AP Psychology Statistics Formulas and Key Concepts | AP心理学统计公式与关键概念梳理

📚 AP Psychology Statistics Formulas and Key Concepts | AP心理学统计公式与关键概念梳理

Statistics is the language psychology uses to make sense of data. On the AP Psychology exam, you are not required to compute complex formulas by hand, but you must understand what each statistic means, when it is used, and what it tells us about behavior.

统计是心理学用来解读数据的语言。在AP心理学考试中，你不需要手算复杂公式，但必须理解每个统计量是什么意思、在什么情况下使用，以及它们能告诉我们关于行为的什么信息。

1. Why Statistics Matter in Psychology | 心理学中统计的重要性

Psychology relies on scientific research. Because human behavior is variable, researchers use statistics to summarize data, identify patterns, and decide whether results are due to chance or real effects.

心理学依赖科学研究。由于人类行为存在变异性，研究者用统计来概括数据、识别模式，并判断结果是偶然还是真实效应。

Descriptive statistics organize and summarize data.

描述统计用于组织和概括数据。
Inferential statistics help draw conclusions about populations from samples.

推断统计帮助从样本推断总体结论。

2. Descriptive Statistics: Measures of Central Tendency | 描述统计：集中趋势量数

Central tendency describes a typical score in a data set. The three main measures are the mean, median, and mode.

集中趋势描述一组数据中的典型分数。三个主要量数是平均数、中数和众数。

Mean μ = (Σx) / N （总体）; x̄ = (Σx) / n （样本）

The mean is the arithmetic average. It is sensitive to extreme scores (outliers).

平均数是算术平均值，对极端分数（异常值）敏感。

The median is the middle score when scores are ordered. It is resistant to outliers.

中位数是有序排列后处于中间位置的分数，不易受异常值影响。

The mode is the most frequent score.

众数是出现次数最多的分数。

If a distribution is skewed, the median is usually the better measure of center.

如果分布偏态，中位数通常是更好的中心量数。
In a symmetrical, normal distribution, mean ≈ median ≈ mode.

在对称的正态分布中，平均数≈中数≈众数。

3. Measures of Variability | 离散程度量数

Variability tells us how spread out scores are. Key measures include range, variance, and standard deviation.

离散程度告诉我们分数有多分散。关键量数包括全距、方差和标准差。

Range = Maximum score − Minimum score

The range is simple but affected by outliers.

全距简单但易受异常值影响。

Population variance σ² = Σ(x − μ)² / N

Standard deviation σ = √[Σ(x − μ)² / N]

The standard deviation indicates, on average, how far scores deviate from the mean. A larger standard deviation means more spread.

标准差表示分数平均偏离平均数多远。标准差越大，说明分数越分散。

For a sample, we often divide by n − 1 to get s² and s. This provides an unbiased estimate of the population variance.

对于样本，我们通常除以 n − 1 得到 s² 和 s，以获得对总体方差的无偏估计。

4. The Normal Distribution and Standard Deviations | 正态分布与标准差

The normal distribution is a symmetric, bell-shaped curve. Many psychological traits approximate this shape.

正态分布是一条对称的钟形曲线，许多心理特质近似这种分布。

In a normal distribution, about 68% of scores fall within ±1 standard deviation of the mean, about 95% within ±2 standard deviations, and about 99.7% within ±3 standard deviations. This is the empirical rule.

在正态分布中，约68%的分数落在平均数±1个标准差之内，约95%落在±2个标准差之内，约99.7%落在±3个标准差之内，这就是经验法则。

Percentages under the normal curve: 34% + 34% = 68%

The distribution also helps us understand percentile ranks. For example, a score at +1 standard deviation is approximately at the 84th percentile.

正态分布还帮助我们理解百分等级。例如，位于+1个标准差的分数大约处于第84百分位。

5. Z-Scores and Percentiles | Z分数与百分位数

A z-score describes a score’s position relative to the mean in standard deviation units.

z分数以一个分数相对于平均数有多少个标准差来描述其位置。

z = (x − μ) / σ or z = (x − x̄) / s

A positive z-score means the score is above the mean.

正的z分数表示分数高于平均数。
A negative z-score means the score is below the mean.

负的z分数表示分数低于平均数。
A z-score of 0 means the score is exactly at the mean.

z分数为0表示分数正好等于平均数。

Percentiles indicate the percentage of scores below a given score. In a normal distribution, z-scores can be converted to percentiles using a standard normal table.

百分位数表示某个分数之下的分数所占百分比。在正态分布中，可以用标准正态表将z分数转换为百分位数。

6. Correlation and Regression | 相关与回归

Correlation measures the direction and strength of the linear relationship between two variables. It ranges from −1 to +1.

