📚 Psychology Formulas and Key Theorems Quick Reference | 心理学公式定理速查手册
This quick-reference guide gathers every essential formula, statistical test, and key theorem you need to master the research-methods component of the CIE IGCSE Psychology course. Use it alongside past papers and your class notes to sharpen your data-analysis skills.
本速查手册汇集了CIE IGCSE心理学课程中研究方法部分所有必备的公式、统计检验与关键定理。配合历年真题与课堂笔记使用，可有效提升你的数据分析能力。
1. Descriptive Statistics: Mean, Median & Mode | 描述性统计：平均数、中位数与众数
The mean is the arithmetic average, obtained by summing all scores and dividing by the number of scores. It is the most sensitive measure of central tendency but can be distorted by outliers.
平均数是算术平均值，将所有分数相加后除以分数个数得出。它是最灵敏的集中趋势量数，但易受极端值影响。
Mean (x̄) = ∑x / n
The median is the middle score when data are arranged in order. For an even number of scores, take the average of the two middle values. It is less affected by extreme scores and is therefore preferred for skewed distributions.
中位数是将数据排序后居中的数值。若分数个数为偶数，则取中间两个数的平均值。它受极端分数影响较小，偏态分布时更适用。
The mode is the most frequently occurring score in a data set. A set can have one mode (unimodal), two (bimodal), or more. It is the only measure suitable for nominal (category) data.
众数是数据集中出现次数最多的数值。一组数据可有一个众数（单峰）、两个（双峰）或更多。它是唯一适用于称名（类别）数据的集中量数。
2. Measures of Dispersion: Range & Standard Deviation | 离散量数：全距与标准差
The range is the simplest measure of dispersion: highest score minus lowest score. It gives a quick sense of spread but ignores the distribution of scores between the extremes.
全距是最简单的离散量数：最高分减去最低分。它能快速反映分布广度，但忽略了极端值之间分数的分布情况。
The standard deviation (SD) quantifies how much, on average, each score deviates from the mean. A smaller SD means scores are tightly clustered around the mean; a larger SD indicates greater spread. In IGCSE Psychology, the formula used assumes a population standard deviation.
标准差（SD）量化每个分数平均偏离平均数的程度。标准差越小，分数越集中在平均数周围；越大则离散程度越高。IGCSE心理学中使用的是总体标准差公式。
s = √(∑(x − x̄)² / n)
Where ∑(x − x̄ Always square the deviations first, sum them, divide by n, and then take the square root.
其中∑(x − x̄)²是离差平方和，n是分数个数，√为平方根。务必先求离差平方，再求和，除以n，最后开平方根。
3. Inferential Statistics: Null Hypothesis & p-value | 推论统计：零假设与p值
A null hypothesis (H₀) states that any observed difference or correlation in the sample is due to chance alone—there is no real effect in the population. The alternative hypothesis (H₁) claims that a genuine effect exists.
零假设（H₀）宣称样本中观察到的任何差异或相关纯属偶然，总体中不存在真实效应。备择假设（H₁）则主张存在真实效应。
a result is deemed statistically significant when p ≤ 0.05, meaning there is less than a 5% likelihood that the outcome arose by chance.
p值是在零假设成立的条件下，获得当前结果（或更极端结果）的概率。心理学中，当 p ≤ 0.05 时，结果被视为具有统计显著性，意味着因偶然因素得到该结果的可能性低于5%。
The critical value is the threshold that the test statistic must exceed (or fall below) to reject the null hypothesis. These values are found in statistical tables and depend on the chosen significance level (usually 0.05), the degrees of freedom, and whether the test is one-tailed or two-tailed.
临界值是检验统计量必须超越（或低于）以便拒绝零假设的阈值。临界值查阅统计表获得，取决于选用的显著性水平（通常为0.05）、自由度以及检验是单尾还是双尾。
4. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验
The Wilcoxon test is used for a repeated-measures or matched-pairs design when the data are at least ordinal. It assesses whether there is a significant difference between two conditions by ranking the size of the difference scores, ignoring the sign, and then summing the ranks of the positive and negative differences separately.
