📚 IGCSE CIE Psychology: Formula & Theorem Quick Reference Handbook | IGCSE CIE 心理学：公式定理速查手册
This quick reference handbook provides a concise summary of essential formulas, statistical rules, and principles used in IGCSE CIE Psychology research methods. Mastering these will help you confidently analyse data, interpret psychological studies, and excel in your examinations.
本速查手册总结了 IGCSE CIE 心理学研究方法中必备的公式、统计规则和原则。掌握这些内容，可以帮助你自信地分析数据、解读心理学研究并在考试中脱颖而出。
1. The Mean | 平均数
The mean, often called the average, is a measure of central tendency. It is calculated by adding together all the scores in a data set and then dividing that sum by the total number of scores. It is the most commonly used average in psychology because it takes every score into account.
平均数，通常被称为均值，是一种集中趋势的度量。它的计算方法是把一组数据中的所有分数相加，然后除以分数的总个数。由于平均数考虑了每一个分数，它是心理学中最常用的平均值。
Mean = Σx / N
scores are 4, 7, 9, 10, 10 and 12, the sum is 52 and N is 6, so the mean is 52 ÷ 6 = 8.67.
在这个公式中，Σx 代表所有单个分数的总和，N 代表分数的总个数。例如，若测试分数为 4, 7, 9, 10, 10 和 12，总和为 52，N 为 6，则平均数为 52 ÷ 6 = 8.67。
A crucial characteristic of the mean is that it is easily distorted by outliers. An extremely high or low score can pull the mean away from the centre of the majority of the data, making it less representative of a typical score.
平均数的一个关键特征是它容易受到异常值的影响。一个极高或极低的分数会将平均数拉离大多数数据的中心，使其代表性降低，无法反映典型的分数水平。
2. The Median | 中位数
The median is the middle score in an ordered list. To find the median, arrange all scores from smallest to largest. If the number of scores (N) is odd, the median is the score located exactly in the middle of the list.
中位数是一组有序数据中居于中间位置的分数。要找出中位数，需要将所有分数从小到大排列。如果分数的个数 (N) 为奇数，中位数就是序列正中间的那个分数。
If N is even, there is no single middle score. In that case, take the two middle scores, add them together and divide by two. Their mean becomes the median of the whole set.
如果 N 为偶数，则没有单一的中间分数。此时，需要取出中间的两个分数，将它们相加后除以二。这两个分数的平均数即为整组数据的中位数。
The median is not influenced by extreme values. Because it depends only on the order of scores, an unusually high outlier will not shift its position as it does the mean. For this reason, the median is often preferred when data contain outliers.
中位数不受极端值的影响。由于它仅取决于分数的排列顺序，异常高的数值不会像干扰平均数那样改变它的位置。因此，当数据包含异常值时，往往优先选用中位数。
3. The Mode | 众数
The mode is the simplest measure of central tendency – it is the score that occurs most frequently in a data set. A set of scores may have one mode (unimodal), two modes (bimodal) or more than two modes (multimodal).
众数是最简单的集中趋势度量，它是一组数据中出现次数最多的分数。一组分数可能有一个众数（单峰）、两个众数（双峰）或多于两个众数（多峰）。
To identify the mode, count how many times each score appears. The score with the highest frequency is the mode. For example, in the set 3, 5, 5, 7, 9, 9, 9, 11, the mode is 9 because it appears three times, more than any other value.
要找出众数，只需统计每个分数出现的次数。出现频率最高的那个分数就是众数。例如，在集合 3, 5, 5, 7, 9, 9, 9, 11 中，众数为 9，因为它出现了三次，次数最多。
Like the median, the mode is not affected by extreme outliers. However, the mode can be less informative in small or highly varied data sets. Sometimes a data set has no mode at all if every score occurs only once.
与中位数一样，众数不受极端异常值的影响。然而，在数据量较少或变异较大的数据集中，众数提供的信息可能较少。有时，如果每个分数都只出现一次，则该组数据根本没有众数。
4. The Range | 范围
The range is a measure of dispersion. It indicates how spread out scores are by showing the difference between the highest and lowest scores. The larger the range, the more variability exists in the data set.
范围是一种离散程度的度量。它通过展示最高分与最低分之间的差异来表示分数的分散程度。范围越大，数据集的变异性越大。
Range = Highest Score – Lowest Score
For instance, if the highest reaction time in a study is 540 ms and the lowest is 210 ms, the range is 330 ms. The range is extremely easy to calculate, but it only considers the two most extreme values and ignores the distribution of the remaining scores.
例如，如果一项研究中反应时间的最高值为 540 毫秒，最低值为 210 毫秒，那么范围就是 330 毫秒。范围非常容易计算，但它只考虑了两个最极端的值，而忽略了其余分数的分布状况。
Because the range is based solely on extremes, a single outlier can dramatically inflate the range and give a misleading impression of variability. Researchers often report the range alongside the median or interquartile range for a more complete picture.
由于范围完全基于极端值，单个异常值就可能显著扩大范围，从而对变异性造成误导性印象。研究人员通常会同时报告范围和中位数或四分位距，以更全面地呈现数据情况。
5. Percentages and Ratios | 百分比与比例
Percentages are used in psychology to express how large one quantity is relative to a whole, standardised to 100. They allow researchers to compare proportions easily across different groups or conditions.
