Quick Reference Handbook of Formulas & Theorems for Year 13 CCEA Psychology | CCEA 13年级心理学公式定理速查手册

📚 Quick Reference Handbook of Formulas & Theorems for Year 13 CCEA Psychology | CCEA 13年级心理学公式定理速查手册

This handbook compiles essential statistical formulas, decision-making rules, and key quantitative concepts tested in Year 13 CCEA Psychology. It serves as a rapid revision tool for descriptive statistics, inferential tests, effect sizes, and significance thresholds used in AS and A2 research methods.

本手册汇编了Year 13 CCEA心理学考核中必备的统计公式、决策规则和关键量化概念。它是描述统计、推断检验、效应量和显著性阈值的快速复习工具，涵盖AS与A2研究方法的所有计算需求。

1. Measures of Central Tendency | 集中趋势量数

The mean is the sum of all scores divided by the number of scores. It is the most powerful measure but is sensitive to extreme values.

平均数等于所有分数之和除以分数的个数。它是最有代表性的量数，但对极端值敏感。

The median is the middle value when data are ordered. It is not affected by outliers and is used with ordinal data.

中位数是排序后位于中间的值。它不受异常值影响，适用于顺序数据。

The mode is the most frequently occurring score. It can be used with nominal data.

众数是出现次数最多的分数。它可用于称名数据。

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2. Measures of Dispersion | 离散量数

The range is the difference between the highest and lowest scores. It provides a quick indication of spread but is heavily influenced by a single extreme value.

全距是最高分与最低分的差值。它能快速反映数据的分布宽度，但极易受单个极端值影响。

Variance and standard deviation quantify how much scores deviate from the mean. The standard deviation is the square root of the variance.

方差和标准差量化了分数偏离平均数的程度。标准差是方差的平方根。

A larger standard deviation indicates greater variability in the dataset.

标准差越大，表示数据集的变异性越大。

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3. Standard Deviation (Sample) Formula | 标准差（样本）公式

s = √( Σ(X – X̄)² ÷ (n – 1) )

X represents each individual score, X̄ is the sample mean, and n is the number of observations. We use n–1 to correct for bias in estimating the population standard deviation.

X代表每一个单独的分数，X̄是样本平均数，n是观测值的个数。使用n–1是为了修正估计总体标准差时的偏差。

Always compute the squared deviations first, sum them, divide by n–1, and then take the square root.

务必先计算离差平方和，除以n–1，然后再开平方根。

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4. Variance Formula | 方差公式

s² = Σ(X – X̄)² ÷ (n – 1)

Variance is the squared standard deviation. It is rarely reported on its own but is necessary for certain inferential tests such as ANOVA and t‑tests.

方差是标准差的平方。它本身很少被单独报告，但某些推断检验（如方差分析和t检验）需要用到它。

The unit of variance is the square of the unit of measurement, making it less intuitive for description.

方差的单位是原始测量单位的平方，因此用来描述数据时不如标准差直观。

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5. Inferential Testing Overview | 推断检验概述

Inferential tests allow psychologists to decide whether observed differences or relationships are statistically significant, i.e. unlikely to have occurred by chance.

推断检验让心理学家能够判断观察到的差异或关系是否具有统计学显著性，即不太可能仅由偶然因素造成。

Every test produces an observed (calculated) value that is compared against a critical value at a chosen significance level, typically p ≤ 0.05.

每一项检验都会得出一个观测值（计算值），然后在选定的显著性水平（通常为p ≤ 0.05）下与临界值进行比较。

The choice of test depends on the level of measurement, research design, and whether data meet parametric assumptions.

选择哪种检验取决于数据的测量水平、研究设计以及数据是否满足参数检验的假设。

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6. Chi‑Squared (χ²) Test | 卡方检验

χ² = Σ( (O – E)² ÷ E )

O is the observed frequency and E is the expected frequency. This test is used for independent groups with nominal (categorical) data.

O为观测频数，E为期望频数。该检验用于独立组设计和称名（分类）数据。

Degrees of freedom (df) are calculated as (number of rows – 1) × (number of columns – 1). If χ² calculated ≥ critical value, the result is significant.

自由度（df）的计算公式为（行数–1）×（列数–1）。若计算所得的χ²值大于或等于临界值，则结果具有显著性。

Always use raw frequencies, not percentages, and ensure expected frequencies are at least 5 for the test to be reliable.

务必使用原始频数而非百分比，并确保每个单元的期望频数至少为5，以保证检验的可靠性。

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7. Sign Test | 符号检验

S = smaller number of + or – signs

The sign test is a non‑parametric alternative for repeated measures or matched pairs designs when data are nominal or ordinal and differences can be coded as positive or negative.

符号检验是用于重复测量或配对设计的非参数检验，适用于数据为称名或顺序且差异可被编码为正号或负号的情况。

Count the number of pluses and minuses. The calculated value S is the frequency of the less common sign. If S ≤ critical value (from a binomial table), the result is significant.

