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Statistics

Introductory statistical reasoning: data, probability, inference, and models.

For: Learners at any starting point, from first-time students to people refreshing a specific skill.

After this page you can: Choose an appropriate introductory route and explain what a statistical result does and does not show.

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Part of: Mathematics

About Statistics

Statistics is the science of collecting, analyzing, interpreting, and presenting data. It helps summarize observed data and use probability to draw conclusions from samples about larger populations. A statistic estimates a population parameter, which is a numerical characteristic of the whole population.

Study it by moving from how data are gathered and summarized to probability, inference, and models. Learn to describe data with graphs and numerical summaries, then ask how sample-based estimates carry uncertainty and how relationships can be represented with regression. Practice interpreting results in context and checking how data were collected and what assumptions a method uses. You can follow these areas in order or choose the one that fits your current question.

Start here

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I’m new to statistics

Start with OpenStax’s introductory text, which introduces statistics through collecting, analyzing, interpreting, and presenting data.

Start with: Introductory Statistics, Section 1.1

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I’m refreshing the foundations

Use MIT’s open course materials to revisit probability and statistics topics with readings, assignments, and exams.

Start with: Introduction to Probability and Statistics

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I have a specific topic in mind

Use OpenIntro’s chapter resources to go directly to topics such as data summaries, probability, inference, or regression.

Start with: OpenIntro Statistics

Learn the foundations

1.Describe and summarize dataDescriptive statistics uses graphs and numerical summaries to organize data and show features such as center, spread, and shape.

Descriptive statistics uses graphs and numerical summaries to organize data and show features such as center, spread, and shape.

  • OpenIntro Statistics

    Its chapter resources include numerical summaries and displays for quantitative and categorical data.

Median

The value that divides ordered data into two halves.

Try it

For the values 2, 4, and 6, find the mean and median.

The mean is 4 and the median is 4.

2.Model randomness with probabilityProbability assigns measures to possible outcomes and provides a mathematical way to reason about randomness.

Probability assigns measures to possible outcomes and provides a mathematical way to reason about randomness.

Probability

A measure of how likely an outcome or event is, on a scale from zero to one.

Conditional probability

The probability of A calculated from the reduced sample space B.

Try it

A fair coin has two equally likely outcomes. What is the probability of heads on one toss?

The probability is one half, or 0.5.

3.Draw conclusions from samplesStatistical inference uses sample data and probability to estimate population quantities and express uncertainty about those estimates.

Statistical inference uses sample data and probability to estimate population quantities and express uncertainty about those estimates.

Sample

A portion or subset of a population selected for study.

Confidence interval

A range of plausible values for a regression function, specified with a probability.

Try it

A survey uses a sample to estimate the average for a larger population. Name one reason to report an interval as well as the estimate.

An interval communicates uncertainty around the sample-based estimate of the population value.

4.Represent relationships with modelsRegression models describe relationships between variables and can be used to estimate responses, with uncertainty considered when interpreting predictions.

Regression models describe relationships between variables and can be used to estimate responses, with uncertainty considered when interpreting predictions.

Regression

A method for representing relationships between variables and predicting a response from explanatory variables.

Try it

A fitted line predicts 10 when x is 2; the observed response at x = 2 is 13. What is the residual, using observed minus predicted?

The residual is 3.

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Books

Glossary

Every free Statistics course and book on RedLectern

Free Statistics online courses

  1. High-Dimensional StatisticsMIT 18.S997 · Browse materials
  2. Introduction to Probability and StatisticsMIT 18.05 · Browse materials
  3. Introduction to Statistical Methods in EconomicsMIT 14.30 · Browse materials
  4. Mathematical StatisticsMIT 18.655 · Browse materials
  5. Prediction: Machine Learning and StatisticsMIT 15.097 · Browse materials
  6. Quantitative Reasoning & Statistical Methods for Planners IMIT 11.220 · Browse materials
  7. Statistical Learning Theory and ApplicationsMIT 9.520 · Browse materials
  8. Statistical Method in EconomicsMIT 14.381 · Browse materials
Show all 20 coursesShow fewer courses
  1. Statistics and Visualization for Data Analysis and InferenceMIT RES.9-0002 · Browse materials
  2. Statistics for ApplicationsMIT 18.650 (formerly 18.443) · Browse materials
  3. Statistics for ApplicationsMIT 18.650 · Browse materials · Video
  4. Statistics for Brain and Cognitive ScienceMIT 9.07 · Browse materials
  5. Topics in Statistics: Nonparametrics and RobustnessMIT 18.465 · Browse materials
  6. Topics in Statistics: Statistical Learning TheoryMIT 18.465 · Browse materials
  7. Introduction to Probability and StatisticsMIT 18.05+summer_2022 · Open on OCW
  8. Introduction to Statistical Method in EconomicsMIT 14.30 · Open on OCW
  9. Probability and Statistics in EngineeringMIT 1.151 · Open on OCW
  10. Statistical Learning Theory and ApplicationsMIT 9.520 · Open on OCW
  11. Statistical Thinking and Data AnalysisMIT 15.075J · Open on OCW
  12. