How to automate AI workflows in Microsoft’s Azure and Fabric, despite marketing rebrands, and passing AI-900 & AI-102 certification exams.
AP Statistics
AP Statistics is College Board’s most popular exam.
The AP Statistics Exam is Thu, May 7, 2026 at 12 PM Local
https://apstudents.collegeboard.org/courses/ap-Statistics
AP® Statistics gives students hands-on experience collecting, analyzing, graphing, and interpreting real-world data. They will learn to effectively design and analyze research studies by reviewing and evaluating real research examples taken from daily life. The next time they hear the results of a poll or study, they will know whether the results are valid. As the art of drawing conclusions from imperfect data and the science of real-world uncertainties, statistics plays an important role in many fields. The equivalent of an introductory college-level course, AP® Statistics prepares students for the AP exam and for further study in science, sociology, medicine, engineering, political science, geography, and business.
Skill Categories:
A. Selecting Statistical Methods = Select methods for collecting and/or analyzing data for statistical
inference.
B. Data Analysis = Describe patterns, trends, associations, and relationships in data.
C. Using Probability and Simulation = to describe probability distributions and define uncertainty in statistical inference. Explore random phenomena.
D. Statistical Argumentation = Develop an explanation or justify a conclusion using evidence from data,
definitions, or statistical inference. Use statistical reasoning to draw appropriate conclusions and justify claims.
REMEMBER: Graphing calculators (TI-85) are allowed during exams. No access to Excel.
https://apcentral.collegeboard.org/courses/ap-statistics
Exam Description PDF (Effective Fall 2020)
| Unit | Weight | Khan | Excel | Python |
|---|
| 1. Exploring One-Variable Data | 15-23%
| AP
| - | - |
| 2. Exploring Two-Variable Data | 5-7%
| AP
| - | - |
| 3. Collecting Data | 12-15%
| AP
| - | - |
| 4. Probability, Random Variables, and Probability Distributions | 10-20%
| AP
| - | - |
| 5. Sampling Distributions | 7-12%
| AP
| - | - |
| 6. Inference for Categorical Data: Proportions | 12-15%
| AP
| - | - |
| 7. Inference for Quantitative Data: Means | 10-18%
| AP
| - | - |
| 8. Inference for Categorical Data: Chi-square | 2-5%
| AP
| - | - |
| 9. Inference for Quantitative Data: Slopes | 2-65%
| AP
| [_]
</td> | - | </tr>
| * Prepare for the 2022* AP Statistics Exam | -
| AP
| - | - |
</table>
## Graphics Presentation Formats
* Bar (bars reflecting the relative sizes of observations from different groups)
* Box and whisker [graphical or text based] (visual representation of data distribution statistics)
* Spread (visualisation of raw data distribution)
* Histogram [graphical or text based] (bars reflecting data distribution shape)
* Control (statistical process control charts)
* Forest (meta-analysis plots used by Cochrane Collaboration)
* Error bar (scatter or line plots reflecting the uncertainty of measures on the vertical axis)
* Ladder (differences between pairs of measurements displayed as rungs of a ladder)
* Normal (normal scores plotted against the raw variable)
* Scatter [graphical or text based] (conventional way to show the relation between two variables)
* Survival (step plots of survivorship or hazard functions)
* ROC (receiver operating characteristic curves of sensitivity vs. 1-specificity)
* Population pyramid (visual representation of the age structure of a population)
see https://www.statsdirect.com/help/references/glossary.htm
## Core Symbols and Abbreviations
| Symbol | Greek | Meaning |
|--------|------|-------------------------------------------|
| α | alpha | Significance level (Type I error rate) |
| β | Beta | Type II error rate (probability of false negative) |
| μ | mu | Population mean (average) |
| n | - | Sample size (N = population size) |
| σ | sigma | Population standard deviation |
| Var or s² | - | Variance (measure of dispersion) |
| P | - | Probability of an event or data sample |
| p̂ | circumflex
U0302 | Sample proportion (estimate of population proportion) |
| R | - | Correlation coefficient (linear relationship strength) |
| t | - | Student’s t-statistic for small-sample inference |
| z | - | Standardized score or test statistic divided by SE |
## Common Statistical Terms
| Term | Definition |
|-------------------------|-----------------------------------------------------------------|
| Alternative hypothesis (H₁) | Hypothesis that there is a significant effect or difference |
| ANOVA | Analysis of variance; tests means across multiple groups |