分别计数正号和负号的个数。计算值S为出现次数较少的那一类符号的频数。如果S小于或等于临界值（查二项分布表），则结果显著。

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8. Wilcoxon Signed‑Rank Test | 威尔科克森符号秩检验

T = smaller sum of signed ranks

This test is used for repeated measures or matched pairs with ordinal data. Differences between pairs are ranked, ignoring zeros, and ranks receive the sign of the original difference.

该检验用于顺序数据的重复测量或配对设计。对配对差值进行排序（忽略零差值），每个秩次赋予原始差值的正负号。

Sum the positive ranks (R⁺) and the negative ranks (R⁻) separately. The observed value T is the smaller of the two sums. To be significant, T must be less than or equal to the critical value.

分别计算正秩和（R⁺）与负秩和（R⁻）。观测值T是两者中较小的那个和。若要达到显著，T必须小于或等于临界值。

Always check the direction of the hypothesis: a one‑tailed test halves the critical p value.

一定要检查假设的方向：单尾检验会将临界p值减半。

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9. Mann‑Whitney U Test | 曼‑惠特尼U检验

U₁ = n₁n₂ + n₁(n₁ + 1) ÷ 2 – R₁

U₂ = n₁n₂ – U₁

The observed U is the smaller of U₁ and U₂. This test compares two independent groups when data are at least ordinal.

观测值U取U₁和U₂中较小的一个。该检验用于比较两个独立组，且数据至少为顺序水平。

R₁ is the sum of ranks for group 1. Rank all scores together. If the calculated U is less than or equal to the critical value, the null hypothesis is rejected.

R₁是第一组的秩和。将所有分数混合排序。若计算出的U值小于或等于临界值，则拒绝虚无假设。

For large samples (n > 20), U can be converted to a z‑score, but CCEA usually assesses the small‑sample table version.

对于大样本（n > 20），U值可转换为z分数，但CCEA通常考察的是小样本查表法。

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10. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数

ρ = 1 – ( 6 Σd² ) ÷ ( n(n² – 1) )

This test measures the strength and direction of a relationship between two ordinal variables. d is the difference in ranks for each pair, and n is the number of pairs.

该检验测量两个顺序变量之间关系的强度和方向。d是每对数据的等级差，n是对数。

A ρ value close to +1 indicates a strong positive correlation, while a value near –1 indicates a strong negative correlation. Compare the absolute value against the critical value at df = n.

ρ值接近+1表示强正相关，接近–1表示强负相关。将ρ的绝对值与自由度为n的临界值进行比较。

Remember that Spearman’s ρ simply assesses monotonic associations; a significant ρ does not prove causation.

请记住，斯皮尔曼ρ只评估单调关联，显著的ρ值并不能证明因果关系。

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11. Probability, Critical Values and Decision Making | 概率、临界值与决策

The p value is the probability of obtaining the observed result (or a more extreme one) if the null hypothesis is true. In Psychology, p ≤ 0.05 is the conventional threshold.

p值是指在虚无假设为真的情况下获得当前观测结果（或更极端结果）的概率。在心理学中，通常以p ≤ 0.05作为显著性阈值。

To be significant, the calculated value must be equal to or more extreme than the critical value. For χ², Spearman’s ρ and U, the calculated value needs to be equal to or larger (or smaller, depending on the test table).

要达到显著性，计算值必须等于或比临界值更极端。对于χ²、Spearman ρ和U检验，要记住不同检验的比较方向：有的要求计算值大于等于临界值，有的要求小于等于。

One‑tailed tests have a directional hypothesis and a more lenient critical value; two‑tailed tests are used with non‑directional hypotheses and require a more extreme value.

单尾检验对应有方向性的假设，临界值更宽松；双尾检验用于无方向性假设，需要更极端的值才能达到显著。

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12. Type I & Type II Errors and Effect Size | Ⅰ型、Ⅱ型错误与效应量

A Type I error occurs when the null hypothesis is true but is wrongly rejected (a false positive). The probability equals the chosen alpha level, e.g. 0.05.

Ⅰ型错误发生在虚无假设为真却被错误拒绝时（假阳性）。其概率等于选定的α水平，如0.05。

A Type II error happens when the null hypothesis is false but is not rejected (a false negative). It is linked to statistical power.

Ⅱ型错误指的是虚无假设为假却未能被拒绝（假阴性）。它与统计检验力有关。

Effect size quantifies the magnitude of a difference or relationship, independent of sample size. A common formula for Cohen’s d is:

效应量量化了差异或关系的大小，且不受样本容量影响。一个常用的Cohen’s d公式为：

d = (M₁ – M₂) ÷ SD_pooled

M₁ and M₂ are group means, and SD_pooled is the pooled standard deviation. Even a significant result has limited real‑world meaning if effect size is trivial.

M₁和M₂是两组平均数，SD_pooled是合并标准差。即使结果显著，如果效应量微不足道，它在现实世界中的意义也很有限。

Always consider the balance between Type I and Type II error risks when evaluating psychological research.

在评估心理学研究时，要始终权衡Ⅰ型和Ⅱ型错误风险的平衡。

Published by TutorHao | Psychology Revision Series | aleveler.com

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