威尔科克森检验用于重复测量或配对设计，且数据至少为顺序量表。它通过将差值的大小排秩（忽略正负号），再分别计算正秩和与负秩和，来评估两种条件间是否存在显著差异。
T = smaller sum of positive or negative ranks
lt to be significant at p ≤ 0.05, the obtained T must be equal to or less than the critical value.
检验统计量T取正秩和与负秩和中的较小值。将T与给定N（非零差值对数）下的临界值比较。若要在 p ≤ 0.05 时显著，计算得到的T必须等于或小于临界值。
Always state the null hypothesis (e.g., 'there is no significant difference between Condition A and Condition B'), note the significance level, and report whether the null is rejected. Remember that the Wilcoxon test is sensitive to the magnitude of differences because it uses ranks.
务必写出零假设（如“条件A与条件B之间无显著差异”），注明显著性水平，并报告是否拒绝零假设。要记住，由于使用秩次，威尔科克森检验对差异的大小是敏感的。
5. Mann-Whitney U Test | 曼-惠特尼U检验
The Mann-Whitney U test is the independent-measures counterpart to the Wilcoxon. It is used when comparing two separate groups of participants on at least ordinal data and does not assume normal distribution.
曼-惠特尼U检验是威尔科克森检验的独立测量版本。它用于比较两组独立被试在至少为顺序数据上的表现，且不要求正态分布。
U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁
U₂ = n₁n₂ + n₂(n₂+1)/2 − R₂
U = min(U₁, U₂)
Where n₁ and n₂ are the sample sizes of group 1 and group 2, and R₁ and R₂ are the sums of ranks for each group after all scores have been ranked together. The smaller U-value is the test statistic.
其中 n₁ 与 n₂ 分别是组1和组2的样本量，R₁ 与 R₂ 是所有分数混合排秩后两组的秩和。取较小的U值为检验统计量。
served U must be equal to or smaller than the critical value. This test is frequently examined; always remember to rank all scores together before calculating R₁ and R₂.
将观察到的U值与曼-惠特尼表临界值进行比较。若要在 p ≤ 0.05 时显著，观察U必须等于或小于临界值。该检验是考查重点，务必记住先将全部分数混合排秩再计算R₁和R₂。
6. Chi-Squared Test for Independence | 独立性的卡方检验
The chi-squared (χ²) test examines whether there is a significant association between two categorical (nominal) variables. It compares observed frequencies (O) in each cell of a contingency table with the frequencies expected (E) if the null hypothesis of no association is true.
卡方（χ²）检验考察两个类别（称名）变量之间是否存在显著关联。它比较列联表每一单元格的观察频次（O）与零假设（无关联）成立时预期的频次（E）。
χ² = ∑ ((O − E)² / E)
olumns − 1). Compare the calculated χ² with the critical value at the chosen significance level. The null hypothesis is rejected if χ² exceeds the critical value.
某一单元格的预期频次 = （行合计 × 列合计） ÷ 总计。自由度（df） = （行数 − 1） × （列数 − 1）。将计算得到的χ²值与选定显著性水平下的临界值比较。若χ²大于临界值，则拒绝零假设。
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 the data are at least ordinal. It does not assume a linear relationship or normal distribution.
斯皮尔曼等级相关系数（rₛ）衡量两个变量之间（数据至少为顺序量表）单调关系的强度与方向，不要求线性关系或正态分布。
rₛ = 1 − (6∑d²) / (n(n² − 1))
Here d is the difference between the ranks of each pair of scores, and n is the number of pairs. The coefficient ranges from −1 (perfect negative correlation) to +1 (perfect positive correlation); 0 indicates no correlation.
其中d是每对分数的秩次差，n为数据对数。系数范围从 −1（完全负相关）到 +1（完全正相关），0表示无相关。
After computing rₛ, compare it with the critical value for Spearman's test. The null hypothesis of no correlation is rejected if the absolute value of rₛ is equal to or greater than the critical value. Remember that correlation does not imply causation—an experimental design is needed to establish cause and effect.