| Chi-square test (χ²) | Test comparing observed and expected frequencies |
| Confidence Interval (CI)| Range estimating the true population parameter |
| Correlation | Strength and direction of relationship between two variables |
| Histogram | Graph displaying frequency distribution of data |
| Kurtosis | Measure of tail heaviness or peakedness of a distribution |
| Normal distribution | Bell-shaped distribution symmetric about the mean |
| Null hypothesis (H₀) | Initial assumption of no effect |
| Outlier | Data point that differs significantly from others |
| p-value | Probability of obtaining results as extreme as observed if H₀ is true |
| Population | Complete group from which samples are drawn |
| Random variable | Variable whose value is subject to randomness |
| Regression analysis | Method estimating relationships between dependent and independent variables |
| Sample | Subset drawn from a population to represent it in analysis |
| Skewness | Measure of asymmetry in the distribution |
| Standard deviation | Square root of variance; average distance from mean |
| Time series | Data collected over time at regular intervals |
| Variance | Average squared deviation from the mean; measures spread |
## Acronyms in Statistics
| Acronym | Full Form | Notes |
|---------|-------------------------------|------------------------------------------|
| ANOVA | Analysis of Variance | Compares group means |
| ANCOVA | Analysis of Covariance | Controls for covariates |
| BIC/AIC | Bayesian/ Akaike Information Criterion | Used for model selection |
| CI | Confidence Interval | Indicates estimate precision |
| CI95 | 95% Confidence Interval | Standard interval for hypothesis testing |
| df | Degrees of Freedom | Number of independent values minus constraints |
| IQR | Interquartile Range | Distance between 25th and 75th percentile (spread of middle 50%) |
| RMSE | Root Mean Square Error | Common measure of model accuracy |
| R² | Coefficient of Determination | Percentage of variance explained by model|
| SD | Standard deviation of a sample |
| SE | Standard error (uncertainty around sample)|
| SEM | Standard Error of the Mean | Average variability of sample means |
| SS | Sum of Squares | Total variation in data |
| SRS | simple random sample | a sample taken so that each member and set of \[n\] members has an equal chance of being in the sample. |
## Khan Academy
https://www.khanacademy.org/math/ap-statistics
Unit 1: Exploring categorical data
* The language of variation: Variables: Exploring categorical data
* Representing a categorical variable with graphs: Exploring categorical data
* Representing two categorical variables: Exploring categorical data
* Statistics for two categorical variables: Exploring categorical data
Unit 2: Exploring one-variable quantitative data: Displaying and describing
* Representing a quantitative variable with dot plots: Exploring one-variable quantitative data: Displaying and describing
* Representing a quantitative variable with histograms and stem plots: Exploring one-variable quantitative data: Displaying and describing
* Describing the distribution of a quantitative variable: Exploring one-variable quantitative data: Displaying and describing
* Comparing distributions of a quantitative variable: Exploring one-variable quantitative data: Displaying and describing
Unit 3: Exploring one-variable quantitative data: Summary statistics
* Measuring center in quantitative data: Exploring one-variable quantitative data: Summary statistics
* More on mean and median: Exploring one-variable quantitative data: Summary statistics
* Measuring variability in quantitative data: Exploring one-variable quantitative data: Summary statistics
* Effects of linear transformations: Exploring one-variable quantitative data: Summary statistics
* More on standard deviation (optional): Exploring one-variable quantitative data: Summary statistics
* Graphical representations of summary statistics: Exploring one-variable quantitative data: Summary statistics
Unit 4: Exploring one-variable quantitative data: Percentiles, z-scores, and the normal distribution
* Percentiles: Exploring one-variable quantitative data: Percentiles, z-scores, and the normal distribution
* Z-scores: Exploring one-variable quantitative data: Percentiles, z-scores, and the normal distribution
* Density curves: Exploring one-variable quantitative data: Percentiles, z-scores, and the normal distribution
* Normal distributions and the empirical rule: Exploring one-variable quantitative data: Percentiles, z-scores, and the normal distribution
* Normal distribution calculations: Exploring one-variable quantitative data: Percentiles, z-scores, and the normal distribution