心理学中使用百分比来表示某个数量相对于整体的相对大小，并以 100 为基准进行标准化。这使研究人员能够轻松比较不同组别或条件下的比例。
Percentage = (Part ÷ Whole) × 100%
ut of 25 participants recalled a word correctly, the percentage correct is (18 ÷ 25) × 100% = 72% onverting raw data into a standardised format when presenting results in graphs or tables.
如果 25 名参与者中有 18 人正确回忆了一个词语，则正确百分比为 (18 ÷ 25) × 100% = 72%。在图表或表格中呈现结果时，百分比还可用于将原始数据转换为标准化格式。
A ratio compares two quantities directly, often written with a colon. For example, a study might report that participants in the experimental condition improved at a ratio of 3:1 compared to controls. Ratios provide a simple way to express the relationship between two values without scaling to 100.
比例直接比较两个数量，通常用冒号书写。例如，一项研究可能报告实验条件下的参与者与对照组相比的改善比例为 3:1。比例提供了一种简单的方式来表示两个数值之间的关系，而无需缩放至 100。
6. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分比的转换
Psychological data are often expressed in different numerical forms. Being able to switch confidently between fractions, decimals and percentages is essential for interpreting tables, graphs and statistical findings.
心理学数据常以不同的数字形式呈现。能够自信地在分数、小数和百分比之间进行转换，对于解读表格、图表和统计结果至关重要。
To change a fraction to a decimal, divide the numerator (top) by the denominator (bottom). For example, 3/8 becomes 0.375. To convert that decimal to a percentage, multiply it by 100, giving 37.5%. To go from a percentage to a decimal, divide by 100; thus 85% becomes 0.85.
要将分数转换为小数，用分子除以分母。例如，3/8 变为 0.375。要将该小数转换为百分比，乘以 100，得到 37.5%。若要从百分比转换为小数，除以 100，因此 85% 变为 0.85。
In data analysis, you may need to express a probability as a decimal, a fraction of participants as a percentage, or scale a proportion. A solid grasp of these conversions ensures accuracy when calculating effect sizes, proportions or descriptive statistics.
在数据分析中，你可能需要将概率表示为小数，将参与者的比例表示为百分比，或对某个比例进行缩放。牢固掌握这些转换可确保在计算效应量、比例或描述性统计时的准确性。
7. Interpreting Scatter Plots | 解读散点图
Scatter plots display the relationship between two co-variables. Each point on the plot represents an individual's paired score on two measures. The overall pattern of points reveals whether and how strongly the variables are related.
散点图展示两个协变量之间的关系。图上的每个点代表一个个体在两个测量指标上的一对分数。点的总体分布模式揭示了变量之间是否存在相关以及相关的强度。
A positive correlation appears as points sloping upwards from left to right. As one variable increases, the other also tends to increase. A negative correlation shows points sloping downwards: high scores on one variable are associated with low scores on the other.
正相关表现为点从左下方向右上方倾斜分布。当一个变量增加时，另一个变量也倾向于增加。负相关表现为点从左上方到右下方倾斜：一个变量的高分与另一个变量的低分相关联。
No correlation results in a random cloud of points with no clear direction. The closer the points lie to a straight line, the stronger the correlation. Even a very tight scatter plot does not prove causation – a third variable may explain the link.
无相关则表现为散点呈无规律云状分布，没有明确方向。点越接近一条直线，相关性越强。即使散点图非常紧凑也不能证明因果关系——第三个变量或许可以解释这种关联。
8. Understanding Correlation Coefficients | 理解相关系数
The correlation coefficient (r) is a numerical summary of the direction and strength of a linear relationship between two variables. It always ranges from –1 to +1. The sign indicates direction, and the absolute value indicates strength.
相关系数 (r) 是对两个变量之间线性关系的方向和强度的数值概括。其范围总是在 –1 到 +1 之间。符号表示方向，绝对值表示强度。
–1 ≤ r ≤ +1
A positive r means a positive correlation; a negative r means a negative correlation. A value of 0 suggests no linear correlation. The table below offers a general guide to interpreting the strength of r, though cut-offs may vary by context.
正的 r 意味着正相关；负的 r 意味着负相关。r 值为 0 表明不存在线性相关。下表提供了一个解读 r 强度的一般指南，不过这些分界值可能因具体情境而异。
| r Value | Strength (English) | 强度 (中文) |
|---|---|---|
| 0.00 to ±0.19 | Very weak / negligible | 非常弱 / 可忽略 |
| ±0.20 to ±0.39 | Weak | 弱 |
| ±0.40 to ±0.59 | Moderate | 中等 |
| ±0.60 to ±0.79 | Strong | 强 |
| ±0.80 to ±1.00 | Very strong | 非常强 |
Always remember: correlation does not imply causation. A strong correlation between, for instance, ice-cream sales and drowning incidents does not mean one causes the other. Both may be influenced by a third factor, such as hot weather.
请始终牢记：相关不意味着因果。例如，冰淇淋销量与溺水事件之间的
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