📚 A-Level CCEA Psychology: Formulae and Theorems Quick Reference Handbook | A-Level CCEA 心理学：公式定理速查手册
This quick reference handbook provides the essential statistical formulae and key theorems required for the A-Level CCEA Psychology specification. Mastering these will enhance your ability to analyse data in research methods and interpret findings accurately, from the first descriptive table you draw to the final significance statement you write.
本速查手册提供 A-Level CCEA 心理学课程所需的基本统计公式与关键定理。掌握这些内容将提升你在研究方法中分析数据并准确解释研究结果的能力，从你绘制的第一张描述性表格，到最终写下的显著性结论，都能更加规范、准确。
1. Descriptive Statistics Overview | 描述统计概览
Descriptive statistics summarise and organise raw data so that patterns become apparent. They include tables, graphs, and summary measures such as central tendency and dispersion. In CCEA coursework and examination questions, you are expected to describe your data clearly before any inferential test is applied, because a mean without a measure of spread tells the reader almost nothing.
描述统计用于总结和整理原始数据，以便显现规律。它们包括表格、图表以及集中趋势和离散程度等汇总测量。在 CCEA 的课程作业与考试题目中，你需要在做任何推断统计检验之前清晰地描述数据，因为一个没有离散指标相伴的平均数几乎无法向读者传达完整信息。
Descriptive statistics fall into two families. The first family gives a single representative value for the data set: the mean, the median and the mode. The second family describes how spread out the scores are: the range, the interquartile range, the variance and the standard deviation. Graphical summaries complete the picture, and the correct graph depends on the level of measurement.
描述统计分为两大族。第一族为数据集提供一个代表性的单一数值：平均数、中位数与众数。第二族描述分数的分散程度：极差、四分位距、方差与标准差。图形汇总让整体情况更完整，而正确的图形取决于测量水平。
- Nominal data → bar chart or pie chart | 称名数据 → 条形图或饼图
- Ordinal data → bar chart of frequencies, or box plot | 顺序数据 → 频数条形图或箱线图
- Interval data → histogram, frequency polygon, scattergraph | 等距数据 → 直方图、频数多边形、散点图
2. Measures of Central Tendency | 集中趋势测量
The three main measures are the mean, median, and mode. The mean, or arithmetic average, is calculated as the sum of all scores divided by the number of scores. The median is the middle value when data are ordered; for an odd number of scores it is the (n+1)/2th value, and for an even number it is the average of the two middle values. The mode is the most frequently occurring score.
三种主要测量指标是平均数、中位数和众数。平均数即算术平均值，等于所有分数之和除以分数个数。中位数是排序后最中间的数值；当分数个数为奇数时，第 (n+1)/2 个数值即为中位数；当为偶数时，取中间两个数值的平均值。众数是出现频率最高的分数。
x̄ = Σx ÷ n
Median position = (n + 1) ÷ 2
Choosing the right measure matters. The mean uses every score and is the most sensitive measure, which is why it is preferred for interval data, but it is distorted by extreme values. The median is resistant to outliers and is therefore appropriate for ordinal data or skewed distributions. The mode is the only measure that works for nominal data, and it is useful when a researcher wants to know the most typical category rather than a mathematical average.
选择正确的指标至关重要。平均数使用了每一个分数，是最敏感的指标，因此适用于等距数据，但它会被极端值扭曲。中位数不受异常值影响，因此适合顺序数据或偏态分布。众数是唯一适用于称名数据的指标，当研究者想知道最具代表性的类别而非数学平均值时，它尤其有用。
| Measure | 指标 | Level of measurement | 适用测量水平 | Key limitation | 主要局限 |
| Mean 平均数 | Interval 等距 | Affected by outliers 受异常值影响 |
| Median 中位数 | Ordinal, interval 顺序、等距 | Ignores exact values 忽略具体数值 |
| Mode 众数 | Nominal, ordinal, interval 称名、顺序、等距 | May not be unique 可能不唯一 |
3. Measures of Dispersion: Range and Interquartile Range | 离散测量：极差与四分位距
Dispersion indicates how spread out the data are. The range is the difference between the highest and lowest values. The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁). The IQR is less affected by outliers than the range, because it describes the middle 50 per cent of the distribution and ignores the extreme tails.
