📚 AP Statistics Core Formulas: A Complete Review | AP统计学核心公式汇总与应用
AP Statistics is less about pure calculation and more about knowing which formula applies in which situation. This review organizes the essential formulas by topic, explains their meaning, and highlights common exam applications.
AP统计学不强调纯计算，而是要求你判断在特定情境下该使用哪个公式。本篇文章按主题整理核心公式，解释其含义，并突出常见考试应用。
1. Descriptive Statistics: Measures of Center and Spread | 描述统计：中心与离散趋势
The most basic summaries involve a data set’s center and variability. The sample mean is the arithmetic center, and the sample standard deviation measures how far values typically lie from the mean.
最基础的数据概括涉及数据中心与离散趋势。样本均值是算术中心，样本标准差衡量数据通常偏离均值多远。
x̄ = (Σxᵢ) / n
s² = Σ(xᵢ – x̄)² / (n – 1)
s = √[Σ(xᵢ – x̄)² / (n – 1)]
The five-number summary (min, Q1, median, Q3, max) is used for boxplots. IQR = Q3 – Q1 flags potential outliers using the 1.5 × IQR rule.
五数概括（最小值、Q1、中位数、Q3、最大值）用于绘制箱线图。IQR = Q3 – Q1，并利用 1.5 × IQR 规则判断潜在异常值。
A z-score standardizes a value: z = (x – x̄) / s (sample) or z = (x – μ) / σ (population). This tells how many standard deviations the value is from the mean.
z 分数将数据标准化：z = (x – x̄) / s（样本）或 z = (x – μ) / σ（总体）。它表示该值距均值多少个标准差。
2. Correlation and Regression | 相关与回归
Correlation r measures the strength and direction of a linear relationship between two quantitative variables. It is unitless and always between -1 and 1.
相关系数 r 衡量两个定量变量之间线性关系的强度与方向。它没有单位，始终在 -1 到 1 之间。
r = Σ(xᵢ – x̄)(yᵢ – ȳ) / [√Σ(xᵢ – x̄)² · √Σ(yᵢ – ȳ)²]
The least-squares regression line is ŷ = b₀ + b₁x, where b₁ is the slope and b₀ is the intercept.
最小二乘回归线为 ŷ = b₀ + b₁x，其中 b₁ 是斜率，b₀ 是截距。
b₁ = r · (sᵧ / sₓ) and b₀ = ȳ – b₁x̄
A residual is the difference between observed and predicted y: residual = observed y – predicted y. Residual plots help assess linearity.
残差是观测 y 与预测 y 之差：残差 = 实际观测值 – 预测值。残差图用于判断线性关系是否合理。
3. Probability Rules and Random Variables | 概率规则与随机变量
Probability rules allow you to compute the likelihood of events. The complement rule, addition rule, and multiplication rule are fundamental.
概率规则用于计算事件发生的可能性。补集法则、加法法则和乘法法则是基础。
P(Aᶜ) = 1 – P(A)
P(A or B) = P(A) + P(B) – P(A and B)
P(A and B) = P(A) · P(B|A) = P(B) · P(A|B)
Conditional probability is given by P(B|A) = P(A and B) / P(A). If P(A and B) = P(A)P(B), then A and B are independent.
条件概率定义为 P(B|A) = P(A and B) / P(A)。若 P(A and B) = P(A)P(B)，则 A 与 B 独立。
For a discrete random variable X, the expected value is a weighted average, and the variance measures spread around the mean.
对于离散随机变量 X，期望值是加权平均，方差衡量围绕均值的离散程度。
μₓ = Σxᵢpᵢ
σ²ₓ = Σ(xᵢ – μₓ)²pᵢ
4. Binomial and Geometric Distributions | 二项分布与几何分布
The binomial distribution counts successes in n independent trials with constant probability p. Each trial must have two outcomes.
二项分布用于统计在 n 次独立试验中成功的次数，每次试验成功概率 p 恒定，且每次结果只有两种可能。
P(X = k) = C(n,k) · pᵏ(1 – p)ⁿ⁻ᵏ, where C(n,k) = n! / [k!(n – k)!]
The mean and standard deviation of a binomial random variable are:
二项随机变量的均值与标准差为：
μ = np and σ = √[np(1 – p)]
The geometric distribution counts trials needed to get the first success. Its probability formula and mean are different.
几何分布用于统计直到首次成功所需的试验次数。其概率公式和均值不同。
P(X = k) = (1 – p)ᵏ⁻¹p and μ = 1/p
5. Sampling Distributions and Central Limit Theorem | 抽样分布与中心极限定理
Sampling distributions describe the behavior of sample statistics. For a sample proportion p̂, if np ≥ 10 and n(1 – p) ≥ 10, the distribution is approximately normal.
抽样分布描述样本统计量的行为。对于样本比例 p̂，若 np ≥ 10 且 n(1 – p) ≥ 10，则其分布近似正态。
μ_p̂ = p and σ_p̂ = √[p(1 – p) / n]
For the sample mean x̄, the Central Limit Theorem states that for large n, the distribution of x̄ is approximately normal regardless of the population shape.
对于样本均值 x̄，中心极限定理表明，当 n 较大时，x̄ 的分布近似正态，无论总体形状如何。
μ_x̄ = μ and σ_x̄ = σ / √n
The condition n ≥ 30 is often used for the CLT, but if the population is strongly skewed, a larger sample may be needed.
CLT 的常用条件是 n ≥ 30，但如果总体严重偏斜，则可能需要更大的样本量。
6. Confidence Intervals for a Proportion | 比例的置信区间
A one-sample z-interval for a proportion estimates the unknown true proportion p. The margin of error depends on the critical value z* and sample size.