求得rₛ后，与斯皮尔曼检验的临界值进行比较。若rₛ的绝对值等于或大于临界值，则拒绝“无相关”的零假设。切记相关不等于因果——要确立因果关系必须采用实验设计。
8. Normal Distribution & Central Limit Theorem | 正态分布与中心极限定理
The normal distribution is a bell-shaped, symmetrical curve where the mean, median, and mode coincide. Many psychological traits (e.g., IQ scores) are approximately normally distributed. The area under the curve tells us the probability of scores falling within certain intervals.
正态分布是一条钟形对称曲线，其平均数、中位数与众数重合。许多心理特质（如智商分数）近似正态分布。曲线下的面积表示分数落在特定区间的概率。
In a normal distribution, roughly 68% of values lie within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. This '68–95–99.7 rule' underpins the reasoning behind significance testing.
在正态分布中，约68%的数值落在平均数加减一个标准差的范围内，约95%落在两个标准差内，99.7%落在三个标准差内。这条“68–95–99.7法则”是显著性检验的推理基础。
The Central Limit Theorem states that, as sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the population distribution. This theorem justifies the use of parametric tests with large samples even when the raw data are not perfectly normal.
中心极限定理指出，随着样本量增大，均值的抽样分布将趋近正态分布，无论总体分布形状如何。这一定理为大样本条件下即使原始数据不完全正态仍可使用参数检验提供了依据。
9. Experimental Design & Reliability Theorems | 实验设计与信度原理
Internal reliability refers to the consistency of a test or questionnaire across its items (e.g., split-half reliability). External reliability concerns the stability of results across time (test-retest) or across different observers (inter-rater reliability).
内部信度指测验或问卷各项目之间的一致性（如分半信度）。外部信度涉及跨时间（重测信度）或跨不同观察者（评分者间信度）结果的稳定性。
Validity is the extent to which a test measures what it claims to measure. Face validity is a superficial judgement; concurrent validity checks the test against an existing validated measure; ecological validity is the degree to which findings can be generalised to real-life settings.
效度指测验在多大程度上测量了它所要测量的东西。表面效度是一种直观判断；同时效度将测验结果与现有的效标工具进行对比；生态效度则指研究结果可推广到现实生活环境的程度。
Standardisation means keeping the procedure exactly the same for all participants, which reduces extraneous variables and increases reliability. Counterbalancing, random allocation, and use of control groups are all design strategies that enhance internal validity by controlling for order effects, participant variables, and confounding variables.
标准化是指对所有被试保持完全相同的程序，从而减少额外变量并提高信度。平衡法、随机分配以及使用控制组都是实验设计策略，通过控制顺序效应、被试变量和混淆变量来增强内部效度。
10. Probability & Critical Value Tables | 概率与临界值表
ing, the conventional cut-off is p = 0.05, meaning the researcher is willing to accept a 5% risk of a Type I error—rejecting a true null hypothesis.
概率（p）用比例表示，范围从0（不可能）到1（必然）。假设检验中通常以 p = 0.05 为界限，表示研究者愿意承受5%的第一类错误风险——即错误地拒绝了一个真实的零假设。
A Type II error occurs when a false null hypothesis is not rejected. Power is the probability of correctly rejecting a false null hypothesis. Increasing sample size, using sensitive measures, and controlling extraneous variables all improve power.
第二类错误发生在未能拒绝一个错误的零假设时。统计检验力是指正确拒绝错误零假设的概率。增大样本量、使用灵敏的测量工具以及控制额外变量均可提高检验力。
Critical value tables for Wilcoxon, Mann-Whitney, Chi-squared, and Spearman' ze or degrees of freedom. Always check whether the calculated statistic needs to be equal to/less than or greater than the critical value, according to the specific test.
威尔科克森、曼-惠特尼、卡方和斯皮尔曼检验的临界值表提供了在给定样本量或自由度下 p = 0.05（或0.01）的阈值。务必根据具体检验的要求，核实计算得到的统计量是需要等于/小于还是大于临界值。
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