离散程度反映数据的分散情况。极差是最大值与最小值的差。四分位距（IQR）是上四分位数（Q₃）与下四分位数（Q₁）的差值。四分位距受异常值的影响小于极差，因为它描述的是分布中间 50% 的数据，忽略了极端的尾部。
Range = Maximum − Minimum
IQR = Q₃ − Q₁
The range is quick to calculate and easy to understand, but it depends on only two scores and can be dramatically inflated by a single outlier. The IQR is a more robust description of spread and pairs naturally with the median when data are ordinal or skewed, for example when participants rate their anxiety on a 1–10 scale.
极差计算快捷、易于理解，但它只依赖两个分数，并且可能被单一异常值大幅拉高。四分位距是更稳健的离散描述，当中位数与顺序数据或偏态数据搭配使用时尤为自然，例如当被试在 1–10 量表上评定焦虑程度时。
4. Standard Deviation and Variance | 标准差与方差
The standard deviation measures the average distance of each score from the mean. For a sample, the deviation of each score from the mean is squared, the squared deviations are summed, the total is divided by (n − 1), and the square root is taken. The variance is simply the square of the standard deviation, so it is expressed in squared units rather than in the original units of measurement.
标准差衡量每个分数与平均数之间的平均距离。对样本而言，先将每个分数与平均数的偏差平方，把所有平方偏差相加，再除以 (n − 1)，最后取平方根。方差就是标准差的平方，因此它的单位是原始测量单位的平方，而不是原始单位。
s = √( Σ(x − x̄)² ÷ (n − 1) )
Variance = s²
For a whole population the same logic applies, but the Greek symbols σ and μ are used and the divisor is N rather than (n − 1). The (n − 1) divisor, known as Bessel's correction, is used for samples because it gives a less biased estimate of the population value. A large standard deviation means the scores are widely spread; a small one means they cluster tightly around the mean.
对于整个总体，逻辑相同，但使用希腊字母 σ 与 μ，除数也变为 N 而非 (n − 1)。除数 (n − 1) 被称为贝塞尔校正，用于样本是因为它能给出对总体数值偏差更小的估计。标准差大意味着分数分布分散；标准差小意味着分数紧密聚集在平均数附近。
σ = √( Σ(x − μ)² ÷ N )
5. Levels of Measurement and Test Selection | 测量水平与统计检验的选择
CCEA Psychology recognises three levels of measurement, and the level you have determines which statistical test you may use. Nominal data are categories that cannot be ranked, such as the number of boys and girls who chose an aggressive response. Ordinal data can be placed in rank order but the intervals between ranks are not equal, such as a 1–10 rating scale. Interval data have equal intervals between consecutive points, such as reaction time in milliseconds or IQ scores.
CCEA 心理学承认三种测量水平，你所拥有的水平决定了你可以使用哪种统计检验。称名数据是无法排序的类别，例如选择攻击性反应的男生与女生人数。顺序数据可以排名，但名次之间的间隔不相等，例如 1–10 的评定量表。等距数据相邻点之间的间隔相等，例如以毫秒计的反应时或智商分数。
Two further decisions complete the choice: whether you are looking for a correlation or a difference, and whether the design is independent groups (unrelated) or repeated measures (related). A useful habit is to state the purpose, the level of measurement and the design in one sentence before you name the test, because examiners award marks for justification, not only for the name.
