Year 11 Edexcel Psychology: Formula & Theorem Quick Reference Guide | Year 11 Edexcel 心理学：公式定理速查手册

📚 Year 11 Edexcel Psychology: Formula & Theorem Quick Reference Guide | Year 11 Edexcel 心理学：公式定理速查手册

In Edexcel GCSE Psychology, data handling is a fundamental skill assessed throughout the research methods component. This quick reference handbook brings together every essential formula, theorem, and decision rule you need to master – from descriptive statistics to inferential testing. Use it to check your calculations, prepare for exam questions, and build confidence in interpreting psychological data.

在Edexcel GCSE心理学中，数据处理是贯穿研究方法部分的核心技能。这本速查手册汇集了你需要掌握的所有关键公式、定理和决策规则，涵盖从描述统计到推断检验的全部内容。用它来检验计算、备考答题，并建立解读心理学数据的信心。

1. Measures of Central Tendency: Mean, Median, Mode | 集中趋势的度量：平均数、中位数、众数

The mean is calculated by summing all the scores in a data set and dividing by the total number of scores. It is sensitive to extreme values, or outliers, and is used with interval/ratio data.

平均数通过将所有分数相加后除以分数总数来计算。它对极端值或异常值敏感，适用于等距/等比数据。

Mean (x̄) = Σx / n

The median is the middle value when scores are arranged in ascending order. If there is an even number of scores, the median is halfway between the two central values. The median is unaffected by outliers and is preferred for ordinal data.

中位数是将分数按升序排列后处于中间位置的数值。如果分数个数为偶数，中位数为中间两个数值的平均数。中位数不受异常值影响，适用于顺序数据。

The mode is the most frequently occurring score in a data set. A set can have one mode (unimodal), two modes (bimodal), or more. The mode is the only measure of central tendency suitable for nominal data.

众数是数据集中出现频率最高的分数。一组数据可以是单众数、双众数或多众数。众数是唯一适用于名义数据的集中趋势度量。

2. Measures of Dispersion: Range and Standard Deviation | 离散程度的度量：极差与标准差

Dispersion tells us how spread out the scores are. The simplest measure is the range, which is the difference between the highest and lowest values. It is quick to compute but heavily affected by outliers.

离散程度揭示分数的分散情况。最简单的度量是极差，即最大值与最小值之差。它计算简便，但极易受异常值影响。

Range = Highest score – Lowest score

The standard deviation (SD) provides a more precise measure of spread by showing how much scores deviate from the mean on average. A small SD indicates that most scores cluster near the mean; a large SD shows greater variability.

标准差（SD）通过反映分数平均偏离均值的程度，提供了一个更精确的分散程度度量。标准差小，说明多数分数聚集在均值附近；标准差大，则变异性高。

s = √[ Σ(x – x̄)² / (n – 1) ]

In the formula, x is each individual score, x̄ is the mean, and n is the number of scores. The squared deviations are summed, divided by n-1 (for a sample), and then the square root is taken.

公式中，x是每个分数，x̄是均值，n是分数个数。先求每个分数与均值之差的平方，求和后除以n-1（对于样本数据），再开平方即得标准差。

3. Percentage Change and Ratios | 百分比变化与比率

Percentage change is used to compare a quantity before and after an intervention or over time. You subtract the original value from the new value, divide by the original, and multiply by 100.

百分比变化用来比较干预前后或一段时间内某个数量的变化。用新值减去原值，除以原值，再乘以100。

Percentage change = [(New value – Original value) / Original value] × 100

A positive result indicates an increase, while a negative result signals a decrease. For example, if a memory score rises from 50 to 65, the percentage change is ((65 – 50) / 50) × 100 = 30% improvement.

正值表示上升，负值表示下降。例如，若记忆分数从50提高到65，百分比变化为((65 – 50) / 50) × 100 = 30%的进步。

Ratios express the relationship between two quantities. In psychology, you might see ratios when reporting the number of participants in different conditions, e.g., a ratio of 2:1 would mean twice as many participants in one group as in the other.

比率表示两个量之间的关系。在心理学中，报告不同条件下的被试数量时常用比率，比如2:1表示一组被试人数是另一组的两倍。

4. The Normal Distribution and Standard Scores | 正态分布与标准分数

Many psychological variables, such as IQ scores, are normally distributed. The curve is bell-shaped and symmetrical, with the mean, median, and mode all at the centre. In a perfect normal distribution, about 68% of values fall within ±1 SD from the mean, and about 95% within ±2 SDs.

