📚 A-Level AQA Psychology: Formulae & Theorems Quick Reference Guide | A-Level AQA 心理学：公式定理速查手册
Mastering the statistical formulae and decision rules is essential for AQA A-level Psychology students aiming for top marks in research methods. This guide distils the key calculations you need, from measures of dispersion to non-parametric tests, with step-by-step presentation.
掌握统计公式和决策规则对于追求高分的AQA A-level心理学学生至关重要。本指南浓缩了你需要的关键计算，从离散量数到非参数检验，并逐步展示。
1. Measures of Central Tendency | 集中趋势度量
The arithmetic mean is the sum of all scores divided by the number of scores. It uses every data point and is the most sensitive measure of central tendency, but it can be pulled by extreme outliers.
算术平均数是所有分数之和除以分数个数。它使用了每一个数据点，是最敏感的集中趋势度量，但容易被极端异常值拉偏。
Mean (x̄) = Σx / n
The median is the middle score when data are arranged in order. It is robust to skewed distributions and is the preferred measure for ordinal data or when outliers are present.
中位数是将数据排序后处于中间位置的数值。它对偏态分布稳健，是顺序数据或存在异常值时的首选度量。
The mode is the most frequently occurring value in a dataset. It is the only measure of central tendency suitable for nominal (category) data, and a dataset may have more than one mode.
众数是数据集中出现次数最多的数值。它是唯一适合称名(类别)数据的集中趋势度量，一个数据集可以有多个众数。
2. Measures of Dispersion: Range, Variance and Standard Deviation | 离散量数：极差、方差和标准差
The range is the simplest measure of spread, calculated as the difference between the highest and lowest scores. It gives a quick sense of data spread but completely ignores the distribution of scores between the extremes.
极差是最简单的离散量数，计算为最高分与最低分之差。它能快速反映数据的散布范围，但完全忽略了极端值之间的分布情况。
Range = Xmax – Xmin
Variance quantifies how far each score deviates from the mean, on average squared. For a sample, we use n-1 in the denominator (Bessel’s correction) to give an unbiased estimate of the population variance.
方差量化了每个分数偏离均值的平均平方距离。对于样本，我们在分母中使用 n-1（贝塞尔校正），以给出总体方差的无偏估计。
s² = Σ(x – x̄)² / (n – 1)
The standard deviation is the square root of the variance, returning the spread to the original units of measurement. It is widely used because it fits naturally with the normal distribution.
标准差是方差的平方根，将离散程度还原到原始测量单位。它被广泛使用，因为它与正态分布自然契合。
s = √[Σ(x – x̄)² / (n – 1)]
A larger standard deviation indicates greater variability among scores; a smaller one indicates that scores cluster tightly around the mean.
较大的标准差表明分数之间的变异较大；较小的标准差表明分数紧密聚集在均值周围。
3. The Sign Test | 符号检验
The Sign Test is a non-parametric test used for a repeated measures or matched pairs design when the data are at least nominal (by recording the direction of difference). It tests whether there is a significant difference between two conditions.
符号检验是一种非参数检验，用于重复测量或配对设计，且数据至少为称名水平（记录差异方向）。它检验两种条件之间是否存在显著差异。
You first calculate the sign of each difference (condition B – condition A) as ‘+’ or ‘-‘. Pairs with zero difference are discarded. The test statistic S is the count of the less frequent sign.
首先计算每对差异的符号（条件B – 条件A），记录为 ‘+’ 或 ‘-‘。差值为零的对被剔除。检验统计量 S 是出现次数较少的符号的个数。
S = min(n⁺, n⁻)
The observed S is compared against a critical value from the binomial distribution table for the number of non-zero difference pairs, N. If S is equal to or smaller than the critical value at a chosen significance level (usually p < 0.05), the result is significant and you reject the null hypothesis.
计算出的 S 与二项分布表中对应于非零差值对数目 N 的临界值进行比较。如果在选定的显著性水平（通常 p < 0.05）下 S 小于或等于临界值，则结果显著，拒绝零假设。
The Sign Test is simple but low in power because it only uses the direction of differences, not their magnitude.
符号检验简单但统计效力较低，因为它只利用了差异的方向，而忽略了差异的大小。
4. Chi-Squared Test (χ²) | 卡方检验
The Chi-Squared test is used for independent groups designs with nominal data (frequency counts). It determines whether the observed frequencies differ significantly from the frequencies expected by chance.
