Statistics Formulas and Concepts

This cheat sheet provides a concise summary of key formulas, concepts, and definitions from introductory statistics, covering data summarization, probability, distributions, inference, and regression.

Core Principles

Statistics is the science of collecting, organizing, summarizing, and analyzing information to draw conclusions and measure confidence.
Data vary, and understanding the sources of variability is a primary goal of statistics.
Descriptive statistics organize and summarize data, while inferential statistics extend sample results to populations.
A parameter summarizes a population, while a statistic summarizes a sample.
Understanding the shape, center, and spread of data distributions is crucial for appropriate analysis.
Hypothesis testing involves formulating null and alternative hypotheses and using sample data to assess their plausibility.

Formulas

$ \bar{x} = \frac{\sum x_i}{n} $
$ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} $
$ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} $
$ z = \frac{x - \mu}{\sigma} $
$ z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} $
$ P(E) = \frac{\text{number of ways E can occur}}{\text{number of possible outcomes}} $
$ P(E \text{ or } F) = P(E) + P(F) $
$ P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F) $
$ P(E^c) = 1 - P(E) $
$ P(E \text{ and } F) = P(E) \cdot P(F) $

Key Terms

Parameter: A numerical summary of a population.
Statistic: A numerical summary of a sample.
Population: The entire group of individuals to be studied.
Sample: A subset of the population that is being studied.
Random Sampling: The process of using chance to select individuals from a population to be included in the sample.
Observational Study: Measures variables without attempting to influence them.
Designed Experiment: Randomly assigns individuals to groups and manipulates an explanatory variable.
Confounding Variable: An explanatory variable whose effect cannot be separated from that of another explanatory variable.
Lurking Variable: An unconsidered variable that affects the response variable.
Type I Error: Rejecting a true null hypothesis.
Type II Error: Failing to reject a false null hypothesis.
P-value: The probability of observing a sample statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.
Confidence Interval: An interval estimate for a population parameter.
Least-Squares Regression Line: The line that minimizes the sum of squared residuals.
Correlation Coefficient (r): Measures the strength and direction of a linear relationship between two quantitative variables.
Coefficient of Determination (R^2): The proportion of the total variation in the response variable explained by the regression line.

Pitfalls to Avoid

Confusing correlation with causation.
Misinterpreting statistical significance as practical significance.
Drawing conclusions from data without considering lurking variables.
Using convenience samples instead of random samples.
Not checking assumptions for statistical tests (e.g., normality, independence).