在正偏态分布中，大部分分数聚集在低分端，尾部向右延伸（均值 > 中位数 > 众数）。在负偏态分布中，大部分分数聚集在高分端，尾部向左延伸（均值 < 中位数 < 众数）。
Recognising skewness helps decide whether to use the median and IQR rather than the mean and standard deviation as more appropriate descriptive statistics.
识别偏态有助于判断是否更适合用中位数和四分位距，而非均值和标准差作为描述性统计指标。
5. Correlation: Spearman's Rho | 相关：斯皮尔曼等级相关系数
Spearman's rho (rs) is a non‑parametric test of correlation for ordinal data or non‑normally distributed interval data. It measures the strength and direction of a monotonic relationship between two co‑variables.
斯皮尔曼等级相关系数 (rs) 是一种非参数相关检验，适用于顺序数据或非正态分布的等距数据。它测量两个协同变量之间单调关系的强度和方向。
rs = 1 – (6 Σd²) / (n (n² – 1))
the difference between the ranks of each pair, and n = the number of pairs. The value of rs ranges from –1 (perfect negative correlation) through 0 (no correlation) to +1 (perfect positive correlation).
其中 d = 每对数据的秩次差，n = 配对数。rs 的取值范围从 –1（完全负相关）到 0（无相关）再到 +1（完全正相关）。
A calculated rs l value at a given significance level (usually p ≤ 0.05), the null hypothesis is rejected.
计算出的 rs 必须与临界值表进行比较。如果在给定的显著性水平下（通常 p ≤ 0.05），观测值等于或超过临界值，则拒绝零假设。
6. The Sign Test | 符号检验
The sign test is a non‑parametric test of difference for repeated measures or matched pairs designs when the data are at least nominal (directional difference). It tests whether the number of plus and minus signs across pairs deviates significantly from chance.
符号检验是一种用于重复测量或配对设计的非参数差异检验，要求数据至少为称名水平（方向性差异）。它检验配对间正号和负号的数量是否显著偏离随机水平。
Step 1: Convert each pair into a '+' or '–' sign by subtracting one score from the other. Ignore any zero‑difference pairs.
S = the count of the less frequent sign.
The calculated S value is compared with the critical value from the sign test table. For the result to be significant, the observed S must be equal to or less than the critical value at the chosen significance level.
计算得到的 S 值与符号检验表的临界值进行比较。要使结果显著，观测 S 值必须等于或小于选定显著性水平下的临界值。
7. Chi-Squared Test (χ²) | 卡方检验
The chi‑squared test is used to analyse nominal data in an independent groups design. It can be applied as a test of association (contingency table) or as a goodness‑of‑fit test. It compares the observed frequencies with the frequencies expected under the null hypothesis.
卡方检验用于分析独立组设计中的称名数据。它可作为关联性检验（列联表）或拟合优度检验使用。原理是比较观测频次与零假设下的期望频次。
χ² = Σ (O – E)² / E
observed frequency, E = expected frequency. The sum is taken over all cells of the table.
O = 观测频次，E = 期望频次。求和覆盖表格中的所有单元格。
Degrees of freedom (test of association): df = (rows – 1) × (columns – 1)
For a goodness‑of‑ nus 1. The calculated χ² value is compared with a critical value at the chosen significance level using the appropriate df. A significant result means the observed distribution differs from chance.
拟合优度检验的自由度 df = 类别数减 1。计算出的 χ² 值与选定显著性水平下、相应自由度的临界值比较。结果显著意味着观测分布偏离随机水平。
8. Mann–Whitney U Test | 曼–惠特尼 U 检验
The Mann–Whitney U test is a non‑parametric test of difference suitable for an independent groups design when data are at least ordinal. It tests whether two independent samples are drawn from the same population by comparing the ranks of all scores.
曼–惠特尼 U 检验是一种非参数差异检验，适用于独立组设计且数据至少为顺序水平。它通过比较所有分数的秩次，检验两个独立样本是否来自同一总体。
U₁ = n₁n₂ + n₁(n₁ + 1) / 2 – R₁
U₂ = n₁n₂ – U₁
n₁ and n₂ are the two sample sizes; R₁ is the sum of the ranks in group 1. The smaller of U₁ and U₂ becomes the observed value, which is then compared to the critical U value. If the observed U is equal to or less than the critical value, the result is significant.
n₁ 和 n₂ 是两个样本的容量；R₁ 是组 1 的秩次和。取 U₁ 和 U₂ 中较小的值作为观测值，再与临界 U 值比较。如果观测 U 值等于或小于临界值，结果显著。
9. Wilcoxon Signed-Ranks Test | 威尔科克森符号秩检验
The Wilcoxon test is a non‑parametric test of difference for a repeated measures or matched pairs design, using ordinal data. It considers both the direction and the magnitude of the differences between paired scores.
威尔科克森检验是一种非参数差异检验，适用于重复测量或配对设计，使用顺序数据。它同时考虑配对分数差异的方向和大小。
Step 1: Calculate the difference for each pair and ignore any zero differences. Rank the absolute differences from smallest to largest.
T = sum of the ranks belonging to the less frequent sign.
