AP Statistics Formula Sheet & Booklet
Complete Unit-by-Unit Reference with All Formulas, Theorems & Key Concepts
Welcome to your comprehensive AP Statistics Formula Booklet! This is your complete, one-stop resource for all AP Statistics formulas, theorems, and key concepts organized by unit. Whether you're preparing for the 2026 AP Exam or need a quick reference sheet, this booklet contains everything you need to succeed.
1
Exploring One-Variable Data (15-23% of Exam)
Sample Mean
\( \bar{x} = \frac{\sum x_i}{n} \)
Average value of all observations in a sample.
Used to find the center of a distribution.
Sample Standard Deviation
\( s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} \)
Measures spread of data around the mean. Divide by (n-1) for sample data.
The sample SD uses (n-1) to make it an unbiased estimator.
Interquartile Range (IQR)
\( IQR = Q_3 - Q_1 \)
Range containing the middle 50% of data.
Outlier Rule: Outlier if value < Q₁ - 1.5(IQR) or value > Q₃ + 1.5(IQR)
Standardized Score (z-score)
\( z = \frac{x - \mu}{\sigma} \)
Measures how many standard deviations a value is from the mean.
z-score tells you position relative to the mean.
Population Standard Deviation
\( \sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}} \)
Measures spread of entire population data.
Use N (not n-1) when you have the entire population.
Empirical Rule (68-95-99.7)
Normal Distribution:
• 68% within 1σ of mean
• 95% within 2σ of mean
• 99.7% within 3σ of mean
• 68% within 1σ of mean
• 95% within 2σ of mean
• 99.7% within 3σ of mean
Approximation for data following normal distribution.
Only applies to approximately normal distributions.
Unit 1 Key Concepts: Center (mean, median, mode), Spread (range, IQR, standard deviation), Shape (skewness, outliers), Normal Distribution, z-scores
2
Exploring Two-Variable Data (5-7% of Exam)
Linear Regression Line
\( \hat{y} = a + bx \)
Equation of the best-fit line for predicting y from x.
Only use for predictions within the range of observed data.
Slope (b)
\( b = r\frac{s_y}{s_x} \)
Change in y for each unit increase in x.
r is correlation coefficient; sy and sx are standard deviations.
y-intercept (a)
\( a = \bar{y} - b\bar{x} \)
Value of y when x = 0.
Regression line always passes through (x̄, ȳ).
Correlation Coefficient (r)
\( r = \frac{1}{n-1}\sum\left(\frac{x_i - \bar{x}}{s_x}\right)\left(\frac{y_i - \bar{y}}{s_y}\right) \)
Measures strength and direction of linear relationship. Range: -1 to +1.
|r| close to 1 = strong linear relationship; |r| close to 0 = weak.
Coefficient of Determination (R²)
\( R^2 = r^2 \)
Proportion of variation in y explained by x. Range: 0 to 1.
If R² = 0.85, then 85% of variation in y is explained by x.
Residual
\( residual = y - \hat{y} = observed - predicted \)
Difference between observed and predicted y values.
Good regression models have residuals randomly scattered around 0.
Unit 2 Key Concepts: Scatterplots, Correlation, Linear Regression, Residuals, R², Causation vs. Correlation
3
Collecting Data (12-15% of Exam)
Sources of Bias
Types:
• Sampling Bias
• Nonresponse Bias
• Response Bias
• Voluntary Response Bias
• Sampling Bias
• Nonresponse Bias
• Response Bias
• Voluntary Response Bias
Systematic errors that affect data collection accuracy.
Random sampling reduces sampling bias.
Sampling Methods
Random Methods:
• Simple Random Sample (SRS)
• Stratified Random Sample
• Cluster Sample
• Systematic Sample
• Simple Random Sample (SRS)
• Stratified Random Sample
• Cluster Sample
• Systematic Sample
Techniques for selecting representative samples.
Random methods reduce bias in sampling.
Experimental Design Principles
Key Principles:
• Control (Control Group)
• Randomization
• Replication
• Blocking
• Control (Control Group)
• Randomization
• Replication
• Blocking
Proper experiment design allows causal conclusions.
Experiments establish causation; observational studies do not.
Completely Randomized Design
Subjects → Random Assignment → Treatment Groups → Compare Results
Subjects randomly assigned to treatment or control.
Randomization balances unknown variables.