相关衡量两个变量之间线性关系的方向和强度，取值范围从−1到+1。

r = (1 / (n − 1)) Σ [((x − x̄) / sₓ) × ((y − ȳ) / sᵧ)]

r close to +1 means a strong positive relationship.

r接近+1表示强正相关。
r close to −1 means a strong negative relationship.

r接近−1表示强负相关。
r near 0 means little or no linear relationship.

r接近0表示几乎没有线性相关。

Regression uses the line of best fit to predict one variable from another. The regression equation is often written as:

回归使用最佳拟合线从一个变量预测另一个变量。回归方程通常写为：

ŷ = a + bx

Remember: correlation does not imply causation. A third variable may explain the relationship.

记住：相关不等于因果。可能存在第三个变量解释这种关系。

7. Inferential Statistics and Sampling | 推断统计与抽样

Inferential statistics allow psychologists to generalize from a sample to a population. A good sample should be random and representative.

推断统计使心理学家能够从样本推广到总体。好的样本应该是随机且有代表性的。

Population: the entire group of interest.

总体：感兴趣的全部群体。
Sample: a subset of the population.

样本：总体的一部分。
Sampling error: the difference between the sample statistic and the population parameter.

抽样误差：样本统计量与总体参数之间的差异。

The larger the sample, the smaller the sampling error tends to be.

样本越大，抽样误差往往越小。

8. Hypothesis Testing and p-Values | 假设检验与p值

Researchers start with a null hypothesis (H₀) that there is no effect or no difference. They then test it against an alternative hypothesis (H₁).

研究者首先提出零假设（H₀），即没有效应或没有差异，然后与备择假设（H₁）进行检验。

p-value: the probability of obtaining the observed result if H₀ is true.

If p < 0.05, the result is considered statistically significant.

如果p < 0.05，结果被认为具有统计显著性。
If p ≥ 0.05, the result is not statistically significant.

如果p ≥ 0.05，结果不具有统计显著性。

A p-value is not the probability that H₀ is true. It is the probability of the data given H₀.

p值不是H₀为真的概率，而是在H₀成立时得到当前数据的概率。

9. t-Tests, ANOVA, and Chi-Square | t检验、方差分析与卡方

Different research designs require different inferential tests.

不同的研究设计需要不同的推断检验。

Independent t-test: t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)

A t-test compares the means of two groups.

t检验比较两个组的平均数。
ANOVA (F-test) compares the means of three or more groups.

方差分析（F检验）比较三个或更多组的平均数。
Chi-square (χ²) tests the relationship between two categorical variables.

卡方（χ²）检验两个分类变量之间的关系。

Chi-square formula: χ² = Σ (O − E)² / E

Degrees of freedom (df) are used to find critical values from statistical tables.

自由度（df）用于在统计表中查找临界值。

10. Effect Size and Statistical Power | 效应量与统计检验力

Statistical significance does not tell us how large an effect is. Effect size does.

统计显著性不能告诉我们效应有多大，效应量可以。

Cohen’s d = (x̄₁ − x̄₂) / s_pooled

d ≈ 0.2 is a small effect.

d ≈ 0.2 是小的效应。
d ≈ 0.5 is a medium effect.

d ≈ 0.5 是中等效应。
d ≈ 0.8 is a large effect.

d ≈ 0.8 是大的效应。

Statistical power is the probability of correctly rejecting a false null hypothesis. Power increases with larger samples and larger effect sizes.

统计检验力是正确拒绝错误零假设的概率。样本量越大、效应量越大，检验力越高。

11. Common Mistakes in AP Psychology Statistics | AP心理学统计常见错误

Students often confuse terms or misinterpret results. Here are common pitfalls to avoid on the exam.

学生经常混淆术语或误解结果。以下是考试中需要避免的常见陷阱。

Believing correlation proves causation.

认为相关可以证明因果。
Using the mean when the data are highly skewed.

数据高度偏态时仍然使用平均数。
Confusing p-value with effect size.

混淆p值与效应量。
Forgetting that a small sample may not represent the population.

忘记小样本可能无法代表总体。

12. Key Formulas Summary | 关键公式总结

The following table summarizes the essential formulas you should recognize for the AP Psychology exam.

下表总结了你在AP心理学考试中需要识别的关键公式。

Statistic	Formula	Purpose
Mean	x̄ = Σx / n	Average score
Standard deviation	s = √[Σ(x − x̄)² / (n − 1)]	Spread around the mean
Z-score	z = (x − μ) / σ	Position relative to mean
Correlation	r between −1 and +1	Strength of linear relationship
Chi-square	χ² = Σ (O − E)² / E	Categorical data association

Practice interpreting these formulas conceptually. The AP exam asks you to choose the right statistic for a scenario and explain what it means.

练习从概念上理解这些公式。AP考试要求你为某个情境选择正确的统计量并解释其含义。

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AP Psychology Statistics Formulas and Key Concepts | AP心理学统计公式与关键概念梳理