📚 AQA Psychology Year 13 Formula & Theorem Quick Reference Handbook | AQA 心理学 Year 13 公式定理速查手册
In AQA A-level Psychology, statistical analysis forms the backbone of your research methods knowledge. Being able to swiftly recall and apply the correct formula is essential for both your examinations and practical investigations. This handbook compiles all the central descriptive and inferential statistics formulas you need, presented in a clear, bilingual format to support your revision.
在 AQA A-level 心理学中，统计分析是研究方法知识的核心。能够快速回忆并正确应用公式，对考试和实践研究都至关重要。本手册汇编了你需要的所有核心描述性和推断性统计公式，以清晰的双语形式呈现，助力你的复习。
1. Mean | 平均值
The mean is the arithmetic average of a data set. It is calculated by summing all the values and dividing by the total number of observations.
平均值是数据集的算术平均数。通过将所有数值相加，再除以观察值的总数来计算。
Mean (x̄) = Σx / n
Where x̄ represents the sample mean, Σx is the sum of all scores, and n is the number of scores. The mean is the most sensitive measure of central tendency because it uses every data point, but it can be distorted by extreme outliers. In psychology, it is ideal for interval or ratio data that are normally distributed.
其中 x̄ 代表样本平均值，Σx 是所有分数的总和，n 是分数的个数。平均值是最灵敏的集中趋势指标，因为它利用了每一个数据点，但极易受极端异常值的影响。在心理学中，它适用于正态分布的间隔或比率数据。
2. Median | 中位数
The median is the middle value when all scores are arranged in ascending order. If there is an even number of observations, the median is the average of the two central numbers.
中位数是将所有分数按升序排列后位于中间的值。如果观察值个数为偶数，中位数则是中间两个数的平均值。
Median position = (n + 1) / 2. Unlike the mean, the median is not pulled by skewed data or outliers, making it the preferred measure of central tendency for ordinal data or when the distribution is not symmetrical. Researchers often report the median for questionnaire scores and reaction times with extreme values.
中位数的位置 = (n + 1) / 2。与平均值不同，中位数不受偏态数据或异常值的拉扯，因此它是顺序数据或分布不对称时首选的集中趋势指标。研究者通常对问卷分数和有极端值的反应时报告中位数。
3. Mode | 众数
The mode is the most frequently occurring score in a data set. There can be one mode (unimodal), two modes (bimodal), or more than two (multimodal).
众数是数据集中出现频率最高的分数。可以有一个众数（单峰）、两个众数（双峰）或多于两个众数（多峰）。
No equation is necessary; the mode is simply the value with the highest frequency. It is the only measure of central tendency suitable for nominal data, such as the most common diagnosis or the most preferred stimulus in a choice task. The mode is quick to identify but can be unrepresentative if several values tie for the highest frequency.
无需方程式；众数就是频率最高的值。它是唯一适用于名词数据的集中趋势指标，例如最常见的诊断或在选择任务中最受偏爱的刺激。众数易于确定，但如果多个值并列最高频率，则可能不具代表性。
4. Range | 极差
The range is a simple measure of dispersion. It is the difference between the highest and the lowest score in a data set.
极差是一种简单的离散程度指标。它是数据集中最高分与最低分之间的差值。
Range = Highest score − Lowest score
Although easy to calculate, the range is highly sensitive to outliers because it only considers the two extreme values. It gives no information about the spread of the central bulk of the data. Psychologists use the range as a quick, preliminary check of variability, especially when presenting descriptive statistics in a summary table alongside the median.
虽然计算简便，但极差对异常值高度敏感，因为它只考虑两个极端值。它不提供关于数据中心部分分布的信息。心理学家使用极差作为一种快速的初步变异性检查，特别是在综合表中与中位数一起呈现描述统计时。
5. Variance and Standard Deviation | 方差与标准差
Variance and standard deviation are the most informative measures of spread, as they account for how every score deviates from the mean. The variance is the average of the squared deviations. For a sample, we divide by n−1 to obtain an unbiased estimate.
方差和标准差是最具信息量的离散程度指标，因为它们考虑了每个分数与平均值的偏差。方差是平方偏差的平均值。对于样本，我们除以 n−1 以获得无偏估计值。
Variance (s²) = Σ(x − x̄)² / (n − 1)
The standard deviation is the square root of the variance. It returns the spread to the original units of measurement, making it directly interpretable.