Matched Pairs Design
Matched Pairs → Random Assignment of Treatments → Compare Results
Controls for confounding variables by pairing similar subjects.
More powerful than completely randomized design when effective pairs can be made.
Randomized Block Design
Subjects → Divide into Blocks → Random Assignment within Blocks → Compare
Blocks subjects by known variable, then randomizes within blocks.
Controls for blocked variable effects.
Unit 3 Key Concepts: Sampling vs. Census, Bias Types, Sampling Methods, Observational Studies, Experimental Design, Confounding Variables
4
Probability, Random Variables & Probability Distributions (10-20% of Exam)
Addition Rule for Probability
\( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
Probability of A or B occurring.
Subtract P(A ∩ B) to avoid double-counting.
Multiplication Rule for Probability
\( P(A \cap B) = P(A) \times P(B|A) \)
Probability of both A and B occurring.
If events are independent: P(A ∩ B) = P(A) × P(B)
Conditional Probability
\( P(A|B) = \frac{P(A \cap B)}{P(B)} \)
Probability of A given that B has occurred.
Read as "probability of A given B".
Complement Rule
\( P(A^c) = 1 - P(A) \)
Probability of NOT A (complement of A).
Useful when easier to calculate probability of complement.
Independence
Events A and B are independent if:
P(A|B) = P(A), or
P(A ∩ B) = P(A) × P(B)
P(A|B) = P(A), or
P(A ∩ B) = P(A) × P(B)
Occurrence of one event doesn't affect probability of other.
Independence is assumed in many statistical tests.
Expected Value (Mean) of Random Variable
\( \mu_X = E(X) = \sum x_i P(x_i) \)
Long-run average value of random variable.
Also called the mean of the probability distribution.
Variance of Random Variable
\( \sigma_X^2 = \sum (x_i - \mu_X)^2 P(x_i) \)
Measure of spread in probability distribution.
Standard deviation is square root of variance.
Binomial Probability
\( P(X = k) = \binom{n}{k}p^k(1-p)^{n-k} \)
Probability of exactly k successes in n trials.
Conditions: Fixed n, two outcomes, constant p, independent trials.
Binomial Mean & Standard Deviation
\( \mu_X = np \)
\( \sigma_X = \sqrt{np(1-p)} \)
\( \sigma_X = \sqrt{np(1-p)} \)
Expected value and spread of binomial distribution.
Use these for binomial random variables.
Geometric Probability
\( P(X = k) = (1-p)^{k-1}p \)
Probability first success on kth trial.
Trials continue until first success occurs.
Geometric Mean & Standard Deviation
\( \mu_X = \frac{1}{p} \)
\( \sigma_X = \frac{\sqrt{1-p}}{p} \)
\( \sigma_X = \frac{\sqrt{1-p}}{p} \)
Expected value and spread of geometric distribution.
Mean is average number of trials until first success.
Normal Distribution
Characteristics:
• Symmetric, bell-shaped curve
• Mean = Median = Mode
• 68-95-99.7 Rule applies
• Continuous probability distribution
• Symmetric, bell-shaped curve
• Mean = Median = Mode
• 68-95-99.7 Rule applies
• Continuous probability distribution
Most common probability distribution in statistics.
Use z-scores to find probabilities with standard normal table.
Unit 4 Key Concepts: Basic Probability Rules, Random Variables, Probability Distributions, Binomial Distribution, Geometric Distribution, Normal Distribution
5
Sampling Distributions (7-12% of Exam)
Central Limit Theorem
For large n:
Distribution of sample means is approximately normal with
\( \mu_{\bar{x}} = \mu \)
\( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \)
Distribution of sample means is approximately normal with
\( \mu_{\bar{x}} = \mu \)
\( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \)
Foundation of statistical inference; applies to sample means and proportions.
n ≥ 30 generally considered "large enough" unless strongly skewed.
Standard Error of Sample Mean
\( SE(\bar{x}) = \frac{s}{\sqrt{n}} \)
Estimate of standard deviation of sample mean.
Use when population standard deviation σ is unknown.
Standard Deviation of Sample Mean
\( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \)
True standard deviation of sample mean (when σ known).
Decreases as sample size increases.
Sampling Distribution of Sample Proportion
\( \mu_{\hat{p}} = p \)
\( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \)
\( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \)
Mean and standard deviation of sample proportion distribution.
