Free statistics reference

Statistics topics

54 reference pages covering the full StatRise curriculum — each with the definition, key formulas, a worked example, and guidance on when to use it. All free to read, no account needed. When you want to go hands-on, every topic links to the matching interactive lesson, calculators, and simulations.

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Module 1 · 8 topics · Free lessons

Descriptive Statistics

Mean, median, mode, variance, spread, and visualization.

  • What is Statistics? — Statistics is the science of collecting, organizing, analyzing, and interpreting data to make informed decisions. It provides tools to transform raw data into…
  • Types of Data — Qualitative (categorical) data describes qualities or categories, like color or brand. Quantitative (numerical) data represents amounts or counts, like height…
  • Measures of Center — Measures of center describe a typical or representative value in a data set. The three main measures are the mean, median, and mode.
  • Measures of Spread — Measures of spread describe how spread out the data is. Common measures include range, variance, and standard deviation.
  • Quartiles & IQR — Quartiles divide sorted data into four equal parts. Q1 (25th percentile) is the median of the lower half. Q2 is the overall median (50th percentile). Q3 (75th…
  • Skewness & Shape — The shape of a distribution describes how data values are spread around the center. Distributions can be symmetric, left-skewed (negatively skewed), or…
  • Data Visualization Basics — Histograms show frequency distributions of quantitative data. Bar charts display counts or percentages for categorical data. Box plots summarize the…
  • Choosing the Right Chart — A scatter plot displays pairs of quantitative values as points on an x-y plane. It is the primary tool for exploring the relationship (direction, form…

Module 2 · 10 topics · Premium lessons

Probability Foundations

Events, rules, conditional probability, and Bayes theorem.

  • What is Probability? — Probability is a number between 0 and 1 that measures how likely an event is to occur. A probability of 0 means impossible, and 1 means certain.
  • Sample Spaces & Events — The sample space (S) is the set of all possible outcomes of an experiment. An event is a subset of the sample space.
  • Complement Rule — The complement of event A, written A′ (or Ā), is the event that A does not occur. Together, A and A′ account for all possible outcomes.
  • Addition Rule — If A and B cannot happen at the same time (mutually exclusive), then P(A ∩ B) = 0, and the formula simplifies to P(A ∪ B) = P(A) + P(B).
  • Multiplication Rule — If A and B are independent (one doesn’t affect the other), then P(B | A) = P(B), and the rule simplifies to P(A ∩ B) = P(A) × P(B).
  • Conditional Probability — P(A | B) is the probability of A occurring given that B has already occurred. It “restricts” the sample space to only those outcomes where B happened.
  • Independence — Two events are independent if the occurrence of one does not change the probability of the other. Formally, A and B are independent if P(A | B) = P(A).
  • Bayes' Theorem — Bayes’ theorem lets you reverse conditional probabilities. If you know P(B | A) and want P(A | B), Bayes’ theorem connects them using the prior probability…
  • Counting Principles — If one event can occur in m ways and a second independent event can occur in n ways, then the two events together can occur in m × n ways.
  • Permutations & Combinations — A permutation is an arrangement where order matters. A combination is a selection where order does not matter.

Module 3 · 9 topics · Premium lessons

Distributions

Normal, binomial, Poisson, and expected value tools.

  • Random Variables — A random variable is a numerical outcome of a random process. It assigns a number to each outcome in the sample space. Denoted by capital letters like X or Y.
  • Discrete vs Continuous — A discrete random variable takes countable values (e.g., 0, 1, 2, ...). Its probabilities are described by a probability mass function (PMF).
  • Probability Distributions — A probability distribution describes all possible values of a random variable and their associated probabilities. It completely specifies the behavior of the…
  • Binomial Distribution — A binomial experiment has a fixed number of trials (n), each trial has exactly two outcomes (success/failure), trials are independent, and the probability of…
  • Poisson Distribution — The Poisson distribution models the number of events occurring in a fixed interval of time or space, when events happen independently at a constant average…
  • Normal Distribution Intro — The normal (Gaussian) distribution is a continuous, symmetric, bell-shaped distribution defined by its mean (μ) and standard deviation (σ). It is the most…
  • Standard Normal & Z-Scores — The standard normal distribution has μ = 0 and σ = 1. Any normal distribution can be converted to standard normal using z-scores.
  • Normal Distribution Applications — To find P(X < value) for a normal distribution: convert to a z-score, then look up the cumulative probability. Most statistics tools and calculators can do…
  • Comparing Distributions — Binomial: fixed n independent trials with constant p. Poisson: count of events in a fixed interval with rate λ. Normal: continuous data that is symmetric and…

Module 4 · 7 topics · Premium lessons

Sampling & CLT

Sampling distributions and central limit intuition.

