Probability and Statistics
Probability and statistics is the branch of mathematics that quantifies uncertainty and draws conclusions from data. It provides the tools to design experiments, measure evidence, estimate parameters, and test claims across science, engineering, and business.
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Intro
Probability and Statistics
Probability and statistics gives you a framework for reasoning about uncertainty. When you measure something, observe a pattern, or make a prediction, probability tells you how confident you should be. Statistics tells you how to get there from data.
The two halves work together. Probability builds models of random processes — coin flips, measurement errors, customer arrivals. Statistics uses those models in reverse: given observed data, what can you conclude about the process that generated it?
Why this matters
Every field that collects data needs statistics. A clinical trial must determine whether a drug works or the result is noise. A manufacturer must decide whether a batch meets specification. A data scientist must choose which features predict an outcome and which are coincidence.
Without statistical reasoning, you are guessing. With it, you can quantify how likely you are to be wrong.
The core loop
Statistical work follows a repeating pattern:
- Ask a question about a population or process.
- Design a study or experiment to collect relevant data.
- Summarize the data (descriptive statistics).
- Build a probability model that could have generated the data.
- Use the model to draw inferences (estimation, testing, prediction).
- Assess how uncertain your conclusions are.
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Where this skill leads
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Sources
- https://www.itl.nist.gov/div898/handbook/eda/section3/eda3.htm
Supports
- Exploratory data analysis techniques and graphical methods
- Probability distributions (normal, t, chi-squared, F) with properties
- Hypothesis testing framework and interpretation of p-values
- Quantitative measures of location, spread, skewness, and kurtosis
- https://ocw.mit.edu/courses/18-05-introduction-to-probability-and-statistics-spring-2022/
Supports
- Combinatorics, random variables, and probability distributions
- Bayesian inference and updating with prior and likelihood
- Hypothesis testing and confidence intervals
- Central Limit Theorem and sampling distributions
- Posterior concentration with increasing sample size
- https://www.khanacademy.org/math/statistics-probability
Supports
- Basic probability rules and counting principles
- Discrete and continuous probability distributions
- Sampling distributions and the Central Limit Theorem
- Confidence intervals and significance tests
- Regression and correlation
- https://online.stat.psu.edu/stat500/
Supports
- Applied probability distributions and their parameters
- Point and interval estimation for means and proportions
- One-sample, two-sample, and paired hypothesis tests
- Simple and multiple linear regression with diagnostics
- Resistance of the median to outliers
- https://seeing-theory.brown.edu/
Supports
- Visual and interactive demonstrations of probability axioms
- Geometric intuition for distributions and sampling
- Interactive Bayesian updating demonstrations
- Visual Central Limit Theorem demonstrations