另外两个决定完成整个选择过程：你要寻找的是相关还是差异，以及设计是独立组（不相关）还是重复测量（相关）。一个有用的习惯是：在说出检验名称之前，先用一句话说明目的、测量水平与设计，因为考官给分的是理由，而不仅仅是名称。
| Test | 检验 | Purpose | 目的 | Measurement | 测量水平 | Design | 设计 |
| Spearman's rho 斯皮尔曼等级相关 | Correlation 相关 | Ordinal 顺序 | Any 任意 |
| Chi-square 卡方 | Association 关联 | Nominal 称名 | Independent 独立 |
| Sign test 符号检验 | Difference 差异 | Nominal 称名 | Repeated 重复 |
| Mann-Whitney U 曼-惠特尼 U | Difference 差异 | Ordinal/interval 顺序或等距 | Independent 独立 |
| Wilcoxon T 威尔科克森 T | Difference 差异 | Ordinal/interval 顺序或等距 | Repeated 重复 |
| Related t-test 相关 t 检验 | Difference 差异 | Interval 等距 | Repeated 重复 |
| Unrelated t-test 独立 t 检验 | Difference 差异 | Interval 等距 | Independent 独立 |
6. Probability, Significance and the 0.05 Level | 概率、显著性与 0.05 水平
, which means there is a 5 per cent probability, or one chance in twenty, that the result arose from chance alone. If the probability is at or below this level, the null hypothesis is rejected and the result is described as significant.
推断统计检验观察到的模式是否可能由偶然造成。心理学中约定俗成的阈值是 p ≤ 0.05，即结果仅由偶然造成的概率为 5%，也就是二十次中有一次。若概率达到或低于该水平，则拒绝零假设，结果被称为显著。
p ≤ 0.05 → Reject H₀ (significant) | 拒绝零假设（显著）
p > 0.05 → Retain H₀ (not significant) | 保留零假设（不显著）
The observed value you calculate is compared with a critical value taken from a table. The critical value depends on the significance level, whether the hypothesis is one-tailed or two-tailed, and the degrees of freedom or number of participants. A strict two-tailed test at 0.01 reduces the risk of a Type I error, which is a false positive: concluding there is an effect when there is none. A Type II error is a false negative: failing to detect a real effect, often because the sample is too small.
你计算出的观察值需要与从表中查得的临界值进行比较。临界值取决于显著性水平、假设是单尾还是双尾，以及自由度或被试人数。在 0.01 水平进行严格的双尾检验可以降低 I 型错误的风险，即假阳性：本无效应却得出结论认为存在效应。II 型错误是假阴性：未能检测到真实效应，通常是因为样本量太小。
7. Correlation and Spearman's Rho | 相关与斯皮尔曼等级相关
Spearman's rank correlation coefficient, usually written as rs or ρ, measures the strength and direction of a relationship between two variables when at least ordinal data are available. Each pair of scores is ranked, the difference between the two ranks in each pair is calculated as d, each difference is squared, and the squared differences are summed. The coefficient always lies between −1 and +1.
斯皮尔曼等级相关系数通常写作 rs 或 ρ，用于在至少拥有顺序数据时衡量两个变量之间关系的强度与方向。先对每对分数排名，计算每对中两个名次的差 d，再将每个差平方并求和。该系数始终介于 −1 与 +1 之间。
ρ = 1 − ( 6Σd² ÷ (n(n² − 1)) )
A coefficient near +1 indicates a strong positive correlation, near −1 a strong negative correlation, and near 0 no meaningful relationship. Interpretation must never slide into causation: a correlation between hours of sleep and examination performance does not prove that sleep causes better performance. Exam answers gain marks when they state the direction, the strength, and the caution about causation in one clear sentence.
系数接近 +1 表示强正相关，接近 −1 表示强负相关，接近 0 表示无实质关系。解释时绝不能滑向因果推断：睡眠时长与考试成绩之间的相关并不能证明睡眠导致了更好的成绩。在考试答案中，用一句清晰的话说明方向、强度以及对因果关系的谨慎态度，可以获得更多分数。
8. Chi-Square (χ²) Test | 卡方检验
The chi-square test examines whether an association exists between two categorical variables. For each cell in a contingency table you compare the
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