Standard error: \( SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \)
Difference of Sample Proportions
\( \mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2 \)
\( \sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} \)
\( \sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} \)
Sampling distribution for difference between two proportions.
Use for two-proportion inference.
Difference of Sample Means
\( \mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2 \)
\( \sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \)
\( \sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \)
Sampling distribution for difference between two means.
Use for two-sample mean inference.
Point Estimators
Unbiased Estimators:
• \( \bar{x} \) estimates μ
• \( \hat{p} \) estimates p
• \( s \) estimates σ (approximately)
• \( \bar{x} \) estimates μ
• \( \hat{p} \) estimates p
• \( s \) estimates σ (approximately)
Sample statistics used to estimate population parameters.
Unbiased means expected value equals parameter.
Unit 5 Key Concepts: Central Limit Theorem, Sampling Distributions, Standard Error, Unbiased Estimators, Variability in Statistics
6
Inference for Categorical Data: Proportions (12-15% of Exam)
General Confidence Interval Formula
\( (statistic) \pm (critical\ value)(SE) \)
General form for all confidence intervals.
Always follows: point estimate ± margin of error.
CI for One Proportion
\( \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \)
Confidence interval for single population proportion.
Conditions: n·p̂ ≥ 10 and n(1-p̂) ≥ 10 (normality condition).
CI for Difference of Two Proportions
\( (\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \)
Confidence interval for difference between two proportions.
Check all four normality conditions for both samples.
General Significance Test
\( test\ statistic = \frac{(statistic) - (parameter)}{SE} \)
General form for all hypothesis tests.
Compare to z or t critical value; calculate p-value.
One-Proportion z-Test
\( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \)
Test for single population proportion against hypothesized value.
Test claims about proportion; uses p₀ (hypothesized value).
Two-Proportion z-Test
\( z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1-\hat{p}_c)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \)
Test for difference between two population proportions.
Where \( \hat{p}_c = \frac{x_1 + x_2}{n_1 + n_2} \) (pooled proportion).
Pooled Sample Proportion
\( \hat{p}_c = \frac{x_1 + x_2}{n_1 + n_2} \)
Combined proportion when testing equality of two proportions.
Used only in two-proportion z-test, not in CI.
Type I and Type II Errors
Type I (α): Reject H₀ when true (false positive)
Type II (β): Fail to reject H₀ when false (false negative)
Type II (β): Fail to reject H₀ when false (false negative)
Possible errors in hypothesis testing.
Significance level (α) is probability of Type I error.
p-Value Interpretation
p-value is: Probability of observing test statistic as extreme or more extreme, given H₀ is true
Measure of evidence against null hypothesis.
Smaller p-value = stronger evidence against H₀. Reject H₀ if p < α.
Unit 6 Key Concepts: Confidence Intervals, Hypothesis Testing, One-Proportion Tests, Two-Proportion Tests, p-Values, Type I & II Errors, Significance Level
7
Inference for Quantitative Data: Means (10-18% of Exam)
CI for One Mean (σ known)
\( \bar{x} \pm z^* \frac{\sigma}{\sqrt{n}} \)
Confidence interval for population mean (population SD known).
Rarely used in practice; σ usually unknown.
CI for One Mean (σ unknown)
\( \bar{x} \pm t^* \frac{s}{\sqrt{n}} \)
Confidence interval for population mean (population SD unknown).
Most common; uses t-distribution with df = n - 1.
Standard Error for Sample Mean
\( SE(\bar{x}) = \frac{s}{\sqrt{n}} \)
Estimate of standard deviation of sample mean.
Use s (sample SD) when σ is unknown.
One-Sample t-Test
\( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \)
Test for population mean against hypothesized value.
Degrees of freedom = n - 1.
Paired t-Test
\( t = \frac{\bar{x}_d - \mu_d}{s_d/\sqrt{n}} \)
Test for mean difference in paired/dependent samples.
Calculate differences first, then test; df = n - 1.
Two-Sample t-Test (Independent)
\( t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \)
Test for difference between two independent sample means.
Use when samples are independent; df calculated from formula or use conservative approach (min(n₁-1, n₂-1)).
Welch's Approximation (df)
\( df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}} \)
Calculates degrees of freedom for two-sample t-tests.
Use technology or conservative df = min(n₁-1, n₂-1).
CI for Difference of Two Means
\( (\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \)
Confidence interval for difference between two population means.
Use t-critical values with appropriate degrees of freedom.