Unit 5: Exploring two-variable quantitative data
* Representing the relationship between two quantitative variables: Exploring two-variable quantitative data
* Correlation: Exploring two-variable quantitative data
* Residuals: Exploring two-variable quantitative data
* Least-squares regression: Exploring two-variable quantitative data
* Analyzing departures from linearity: Exploring two-variable quantitative data
Unit 6: Collecting data
* Introduction to planning a study: Collecting data
* Potential problems with sampling: Collecting data
* Random sampling and data collection: Collecting data
* Introduction to experimental design: Collecting data
* Inference and experiments: Collecting data
Unit 7: Probability
* Estimating probabilities using simulation: Probability
* Mutually exclusive events and unions of events: Probability
* Conditional probability: Probability
* Independent versus dependent events and the multiplication rule: Probability
Unit 8: Random variables and probability distributions
* Introduction to random variables and probability distributions: Random variables and probability distributions
* Mean and standard deviation of random variables: Random variables and probability distributions
* Transforming random variables: Random variables and probability distributions
* Combining random variables: Random variables and probability distributions
* Introduction to the binomial distribution: Random variables and probability distributions
* Parameters for a binomial distribution: Random variables and probability distributions
* The geometric distribution: Random variables and probability distributions
Unit 9: Sampling distributions
* The normal distribution, revisited: Sampling distributions
* The central limit theorem: Sampling distributions
* Biased and unbiased point estimates: Sampling distributions
* Sampling distributions for sample proportions: Sampling distributions
* Sampling distributions for differences in sample proportions: Sampling distributions
* Sampling distributions for sample means: Sampling distributions
* Sampling distributions for differences in sample means: Sampling distributions
Unit 10: Inference for categorical data: Proportions
* Introduction to confidence intervals: Inference for categorical data: Proportions
* Confidence intervals for proportions: Inference for categorical data: Proportions
* The idea of significance tests: Inference for categorical data: Proportions
* Setting up a test for a population proportion: Inference for categorical data: Proportions
* Carrying out a test for a population proportion: Inference for categorical data: Proportions
* Concluding a test for a population proportion: Inference for categorical data: Proportions
* Potential errors when performing tests: Inference for categorical data: Proportions
* Confidence intervals for the difference of two proportions: Inference for categorical data: Proportions
* Testing for the difference of two population proportions: Inference for categorical data: Proportions
Unit 11: Inference for quantitative data: Means
* Constructing a confidence interval for a population mean: Inference for quantitative data: Means
* Setting up a test for a population mean: Inference for quantitative data: Means
* Carrying out a test for a population mean: Inference for quantitative data: Means
* Confidence intervals for the difference of two means: Inference for quantitative data: Means
* Testing for the difference of two population means: Inference for quantitative data: Means
Unit 12: Inference for categorical data: Chi-square
* Chi-square test for goodness of fit: Inference for categorical data: Chi-square
* Chi square tests for relationships (homogeneity or independence): Inference for categorical data: Chi-square
Unit 13: Inference for quantitative data: slopes
* Confidence intervals for the slope of a regression model: Inference for quantitative data: slopes
* Testing for the slope of a regression model: Inference for quantitative data: slopes
Unit 14: Prepare for the 2022 AP®︎ Statistics Exam
https://www.khanacademy.org/math/statistics-probability
Unit 1: Analyzing categorical data
* Analyzing one categorical variable: Analyzing categorical data
* Two-way tables: Analyzing categorical data
* Distributions in two-way tables: Analyzing categorical data
Unit 2: Displaying and comparing quantitative data
* Displaying quantitative data with graphs: Displaying and comparing quantitative data
* Describing and comparing distributions: Displaying and comparing quantitative data
* More on data displays: Displaying and comparing quantitative data
Unit 3: Summarizing quantitative data
* Measuring center in quantitative data: Summarizing quantitative data