许多心理变量，如智商分数，呈正态分布。曲线为钟形对称，均值、中位数和众数重合于中心。在完美的正态分布中，约68%的数值落在均值±1个标准差内，约95%落在±2个标准差内。

A standard score, often called a z-score, tells you how many standard deviations a raw score is from the mean. It allows comparisons across different distributions.

标准分数，常被称为z分数，表示一个原始分数距离均值多少个标准差，便于在不同分布间进行比较。

z = (x – μ) / σ

Here, μ is the population mean and σ is the population standard deviation. For samples, you may substitute x̄ and s. A z-score of +1.5 means the score is 1.5 standard deviations above the mean.

此处，μ为总体均值，σ为总体标准差。对于样本，可用x̄和s代替。一个+1.5的z分数表示该分数比均值高出1.5个标准差。

5. Hypothesis Testing: Null and Alternative Hypotheses | 假设检验：零假设与备择假设

Every inferential statistical test begins by formulating two hypotheses. The null hypothesis states that there is no effect, no difference, or no correlation in the population. The alternative (or experimental) hypothesis asserts that an effect, difference, or correlation does exist.

任何推断统计检验都以提出两个假设开始。零假设声称总体中不存在效应、差异或相关性。备择（实验）假设则断言效应、差异或相关性确实存在。

Psychological research often uses a directional (one-tailed) hypothesis when previous evidence points to a specific direction of effect; otherwise, a non-directional (two-tailed) hypothesis is used. This choice affects how critical values are read from statistical tables.

心理学研究中，如果已有证据指向特定的效应方向，常使用定向（单尾）假设；否则使用非定向（双尾）假设。这一选择会影响如何从统计表中读取临界值。

The decision rule: if the calculated test statistic exceeds the critical value at a chosen significance level (typically p < 0.05), the null hypothesis is rejected in favour of the alternative. If not, the null is retained.

决策规则为：如果计算出的检验统计量超过了选定显著性水平（通常为p < 0.05）下的临界值，则拒绝零假设，支持备择假设。否则保留零假设。

6. 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 sets of ordinal data. It is used when the data are ranks or not normally distributed.

斯皮尔曼等级相关系数（rₛ）衡量两组顺序数据之间单调关系的强度和方向。适用于等级数据或非正态分布数据。

rₛ = 1 – (6 Σ d²) / (n (n² – 1))

Where d is the difference between the ranks for each pair, and n is the number of pairs. After ranking each variable separately, calculate d for each pair, square them, and sum to get Σ d². Plug the values into the formula.

式中，d为每对数据的等级差，n为数据对个数。先分别对每个变量进行等级排列，计算每对的等级差d，平方后求和得Σ d²，再代入公式。

The value of rₛ ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation). A result of 0 indicates no correlation. Compare the calculated rₛ against a critical value table to determine significance.

rₛ的取值在-1（完全负相关）到+1（完全正相关）之间。结果为0表示无相关。将计算出的rₛ与临界值表进行比较，以判断显著性。

7. Chi-Square Test of Association | 卡方关联性检验

The chi-square test (χ²) is used to examine whether there is a significant association between two nominal (categorical) variables. It compares the observed frequencies in each category against the frequencies expected if the null hypothesis were true.

卡方检验（χ²）用于考查两个名义（分类）变量之间是否存在显著关联。它比较每个类别中的观察频次与零假设为真时的期望频次。

χ² = Σ ( (O – E)² / E )

O represents the observed frequency, and E is the expected frequency, calculated as (row total × column total) / grand total for each cell of a contingency table. The test statistic is the sum of these values across all cells.

O代表观察频次，E是期望频次，对于列联表的每个单元格，E = (行总计 × 列总计) / 总计。检验统计量为所有单元格( O – E )² / E 的求和。

The degrees of freedom (df) for a chi-square test of association are calculated as (number of rows – 1) × (number of columns – 1). You then compare the computed χ² to the critical value at the appropriate alpha level.

关联性卡方检验的自由度（df）计算方法为（行数 – 1）×（列数 – 1）。将计算出的χ²值与相应alpha水平下的临界值进行比较。

8. Mann-Whitney U Test for Independent Groups | 独立组的曼-惠特尼U检验

The Mann-Whitney U test compares two independent groups when the dependent variable is ordinal or interval/ratio data that do not meet parametric assumptions. It tests whether there is a significant difference between the groups’ distributions.