Conditions for t-Procedures
Requirements:
• Random sample
• Independence (10% rule)
• Normality (n ≥ 30 or check for severe skewness/outliers)
• Random sample
• Independence (10% rule)
• Normality (n ≥ 30 or check for severe skewness/outliers)
Must be satisfied for valid t-inference procedures.
Always check conditions before using t-procedures.
Unit 7 Key Concepts: t-Distribution, One-Sample t-Tests, Two-Sample t-Tests, Paired t-Tests, Confidence Intervals for Means, Degrees of Freedom
8
Inference for Categorical Data: Chi-Square (2-5% of Exam)
Chi-Square Test Statistic
\( \chi^2 = \sum \frac{(Observed - Expected)^2}{Expected} \)
Measures how well observed data fit expected distribution.
Large χ² indicates significant difference from expected; df based on table structure.
Goodness of Fit Test
Hypotheses:
H₀: Distribution follows specific pattern
Hₐ: Distribution doesn't follow pattern
H₀: Distribution follows specific pattern
Hₐ: Distribution doesn't follow pattern
Tests whether categorical data follow expected distribution.
df = number of categories - 1.
Test of Independence/Homogeneity
H₀: Variables are independent (or proportions equal)
Hₐ: Variables are associated (or proportions differ)
Hₐ: Variables are associated (or proportions differ)
Tests relationship between two categorical variables.
df = (rows - 1)(columns - 1) for contingency tables.
Expected Frequency
\( Expected = \frac{(Row\ Total)(Column\ Total)}{Table\ Total} \)
Calculated frequency assuming independence.
All expected frequencies should be ≥ 5 for valid test.
Contingency Table Analysis
| Column 1 | Column 2 | Total | |
|---|---|---|---|
| Row 1 | Count | Count | Row Total |
| Row 2 | Count | Count | Row Total |
| Total | Col Total | Col Total | Grand Total |
Organized data display for two categorical variables.
Calculate expected frequencies for each cell.
Conditions for Chi-Square
Requirements:
• Random sample
• Independence
• All expected frequencies ≥ 5
• Random sample
• Independence
• All expected frequencies ≥ 5
Must be satisfied for valid chi-square test.
Combine categories if expected < 5 (sometimes).
Unit 8 Key Concepts: Chi-Square Distribution, Goodness of Fit Tests, Tests of Independence/Homogeneity, Contingency Tables, Expected Frequencies
9
Inference for Quantitative Data: Slopes (2-5% of Exam)
Standard Error of the Slope
\( SE(b) = \frac{s_e}{\sqrt{\sum(x_i - \bar{x})^2}} \)
Standard deviation of slope estimate from sample to sample.
se (residual standard error) = \( \sqrt{\frac{\sum(y_i - \hat{y}_i)^2}{n-2}} \)
CI for Slope of Regression Line
\( b \pm t^* \cdot SE(b) \)
Confidence interval for true regression slope (β).
df = n - 2; use t-distribution with n - 2 degrees of freedom.
Slope t-Test
\( t = \frac{b - \beta_0}{SE(b)} \)
Test for significance of regression slope.
Usually tests H₀: β = 0 (no relationship); df = n - 2.
Regression Equation with Inference
Point Estimate: \( \hat{y} = a + bx \)
With Interval: \( \hat{y} \pm t^* \cdot se \)
With Interval: \( \hat{y} \pm t^* \cdot se \)
Make predictions with confidence intervals from regression.
Prediction interval is wider than confidence interval for mean.
Confidence Interval vs Prediction Interval
CI for mean y: \( \hat{y} \pm t^* \sqrt{\frac{s_e^2}{n} + \frac{(x^*-\bar{x})^2}{\sum(x_i-\bar{x})^2}} \)
PI for individual y: \( \hat{y} \pm t^* \sqrt{s_e^2 + \frac{s_e^2}{n} + \frac{(x^*-\bar{x})^2}{\sum(x_i-\bar{x})^2}} \)
PI for individual y: \( \hat{y} \pm t^* \sqrt{s_e^2 + \frac{s_e^2}{n} + \frac{(x^*-\bar{x})^2}{\sum(x_i-\bar{x})^2}} \)
CI estimates average y for given x; PI estimates individual y values.
PI always wider because individual values vary more than averages.