* More on mean and median: Summarizing quantitative data
* Interquartile range (IQR): Summarizing quantitative data
* Variance and standard deviation of a population: Summarizing quantitative data
* Variance and standard deviation of a sample: Summarizing quantitative data
* More on standard deviation: Summarizing quantitative data
* Box and whisker plots: Summarizing quantitative data
* Other measures of spread: Summarizing quantitative data
Unit 4: Modeling data distributions
* Percentiles: Modeling data distributions
* Z-scores: Modeling data distributions
* Effects of linear transformations: Modeling data distributions
* Density curves: Modeling data distributions
* Normal distributions and the empirical rule: Modeling data distributions
* Normal distribution calculations: Modeling data distributions
* More on normal distributions: Modeling data distributions
Unit 5: Exploring bivariate numerical data
* Introduction to scatterplots: Exploring bivariate numerical data
* Correlation coefficients: Exploring bivariate numerical data
* Introduction to trend lines: Exploring bivariate numerical data
* Least-squares regression equations: Exploring bivariate numerical data
* Assessing the fit in least-squares regression: Exploring bivariate numerical data
* More on regression: Exploring bivariate numerical data
Unit 6: Study design
* Statistical questions: Study design
* Sampling and observational studies: Study design
* Sampling methods: Study design
* Types of studies (experimental vs. observational): Study design
* Experiments: Study design
Unit 7: Probability
* Basic theoretical probability: Probability
* Probability using sample spaces: Probability
* Basic set operations: Probability
* Experimental probability: Probability
* Randomness, probability, and simulation: Probability
* Addition rule: Probability
* Multiplication rule for independent events: Probability
* Multiplication rule for dependent events: Probability
* Conditional probability and independence: Probability
Unit 8: Counting, permutations, and combinations
* Counting principle and factorial: Counting, permutations, and combinations
* Permutations: Counting, permutations, and combinations
* Combinations: Counting, permutations, and combinations
* Combinatorics and probability: Counting, permutations, and combinations
Unit 9: Random variables
* Discrete random variables: Random variables
* Continuous random variables: Random variables
* Transforming random variables: Random variables
* Combining random variables: Random variables
* Binomial random variables: Random variables
* Binomial mean and standard deviation formulas: Random variables
* Geometric random variables: Random variables
* More on expected value: Random variables
* Poisson distribution: Random variables
Unit 10: Sampling distributions
* What is a sampling distribution?: Sampling distributions
* Sampling distribution of a sample proportion: Sampling distributions
* Sampling distribution of a sample mean: Sampling distributions
Unit 11: Confidence intervals
* Introduction to confidence intervals: Confidence intervals
* Estimating a population proportion: Confidence intervals
* Estimating a population mean: Confidence intervals
* More confidence interval videos: Confidence intervals
Unit 12: Significance tests (hypothesis testing)
* The idea of significance tests: Significance tests (hypothesis testing)
* Error probabilities and power: Significance tests (hypothesis testing)
* Tests about a population proportion: Significance tests (hypothesis testing)
* Tests about a population mean: Significance tests (hypothesis testing)
* More significance testing videos: Significance tests (hypothesis testing)
Unit 13: Two-sample inference for the difference between groups
* Mastery unavailable
* Comparing two proportions: Two-sample inference for the difference between groups
* Comparing two means: Two-sample inference for the difference between groups
* Unit 14: Inference for categorical data (chi-square tests)
* Chi-square goodness-of-fit tests: Inference for categorical data (chi-square tests)
* Chi-square tests for relationships: Inference for categorical data (chi-square tests)
Unit 15: Advanced regression (inference and transforming)
* Mastery unavailable
* Inference about slope: Advanced regression (inference and transforming)
* Nonlinear regression: Advanced regression (inference and transforming)
Unit 16: Analysis of variance (ANOVA)
* Mastery unavailable
* Analysis of variance (ANOVA)
## More Online Tutorial videos
https://brilliant.org/welcome/?sem=statistics
https://www.youtube.com/@datamlistic/shorts and videos explaining match used for Machine Learning.