卡方检验用于独立组设计且数据为称名水平（频次计数）。它确定观察到的频次是否与随机期望频次存在显著差异。
Expected frequencies are calculated assuming no association: E = (row total × column total) / grand total. The formula for the test statistic sums the weighted squared differences between observed (O) and expected (E) frequencies.
期望频次是在假定没有关联的前提下计算出来的：E = (行总计 × 列总计) / 总计。检验统计量的公式将观察频次(O)与期望频次(E)之差的加权平方求和。
χ² = Σ[(O – E)² / E]
The degrees of freedom (df) for the test of independence are calculated as df = (number of rows – 1) × (number of columns – 1). The obtained χ² is compared with a critical value from χ² distribution tables at the chosen df and significance level.
独立性检验的自由度(df)计算为 df = (行数 – 1) × (列数 – 1)。将计算出的 χ² 与卡方分布表中选定自由度和显著性水平下的临界值进行比较。
If the computed χ² is greater than or equal to the critical value, the association is statistically significant. The test requires that no more than 20% of expected frequencies are below 5, and none are below 1.
如果计算出的 χ² 大于或等于临界值，则关联具有统计显著性。该检验要求不超过20%的期望频次低于5，且没有期望频次低于1。
5. Mann-Whitney U Test | 曼-惠特尼U检验
The Mann-Whitney U test is a non-parametric alternative to the independent t-test. It is used with an independent groups design and at least ordinal data to test whether two samples come from the same population.
曼-惠特尼U检验是独立t检验的非参数替代方法。它用于独立组设计且至少为顺序数据，检验两个样本是否来自同一总体。
All scores from both groups are ranked together. The ranks for group 1 (R₁) are summed, and a U statistic is computed for each group. The smaller of the two U values is used as the test statistic.
两组的全部分数被混合排序。将组1的秩次求和得到 R₁，然后分别为每组计算U统计量，取两个U值中较小的一个作为检验统计量。
U₁ = n₁n₂ + n₁(n₁ + 1)/2 – R₁
U₂ = n₁n₂ – U₁
U = min(U₁, U₂)
The observed U is compared with the critical value from Mann-Whitney tables using the two sample sizes (n₁ and n₂). If U is equal to or less than the critical value at p < 0.05, the difference is statistically significant.
将观察到的U值与曼-惠特尼表中根据两个样本大小 (n₁和n₂) 查出的临界值比较。如果在 p < 0.05 水平下 U 小于或等于临界值，则差异具有统计显著性。
6. Wilcoxon Signed-Ranks Test | 威尔科克森符号秩检验
The Wilcoxon test is the non-parametric equivalent of the paired t-test, designed for a repeated measures or matched pairs design with at least ordinal data. It uses both the direction and the magnitude of differences.
威尔科克森检验是配对t检验的非参数等价方法，用于重复测量或配对设计且至少为顺序数据。它同时利用差异的方向和大小。
For each pair, the difference (B – A) is calculated, and the absolute differences are ranked, ignoring the sign. The ranks are then assigned the original sign of the difference. The test statistic T is the smaller of the sum of positive ranks and the sum of negative ranks.
计算每对数据的差值(B – A)，忽略符号后对差值的绝对值进行排序。然后将秩次赋予差值的原始符号。检验统计量 T 为正秩和与负秩和中较小的一个。
T = min(ΣR⁺, ΣR⁻)
Zero differences are discarded, reducing the effective N. The observed T is compared with the critical value from the Wilcoxon table for the number of non-zero pairs. If T is less than or equal to the critical value, the result is significant.
差值为零的对被剔除，从而减少了有效 N。将观察到的 T 值与威尔科克森表中对应非零对数的临界值进行比较。若 T 小于或等于临界值，则结果显著。
7. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rho (rs) measures the strength and direction of a monotonic relationship between two variables when data are at least ordinal. It is often used when parametric assumptions are violated.
斯皮尔曼等级相关系数 (rs) 测量两个变量之间单调关系的强度和方向，要求数据至少为顺序水平。通常用于参数假设不满足时。
Both variables are ranked separately, and the difference D between each pair of ranks is calculated. The formula squares those differences and sums them before incorporating them into the correlation coefficient.
对两个变量分别排序，计算每对秩次的差值 D。公式将这些差值平方并求和，然后代入相关系数公式。
rs = 1 – (6 ΣD²) / (n(n² – 1))
The obtained rs ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation). A value near zero indicates no monotonic relationship. To determine significance, compare the absolute value of rs with the critical value for Spearman’s test at n pairs.
