Statistics
Statistics & ProbabilityStatistics is the branch of mathematics concerned with collecting, organizing, analyzing, interpreting, and presenting data.
Definition
Statistics is the study of collecting, organizing, and making sense of numbers and information. It helps us answer questions like "What is typical?" or "What is likely to happen?"
Example
A teacher collects test scores from all students, finds the average, and uses that information to decide whether to review a topic. That process is statistics in action.
Key Insight
Statistics turns raw numbers into useful knowledge. Without it, a pile of data is just noise.
Definition
Statistics is the mathematical science of collecting, organizing, summarizing, analyzing, and drawing conclusions from data. It is divided into descriptive statistics (summarizing data) and inferential statistics (drawing conclusions about a population from a sample).
Example
A researcher surveys $400$ voters and finds that $54\%$ support a candidate. Descriptive statistics summarizes this result; inferential statistics uses it to estimate the broader population's opinion.
Key Insight
The two branches work together: description tells you what you observed, inference tells you what it likely means beyond your sample.
Definition
Statistics encompasses the theory and methods for drawing inferences under uncertainty. Frequentist statistics treats probability as a long-run frequency, while Bayesian statistics treats it as a degree of belief updated via Bayes' theorem. Both branches rely on probability theory, measure theory, and mathematical optimization.
Example
Maximum likelihood estimation (MLE) finds the parameter value $\theta$ that maximizes the likelihood function $L(\theta \mid \text{data})$, providing a principled estimator with known asymptotic properties (consistency, efficiency).
Key Insight
Modern statistics underpins machine learning, clinical trials, econometrics, and scientific inference. The choice of statistical framework (frequentist vs. Bayesian) shapes both the methodology and interpretation of results.