标准差是方差的平方根。它将离散程度还原到原始测量单位，使其可直接解读。
Standard Deviation (s) = √[ Σ(x − x̄)² / (n − 1) ]
A small standard deviation indicates that data points cluster tightly around the mean, while a large standard deviation indicates wide dispersion. These statistics are essential for parametric tests and for calculating standard error. In AQA psychology, you will use them to describe data sets and to check assumptions before applying tests like Pearson’s r or a t-test.
较小的标准差表明数据点紧密聚集在平均值周围，而较大的标准差则表明离散程度大。这些统计量对于参数检验和计算标准误至关重要。在 AQA 心理学中，你将使用它们描述数据集，并在应用如皮尔逊 r 或 t 检验之前检查假设。
6. Sign Test | 符号检验
The Sign Test is a non-parametric test used with a repeated measures or matched pairs design when the data are at least nominal (difference in categories like + / −). It simply counts the direction of difference between pairs of scores.
符号检验是一种非参数检验，适用于重复测量或配对组设计，且数据至少为名义级（如 + / − 的类别差异）。它只计算成对分数之间的差异方向。
Observed value (S) = min(number of plus signs, number of minus signs)
Steps: 1) For each pair, record whether the difference is positive, negative, or zero (ties are discarded). 2) Count the total number of plus signs and the total number of minus signs. 3) The smaller of these two counts is the calculated Sign Test statistic, S. Compare S against the critical value from the Sign Test table for the given N (number of non-zero differences) and significance level (usually 0.05, one-tailed or two-tailed). If S is less than or equal to the critical value, the result is significant and the null hypothesis can be rejected.
步骤：1) 对每一对数据，记录差异为正、负还是零（剔除平局）。2) 计算正号的总数和负号的总数。3) 这两个计数中较小的一个就是符号检验的统计量 S。将 S 与符号检验表在给定 N（非零差异的数量）和显著性水平（通常 0.05，单尾或双尾）下的临界值进行比较。如果 S ≤ 临界值，则结果显著，可拒绝零假设。
7. Chi-Squared Test of Association | 卡方关联检验
The chi-squared test is used with independent groups design and nominal (categorical) data to see whether there is a significant association between two variables. The data are arranged in a contingency table, and the test compares observed frequencies (O) with the frequencies expected if the null hypothesis were true (E).
卡方检验用于独立组设计和名义（分类）数据，以检验两个变量之间是否存在显著关联。数据排列在列联表中，检验将观察频数 (O) 与在零假设成立时预期的频数 (E) 进行比较。
χ² = Σ [ (O − E)² / E ]
Expected frequency for each cell = (row total × column total) / grand total. The degrees of freedom (df) are calculated as (number of rows − 1) × (number of columns − 1). Once χ² is calculated, it is compared against the critical value from the chi-squared table at the chosen level of significance and the appropriate df. If the calculated χ² is greater than the critical value, the result is significant, suggesting an association between the variables.
每个单元格的期望频数 = (行合计 × 列合计) / 总计。自由度 (df) 计算为 (行数 − 1) × (列数 − 1)。计算出 χ² 后，将其与卡方分布表在选定显著性水平和相应自由度下的临界值进行比较。如果计算得到的 χ² 大于临界值，则结果显著，表明变量间存在关联。
8. 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 to compare differences between two independent groups when the dependent variable is at least ordinal. All scores from both groups are ranked together, ignoring group membership.
曼-惠特尼 U 检验是独立 t 检验的非参数替代方案。当因变量至少为顺序级时，用于比较两个独立组之间的差异。两组所有分数合并在一起进行排序，不考虑组别。
U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁
U₂ = n₁n₂ + n₂(n₂+1)/2 − R₂
Where n₁ and n₂ are the sample sizes, R₁ is the sum of ranks for group 1, and R₂ is the sum of ranks for group 2. The test statistic U is the smaller of U₁ and U₂. This smaller U is compared with the critical value from the Mann-Whitney table for the two sample sizes at the chosen alpha. If the observed U is less than or equal to the critical value, the difference is statistically significant.
其中 n₁ 和 n₂ 为样本量，R₁ 为组 1 的秩和，R
Published by TutorHao | Year 13 Psychology Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导