得到的 rs 取值范围为 -1（完全负相关）到 +1（完全正相关），接近零则表明没有单调关系。判断显著性时，将 rs 的绝对值与斯皮尔曼检验在 n 对时的临界值进行比较。
If the absolute rs is greater than or equal to the critical value at p < 0.05, the correlation is statistically significant.
如果 rs 的绝对值大于或等于 p < 0.05 水平下的临界值，则相关性具有统计显著性。
8. Degrees of Freedom and Critical Values | 自由度与临界值
Degrees of freedom (df) represent the number of independent pieces of information available to estimate a statistic. They vary by test and are crucial for reading statistical tables correctly.
自由度(df)代表用于估计统计量时可用的独立信息数量。不同检验的自由度不同，它们对于正确查阅统计表至关重要。
For the Chi-Squared test, df = (rows – 1)(columns – 1). For the Sign Test, the table entry uses the number of non-zero pairs N; there is no separate df parameter. Similarly, the Wilcoxon and Mann-Whitney tests use N or n₁,n₂ directly to locate critical values.
对于卡方检验，df = (行数 – 1)(列数 – 1)。对于符号检验，表查表使用非零对数目 N，没有单独的自由度参数。同样，威尔科克森检验和曼-惠特尼检验直接使用 N 或 n₁、n₂ 来查找临界值。
In Spearman’s test, the critical value depends on the sample size n, which acts like a df-like parameter; for large samples a t-approximation with df = n-2 can be used, but at A-level the critical value table is provided directly.
在斯皮尔曼检验中，临界值取决于样本容量 n，类似于一个自由度参数；大样本时可采用 df = n-2 的 t 近似，但在 A-level 中会直接提供临界值表。
9. Probability, Significance Levels and Decision Errors | 概率、显著性水平与决策错误
In psychology, a result is typically considered statistically significant when the probability of obtaining it by chance is less than 5% (p < 0.05). This threshold is called the alpha level (α).
在心理学中，当结果由偶然因素造成的概率小于5%（p < 0.05）时，通常认为具有统计显著性。这个阈值称为 alpha 水平 (α)。
A Type I error (false positive) occurs when the null hypothesis is incorrectly rejected; the probability of making a Type I error equals the alpha level. A Type II error (false negative) occurs when a false null hypothesis is not rejected; its probability is denoted β.
I类错误（假阳性）发生在零假设被错误地拒绝时；犯I类错误的概率等于 alpha 水平。II类错误（假阴性）发生在错误的零假设没有被拒绝时；其概率记为 β。
Power is the probability of correctly detecting a genuine effect, equal to 1 – β. Larger sample sizes increase power and reduce the risk of both types of error.
统计检验力是正确检测出真实效应的概率，等于 1 – β。较大的样本量可以增加检验力并减少两类错误的风险。
10. Normal Distribution and Z-Scores | 正态分布与Z分数
Many psychological variables approximate a normal distribution: a symmetrical, bell-shaped curve where the mean, median, and mode coincide. In a normal distribution, roughly 68% of scores lie within ±1 standard deviation of the mean, 95% within ±1.96 SD, and 99.7% within ±3 SD.
许多心理学变量近似正态分布：一条对称的钟形曲线，均值、中位数和众数重合。在正态分布中，约68%的分数落在均值±1个标准差内，95%落在±1.96个标准差内，99.7%落在±3个标准差内。
A z-score transforms an individual score into standard deviation units relative to the mean. This allows comparison across different distributions or tests.
Z分数将一个个别分数转换为相对于均值的标准差单位。这允许跨不同分布或测验进行比较。
z = (x – μ) / σ
While direct z-score calculations are not always required at A-level, understanding the normal curve underpins the logic of parametric tests and helps justify why many statistical models assume normality.
虽然A-level并不总是要求直接计算Z分数，但理解正态曲线是参数检验逻辑的基础，并有助于解释为什么许多统计模型假定正态性。
11. Decision Algorithms for Choosing a Statistical Test | 选择统计检验的决策算法
To select the correct inferential test, ask three sequential questions: (1) Am I testing for a difference or a correlation? (2) What is the experimental design – independent groups, repeated measures, or matched pairs? (3) What is the level of measurement – nominal, ordinal, or interval?
要选择正确的推断检验，依次问三个问题：(1) 我在检验差异还是相关？(2) 实验
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