Conditions for Regression Inference
Requirements (LINE):
• Linearity: Relationship is linear
• Independence: Residuals independent
• Normality: Residuals approximately normal
• Equal Variance: Constant residual spread
• Linearity: Relationship is linear
• Independence: Residuals independent
• Normality: Residuals approximately normal
• Equal Variance: Constant residual spread
Must be satisfied for valid regression inference.
Check with residual plot (should show random scatter).
Residual Analysis
Residuals should:
• Show random scatter around y = 0
• Be approximately normally distributed
• Have constant spread across x values
• Show random scatter around y = 0
• Be approximately normally distributed
• Have constant spread across x values
Visual check of regression model assumptions.
Patterns in residual plot indicate violations of conditions.
Outliers and Influential Points
Outlier: Unusual y value (large residual)
Influential Point: Significantly affects regression line
Influential Point: Significantly affects regression line
Special observations requiring investigation.
High leverage points (extreme x) with large residuals are most influential.
Unit 9 Key Concepts: Regression Inference, Slope Tests, Confidence Intervals for Slope, Prediction Intervals, Regression Conditions (LINE), Residual Analysis
Frequently Asked Questions About AP Statistics Formulas
Q: Do I need to memorize all these formulas for the AP Statistics exam?
A: No! The College Board provides a formula sheet during the exam that includes most formulas you'll need. However, understanding when and how to use each formula is critical. Focus on mastering the concepts and applications rather than pure memorization. A solid understanding will help you work faster during the exam and make fewer mistakes.
Q: How do I decide which statistical test to use?
A: Ask yourself these questions: (1) Am I analyzing categorical or quantitative data? (2) Am I testing one population or comparing two? (3) Are the samples independent or paired? (4) For proportions, use z-tests; for means, use t-tests; for categorical use chi-square. Creating a decision tree helps organize your thinking.
Q: What does a p-value really mean?
A: A p-value is the probability of observing a test statistic as extreme as yours, assuming H₀ is true. Smaller p-values provide stronger evidence against the null hypothesis. If p < α (typically 0.05), reject H₀. Remember: it's NOT the probability that H₀ is true!
Q: What's the difference between z and t distributions?
A: Use z-distributions when the population standard deviation (σ) is known or when working with proportions and large samples. Use t-distributions when the population standard deviation is unknown and you're estimating it with sample SD (s). The t-distribution has heavier tails and accounts for additional uncertainty; it approaches the z-distribution as sample size increases.
Q: What conditions must I check before using a particular test?
A: Always check: (1) Random/representative sample, (2) Independence (n < 10% of population), (3) Appropriate sample size or distribution shape (normality). Write these in your work! Violating conditions can invalidate your results. For specific tests: proportion tests need n·p̂ ≥ 10, mean tests need n ≥ 30 or normal data, chi-square needs all expected frequencies ≥ 5.
Q: How are confidence level and sample size related?
A: Increasing sample size decreases margin of error (narrower CI), making your estimate more precise. Increasing confidence level (95% to 99%) increases margin of error (wider CI), making your interval more likely to contain the parameter. It's a trade-off: larger n gives both narrower intervals and higher confidence, but cost increases with sample size.
Q: What are Type I and Type II errors, and which is worse?
A: Type I error (α): Rejecting H₀ when it's true (false positive). Type II error (β): Failing to reject H₀ when it's false (false negative). Which is worse depends on context. If testing a new drug (H₀ = not effective), Type I error (approving ineffective drug) is dangerous. If testing environmental contamination, Type II error (missing actual contamination) might be worse.
Q: When should I use paired t-test vs. two-sample t-test?
A: Use paired t-test when: (1) Same subjects measured twice (before/after), (2) Matched pairs (twins, siblings), or (3) Dependent samples. Use two-sample t-test when: samples are independent with different subjects. Always identify which type first—using the wrong test is a major error!
Q: What does R² tell me about my regression model?
A: R² (coefficient of determination) tells you the proportion of variation in y explained by x. If R² = 0.82, then 82% of variation in y is explained by the linear model; 18% is due to other factors. Higher R² means better fit, but a strong R² doesn't guarantee causation or that the model is appropriate.
Q: How do I interpret regression slope in context?
A: If slope = 2.5 and units are (y in dollars, x in years), say: "For each additional year, the predicted value increases by $2.50" or "A one-year increase in x is associated with a $2.50 increase in predicted y." Always include units and be careful about causation language in observational studies.