曼-惠特尼U检验用于比较两个独立组，当因变量为顺序数据或不满足参数假设的等距/等比数据时使用。它检验两组分布是否存在显著差异。

U₁ = n₁ n₂ + n₁ (n₁ + 1) / 2 – R₁
U₂ = n₁ n₂ – U₁
U = the smaller of U₁ and U₂

Here, n₁ and n₂ are the sample sizes of the two groups, and R₁ is the sum of ranks assigned to the smaller group (or to group 1 if sizes are equal). All scores from both groups are ranked together, ignoring group membership.

此处，n₁和n₂是两组的样本量，R₁是分配给较小一组（若样本量相等则分配给组1）的秩和。将两组所有数据混合在一起进行等级排列，忽略组别。

The obtained U is compared with a critical value from the Mann-Whitney table. For the result to be significant at a given level, the calculated U must be less than or equal to the critical value.

计算出的U值与曼-惠特尼临界值表比较。要使结果在某一水平上显著，计算得到的U值必须小于或等于临界值。

9. Wilcoxon Signed-Rank Test for Repeated Measures | 重复测量设计的威尔科克森符号秩检验

The Wilcoxon signed-rank test is the non-parametric equivalent of the paired t-test. It is used when the same participants are tested in two conditions (repeated measures) or matched pairs, and the data are at least ordinal.

威尔科克森符号秩检验是配对t检验的非参数等价方法。当同一组被试在两种条件下接受测试（重复测量）或配对设计，且数据至少为顺序数据时使用。

W = the smaller of ΣR+ and ΣR-

Compute the difference between each pair. Ignore any zero differences (reduce n accordingly). Rank the absolute differences from 1 to n. Assign each rank a sign (+) if the difference was positive and (-) if negative. Sum the ranks for the positive signs (ΣR+) and for the negative signs (ΣR-). The test statistic W is the smaller sum.

计算每对数据的差值，忽略差值为零的对（相应减少n）。将差值的绝对值从1到n进行等级排列，根据原差值的正负赋予秩次“+”或“-”号。分别求出正秩和（ΣR+）与负秩和（ΣR-）。检验统计量W取其中较小的和。

Compare W to the critical value in the Wilcoxon table. If the calculated W is less than or equal to the critical value, the result is statistically significant, and the null hypothesis can be rejected.

将W值与威尔科克森临界值表比较。若计算出的W小于或等于临界值，则结果具有统计显著性，可以拒绝零假设。

10. Interpreting Significance Levels and Critical Values | 解读显著性水平与临界值

The significance level (denoted by alpha, α) is the probability of rejecting the null hypothesis when it is actually true – a Type I error. In GCSE Psychology, the conventional level is p ≤ 0.05, meaning there is a 5% risk of a false positive.

显著性水平（记为alpha, α）是指当零假设实际为真时拒绝它的概率——即I类错误。在GCSE心理学中，常用水平为p ≤ 0.05，意味着出现假阳性的概率为5%。

A table summarising which inferential test to choose can help consolidate your knowledge.

下面这张总结表能帮你快速选择合适的推断检验。

Type of data / 数据类型	Design / 设计	Test / 检验
Ordinal / 顺序数据	Correlation / 相关性	Spearman’s rho
Nominal / 名义数据	Independent groups / 独立组	Chi-square
Ordinal / 顺序数据	Independent groups / 独立组	Mann-Whitney U
Ordinal / 顺序数据	Repeated measures / 重复测量	Wilcoxon signed-rank

When using critical value tables, always check that you are using the correct row for the number of participants (n or df), the correct column for a one-tailed or two-tailed hypothesis, and the appropriate α level. If the obtained statistic equals or exceeds the critical value (or, for Mann-Whitney U and Wilcoxon, is less than or equal to it), the result is significant.

使用临界值表时，务必核对被试数量（n或df）对应的行、单尾或双尾假设对应的列，以及正确的α水平。如果计算出的统计量等于或超过临界值（对于曼-惠特尼U检验和威尔科克森检验是等于或小于临界值），则结果显著。

Understanding these formulas and decision rules is essential not only for answering exam questions but also for evaluating the credibility of psychological research. Keep this guide handy whenever you practise data analysis.

理解这些公式和决策规则，不仅对于解答考试题目至关重要，对于评判心理学研究的可信度也同样关键。每次练习数据分析时，请随手查阅本指南。

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Year 11 Edexcel Psychology: Formula & Theorem Quick Reference Guide | Year 11 Edexcel 心理学：公式定理速查手册