Statistics & Probability Cheat Sheet
Your complete reference for first-year university statistics and probability. From descriptive statistics and probability distributions to hypothesis testing — all the formulas and concepts you need to succeed.
Descriptive Statistics
6 topics
Mean, Median, and Mode
Central tendency measures: arithmetic mean, median for ordered data, and mode for most frequent values.
Variance and Standard Deviation
Measures of spread and dispersion: population vs sample variance, standard deviation interpretation.
Range and Interquartile Range
Simple measures of spread: range, quartiles, IQR, and identifying outliers using the 1.5×IQR rule.
Percentiles and Quartiles
Dividing data into equal parts: quartiles, percentiles, and their applications in data analysis.
Data Visualization
Graphical representations: histograms, box plots, scatter plots, bar charts, and pie charts.
Skewness and Kurtosis
Shape of distributions: measuring asymmetry (skewness) and tail heaviness (kurtosis).
Probability Basics
6 topics
Probability Fundamentals
Basic probability concepts: sample spaces, events, probability axioms, and complement rule.
Conditional Probability
Probability given prior information: P(A|B), independence, and the multiplication rule.
Bayes' Theorem
Updating probabilities with new evidence: prior, likelihood, posterior, and applications.
Counting Principles
Permutations, combinations, factorial notation, and the multiplication principle.
Random Variables
Discrete and continuous random variables, probability mass and density functions.
Expected Value and Variance
Mean and variance of random variables: E(X), Var(X), and their properties.
Probability Distributions
6 topics
Binomial Distribution
Distribution of successes in n independent trials: probability formula and applications.
Poisson Distribution
Modeling rare events: count data, rate parameter λ, and relationship to binomial.
Normal Distribution
The bell curve: properties, standard normal distribution, z-scores, and the 68-95-99.7 rule.
Standard Normal and Z-Scores
Standardization: converting to z-scores, using z-tables, and finding probabilities.
Student's t-Distribution
Distribution for small samples: degrees of freedom, t-tables, and comparison to normal.
Chi-Square Distribution
Distribution for variance and categorical data: degrees of freedom and critical values.
Sampling & Estimation
5 topics
Sampling Distributions
Distribution of sample statistics: sampling variability and the sampling distribution of the mean.
Central Limit Theorem
The foundation of inference: sample means approach normality regardless of population shape.
Point Estimation
Estimating population parameters: unbiased estimators, maximum likelihood, and method of moments.
Confidence Intervals
Interval estimation: margin of error, confidence level, and constructing intervals for means and proportions.
Sample Size Determination
Planning studies: calculating required sample size for desired precision and power.
Hypothesis Testing
6 topics
Hypothesis Testing Basics
Null and alternative hypotheses, test statistics, p-values, and decision rules.
Type I and Type II Errors
False positives and negatives: significance level α, power, and the tradeoff between errors.
Z-Test for Means
Testing population means when σ is known: one-sample and two-sample z-tests.
T-Test for Means
Testing means with unknown σ: one-sample, independent two-sample, and paired t-tests.
Chi-Square Tests
Tests for categorical data: goodness-of-fit and test of independence.
Analysis of Variance (ANOVA)
Comparing multiple group means: F-statistic, between and within group variance.
Regression & Correlation
4 topics
Correlation
Measuring linear relationships: Pearson's r, Spearman's ρ, and interpreting correlation coefficients.
Simple Linear Regression
Modeling relationships: least squares method, regression equation, and interpreting slope and intercept.
Coefficient of Determination
R-squared: measuring goodness of fit, explained vs unexplained variance.
Inference in Regression
Testing regression coefficients: standard errors, confidence intervals, and hypothesis tests for β.
Ready to Master Statistics?
Bookmark this page and use it as your go-to reference for all statistics formulas and concepts. Each topic includes detailed explanations and worked examples.