  • Population vs Sample — A population is the entire group of individuals or items that you want to study. Population values are called parameters and are typically denoted by Greek…
  • Sampling Methods — Every possible sample of size n has an equal chance of being selected. This is the gold standard of sampling methods.
  • Sampling Distributions — The sampling distribution of a statistic is the distribution of that statistic across all possible samples of the same size from the same population.
  • Central Limit Theorem — For independent random observations from a population with mean μ, standard deviation σ, and finite variance, the sampling distribution of x̄ approaches a…
  • CLT Applications — The CLT allows us to compute probabilities about sample means using the normal distribution, even when the population itself is not normal.
  • Standard Error — The standard error is the standard deviation of the sampling distribution. It measures how much a sample statistic typically varies from sample to sample.
  • CLT and Sample Size — The “n ≥ 30” rule is a convenient guideline, not a strict law. The required sample size depends on the population’s shape: symmetric populations need fewer…

Module 5 · 12 topics · Premium lessons

Inference

Confidence intervals, hypothesis tests, and errors.

  • Point vs Interval Estimates — A point estimate is a single value used to estimate a population parameter. For example, x̄ is a point estimate for μ, and p̂ is a point estimate for p.
  • Confidence Intervals Intro — The confidence level (e.g., 95%) is the long-run proportion of intervals that would capture the true parameter if we repeated the sampling process many times.
  • CI for Means (σ known) — Requires: population σ is known, the population is normal or n ≥ 30 (by CLT), and the sample is random.
  • CI for Means (σ unknown) — The t-distribution is similar to the standard normal but has heavier tails. It accounts for the extra uncertainty from estimating σ with s. As n grows, t…
  • CI for Proportions — Requires: random sample, np̂ ≥ 10 and n(1 − p̂) ≥ 10 (success/failure condition), and sample is less than 10% of the population.
  • Sample Size Determination — n = (z* × σ / ME)²
  • Hypothesis Testing Intro — A hypothesis test is a formal procedure for using sample data to evaluate a claim about a population parameter. It assesses whether observed data is…
  • Null & Alternative Hypotheses — The null hypothesis is the default claim of no effect or no difference. It always contains an equality (=, ≤, or ≥).
  • Test Statistics & P-values — A test statistic measures how far the sample result is from the null hypothesis value, in standardized units. Common test statistics are z, t, and χ².
  • Z-Test — Use the z-test when the population standard deviation σ is known, the sample is random, and the population is normal or n ≥ 30.
  • T-Test — One-sample: compare a sample mean to a known value. Two-sample: compare two independent sample means. Paired: compare means of matched or before/after data.
  • Chi-Square Test — Goodness-of-fit: test if observed frequencies match expected frequencies from a theoretical distribution. Independence: test if two categorical variables are…

Module 6 · 8 topics · Premium lessons

Regression & Correlation

Linear models, residuals, correlation, and R squared.

  • Scatter Plots — A scatter plot displays the relationship between two quantitative variables by plotting data as points on an x-y coordinate plane. The explanatory variable…
  • Correlation — Correlation measures the strength and direction of the linear relationship between two quantitative variables. It is denoted by r and ranges from −1 to +1.
  • Pearson's r — The formula standardizes each variable by subtracting its mean and dividing by its standard deviation, then averages the products. Positive products (both…
  • Simple Linear Regression — Simple linear regression fits a straight line ŷ = a + bx that best describes the linear relationship between an explanatory variable x and a response variable…
  • Least Squares Method — The least squares method finds the line that minimizes the sum of the squared residuals (SSE = Σ(yᵢ − ŷᵢ)²). This produces the best-fitting line in terms of…
  • R-Squared — R² is the proportion of the total variation in y that is explained by the linear relationship with x. For the usual in-sample least-squares regression with an…
  • Residual Analysis — Plot residuals on the y-axis against x (or ŷ) on the x-axis. A good linear model shows randomly scattered residuals with no pattern. Patterns indicate the…
  • Making Predictions — Interpolation predicts within the range of observed x values and is generally reliable. Extrapolation predicts outside that range and is risky because the…

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