## Study Guides
https://www.amazon.com/Practice-Statistics-AP%C2%AE-Course/dp/1319409342/
$142.99 Kindle, $153.30 hardcover The Practice of Statistics for the AP® Course Seventh Edition
by Daren Starnes (Author), Josh Tabor (Author)
by Martin Sternstein Ph.D. (Author)
https://www.amazon.com/Statistics-Flashcards-Fifth-Up-Date/dp/1506291376/
$29.99 AP Statistics Flashcards, Fifth Edition: Up-to-Date Practice (Barron's AP Prep) Fifth Edition
* 20.99 https://www.amazon.com/AP-Statistics-Premium-2026-Comprehensive/dp/1506296572/
AP Statistics Premium, 2026: Prep Book with 9 Practice Tests + Comprehensive Review + Online Practice (Barron's AP Prep) by Martin Sternstein Ph.D. (Author)
Available in Kindle & Paperback
9 full‑length practice tests‑‑5 in the book,
$19.89 https://www.amazon.com/Princeton-Review-Statistics-Premium-Prep/dp/0593518284/
Princeton Review AP Statistics Premium Prep, 21st Edition: 5 Practice Tests + Digital Practice Online + Content Review (College Test Preparation) 21st Edition
by The Princeton Review (Author)
5 full-length practice tests with answer explanations, timed online practice, and thorough content reviews.
https://www.thriftbooks.com/w/advanced-statistics-demystified_larry-j-stephens_larry-stephens/657200/item/23581914/
$6.29 Advanced Statistics Demystified
By Larry J. Stephens
Answers in Excel and Minitab!
https://www.thriftbooks.com/w/statistics-demystified_stan-gibilisco/600060/#edition=3661527&idiq=1636821
Statistics Demystified
By Stan Gibilisco
https://www.thriftbooks.com/w/business-statistics-demystified_sid-kemp_steven-m-kemp/364146/#edition=4759856&idiq=867710
$7.59 Business Statistics Demystified
By Steven M. Kemp and Sid Kemp
## Student Tutoring Offers
https://www.apexlearningvs.com/course/ap-statistics/
charges $700/year, $380/semester
https://www.superprof.com/le/3336/5583/
https://www.princetonreview.com/college/ap-test-prep
## Praxis exam for teachers
https://www.reddit.com/r/mathteachers/comments/1hpfxfs/studying_for_praxis_advice/
https://praxis.ets.org/test/5165.html
The Praxis test for Mathematics by ETS is $130.00
span the secondary mathematics curriculum including content related to (I) Number & Quantity and Algebra, (II) Functions and Calculus, (III) Geometry, and (IV) Statistics & Probability. A full list of the mathematics topics covered is provided in Content Topics. Test takers will find that approximately 25 percent of the questions call for application of mathematics within a teaching scenario or an instructional task.
159 is roughly the same as getting 59% right.
https://praxis.ets.org/on/demandware.store/Sites-ETS-Praxis-Site/en_US/Search-Show
$24.95 Practice Test: Mathematics (5165)
https://www.youtube.com/watch?v=JWcpNmX_5s8&list=PLKp13uNv0Y54oxRfYkbwBpcD2tl3AQYsY
DEPRECATED: Core Academic Skills for Educators: Mathematics (5733)
https://study.com/buy/test-prep/praxis.html
https://teachercertification.com/buy/praxis/?src=ppc_adwords_nonbrand&rcntxt=aws&crt=575755744010&kwd=praxis%20org&kwid=kwd-369386489182&agid=133520178153&mt=p&device=c&network=s&gad_source=5&gad_campaignid=15845343338&gclid=EAIaIQobChMIydrEyNW4kAMVuwutBh2LCiPyEAAYASAAEgLKBvD_BwE
Statistics & Probability Practice Problems for Praxis Mathematics (Praxis 5165)
Video Playlist
* Calculus: quadratic equation
* VIDEO
https://teachercertification.com/buy/praxis/
## Teacher training
https://skewthescript.org/skewu-online-new-stats-teachers
Nonprofit, low-cost alternatives like SkewU Online (Skew the Script) — a 3-week, self-paced online course for new AP Statistics teachers that issues a training certificate.
https://blog.mathmedic.com/post/new-to-teaching-ap-stats-start-here
Math Medic’s APSO workshops designed to build both content mastery and teaching strategy.
https://moreland.edu/resources/blog-insights/how-to-get-your-u-s-teaching-certification-from-anywhere-in-the-world
https://uwf.edu/soe/teacherready/