The observed T is compared with the critical value from the Wilcoxon table. For significance at the usual 5% level, the observed T must be equal to or less than the critical value. Note that ties in ranks require assigning the mean rank.
观测值 T 与威尔科克森检验表中的临界值比较。要在常用的 5% 水平上显著，观测 T 值必须等于或小于临界值。注意，秩次并列时需赋予平均秩次。
10. Choosing a Statistical Test | 选择统计检验
AQA requires you to justify your choice of statistical test using three key considerations: the purpose of the test (difference or correlation), the experimental design (independent groups, repeated measures or matched pairs), and the level of measurement (nominal, ordinal or interval).
AQA 要求你根据三个关键因素为统计检验的选择提供理由：检验的目的（差异或相关）、实验设计（独立组、重复测量或配对设计）以及测量水平（称名、顺序或等距）。
| Purpose | Design | Level of Measurement | Test |
|---|---|---|---|
| Difference | Independent groups | Nominal | Chi-squared |
| Difference | Independent groups | Ordinal (or interval, non‑normal) | Mann‑Whitney U |
| Difference | Repeated measures / matched pairs | Nominal (direction only) | Sign test |
| Difference | Repeated measures / matched pairs | Ordinal (or interval, non‑normal) | Wilcoxon |
| Correlation | — | Ordinal (or non‑normal interval) | Spearman's rho |
Always state the level of measurement explicitly in your answer. If data are interval and normally distributed, parametric tests might be possible, but for Year 12 AQA the focus is on non‑parametric alternatives because real psychological data often violate parametric assumptions.
在作答中务必明确说明测量水平。若数据为等距且正态分布，可使用参数检验，但 Year 12 AQA 的重点在于非参数替代方案，因为实际心理学数据常违背参数假设。
11. Levels of Measurement | 测量水平
The level of measurement determines which descriptive statistics and which inferential test can be used. Nominal data consist of categories with no inherent order; ordinal data are rank‑ordered but the intervals between ranks are not necessarily equal; interval data have equal intervals and a mean that is meaningful.
测量水平决定了可以使用哪些描述性统计量和推断检验。称名数据由无内在顺序的类别组成；顺序数据有排序，但秩次之间的间距不一定相等；等距数据具有相等的间隔，均值具有意义。
- Nominal: Frequency counts (e.g. number of participants choosing a particular category). Only the mode is appropriate.
- 顺序: 可计数频次（例如，选择某一类别的参与者人数）。通常只适合使用众数。
- Ordinal: Ranked data (e.g. rating scales, class positions). The median, range and IQR are used.
- 顺序: 等级数据（例如，评分量表、班级排名）。使用中位数、全距和四分位距。
- Interval: Data measured on a scale with equal units and no true zero (e.g. temperature in Celsius, standardised test scores). The mean and standard deviation are justified.
- 等距: 在具有相等单位且无绝对零点的量表上测量的数据（如摄氏温度、标准化考试分数）。适合使用均值和标准差。
In practice, many psychological measures are treated as ordinal even when the scale appears to have equal intervals, because one cannot assume that the gaps between scale points are psychologically equal. This leads to the widespread application of non‑parametric tests.
实际上，即使量表看起来有等距间隔，许多心理学测量仍被当作顺序数据处理，因为不能假定量表点之间的心理距离相等。这导致非参数检验被广泛应用。
12. Probability, Significance and Errors | 概率、显著性与错误
In null hypothesis significance testing, we assume there is no effect or no difference in the population (the null hypothesis). The calculated test statistic is compared against a critical value; if the observed value falls in the critical region, we reject the null hypothesis and accept the alternative hypothesis.
在零假设显著性检验中，我们假设总体中无效应或无差异（零假设）。将计算出的检验统计量与临界值比较；如果观测值落在临界区域内，则拒绝零假设并接受备择假设。
The conventional significance level in psychology is p ≤ 0.05 (5%). This means there is a 5% hesis were true. A stricter level (e.g. p ≤ 0.01) may be used to reduce the risk of error.
心理学中传统的显著性水平为 p ≤ 0.05（5%）。这意味着，如果零假设为真，有 5% 的概率仅凭机遇得到当前的观测结果。可以使用更严格的标准（如 p ≤ 0.01）以降低错误风险。
Type I error: Rejecting the null hypothesis when it is actually true (a false positive). The probability of a Type I error equals the significance level.
第一类错误：零假设为真时错误地拒绝了它（假阳性）。第一类错误的概率等于显著性水平。
Type II error: Failing to reject the null hypothesis when the alternative hypothesis is true (a false negative). This is more likely when the sample size is small or the effect size is weak.
第二类错误：备择假设为真时未能拒绝零假设（假阴性）。当样本量较小或效应量较弱时，更容易发生第二类错误。
e between the two types of error. Researchers also look at the calculated effect size to interpret the practical significance of their findings, though effect size formulas are mainly covered in Year 13.
使用标准的 p ≤ 0.05 是在两类错误之间的权衡。研究者也会参考计算出的效应量来解释研究结果的实际意义，不过效应量公式主要在 Year 13 中涉及。
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