单样本比例 z 区间用于估计未知总体比例 p。误差范围取决于临界值 z* 和样本量。
p̂ ± z* · √[p̂(1 – p̂) / n]
Conditions: independent random sample, n p̂ ≥ 10, and n(1 – p̂) ≥ 10.
条件：独立随机样本，n p̂ ≥ 10，且 n(1 – p̂) ≥ 10。
To find the required sample size for a desired margin of error m, use a conservative p* = 0.5 when no prior estimate exists.
若要达到预定误差范围 m，计算所需样本量时，如果没有先验估计，通常用保守的 p* = 0.5。
n = (z* / m)² · p*(1 – p*)
7. Confidence Intervals for a Mean | 均值的置信区间
When σ is unknown, we use the t-distribution. The one-sample t-interval for a population mean μ is centered at the sample mean.
当总体标准差 σ 未知时，我们使用 t 分布。总体均值 μ 的单样本 t 区间以样本均值为中心。
x̄ ± t* · s / √n, with df = n – 1
The critical value t* depends on the confidence level and degrees of freedom. The interval becomes narrower as n increases.
临界值 t* 取决于置信水平和自由度。随着 n 增大，区间会变窄。
For a desired margin of error with known σ, the sample size formula is n = (z*σ / m)²; use t* only when estimating σ from data introduces extra uncertainty.
对于已知 σ 且希望达到误差范围 m，样本量公式为 n = (z*σ / m)²；当用数据估计 σ 时，使用 t* 会引入额外不确定性。
8. Hypothesis Testing for a Proportion | 比例的假设检验
A hypothesis test for a proportion uses a z-test statistic based on the null hypothesis proportion p₀, not on p̂.
比例假设检验使用 z 检验统计量，其计算基于原假设中的比例 p₀，而不是 p̂。
z = (p̂ – p₀) / √[p₀(1 – p₀) / n]
The conditions are: random sample, n p₀ ≥ 10, and n(1 – p₀) ≥ 10. Use the p-value to decide whether to reject H₀.
条件为：随机样本，n p₀ ≥ 10，且 n(1 – p₀) ≥ 10。利用 p 值判断是否拒绝 H₀。
Common errors: using p̂ in the standard error for the test, or failing to check the conditions before applying the formula.
常见错误：在检验的标准误中使用 p̂，或未先检验条件就套用公式。
9. Hypothesis Testing for a Mean | 均值的假设检验
For a mean with unknown σ, the t-test statistic compares the sample mean to the hypothesized value μ₀.
当 σ 未知时，t 检验统计量将样本均值与假设值 μ₀ 进行比较。
t = (x̄ – μ₀) / (s / √n), with df = n – 1
If the samples are paired (e.g., before/after), use the differences dᵢ and test the mean difference μ_d. The test becomes a one-sample t-test on the differences.
如果样本是配对的（如前后测量），使用差值 dᵢ 并检验平均差 μ_d。此时检验转化为对差值进行单样本 t 检验。
t = (d̄ – μ₀) / (s_d / √n)
Conditions: random sample, the sample data should be approximately normal (n large or no strong skew/outliers).
条件：随机样本，样本数据应近似正态（n 较大或没有严重偏斜/异常值）。
10. Chi-Square Tests | 卡方检验
The chi-square statistic compares observed counts to expected counts under the null hypothesis. It is used for categorical data.
卡方统计量比较观测频数与原假设下的期望频数，用于分类数据。
χ² = Σ (Observed – Expected)² / Expected
For goodness-of-fit, df = categories – 1. For tests of independence or homogeneity, df = (rows – 1)(columns – 1).
适合性检验的自由度 df = 类别数 – 1。独立性或齐性检验的自由度 df = (行数 – 1)(列数 – 1)。
All expected counts should be at least 5. The chi-square test is always right-tailed, and each observation can belong to only one category.
所有期望频数应至少为 5。卡方检验始终是右尾检验，且每个观测只能属于一个类别。
11. Inference for Linear Regression | 线性回归的推断
In regression inference, we test the slope β₁ to see if there is a statistically significant linear relationship between x and y.
在回归推断中，我们检验斜率 β₁ 以判断 x 与 y 之间是否存在统计上显著的线性关系。
The standard error of the slope measures how much the sample slope b₁ would vary from sample to sample.
斜率的标准误衡量样本斜率 b₁ 在不同样本间的波动程度。
SE₍b₁₎ = s / √(Σ(xᵢ – x̄)²), where s = √[SSE / (n – 2)]
A confidence interval for the true slope β₁ is b₁ ± t* · SE₍b₁₎, with df = n – 2. The test statistic is t = (b₁ – β₀) / SE₍b₁₎, usually testing β₀ = 0.
真实斜率 β₁ 的置信区间为 b₁ ± t* · SE₍b₁₎，自由度 df = n – 2。检验统计量为 t = (b₁ – β₀) / SE₍b₁₎，通常检验 β₀ = 0。
12. Choosing the Right Inference Procedure | 如何选择正确的推断方法
When planning an exam response, first identify the type of data (categorical or quantitative), the number of samples, and whether you are estimating or testing.
作答时，首先判断数据类型（分类还是定量）、样本数量，以及目标是估计还是检验。
| Data | 1 sample | 2 samples |
| Categorical proportion | z interval/z test | 2-proportion z test |
| Quantitative mean | t interval/t test | 2-sample t test or paired t test |
| Categorical counts (multiple categories) | Chi-square goodness of fit | Chi-square independence/homogeneity |
Always justify the chosen procedure by naming the test and stating the conditions. Memorizing these formulas is useless without connecting them to their underlying assumptions.
务必说明所选方法，并写出适用条件。如果不知道公式背后的假设，单纯记忆公式是没有意义的。
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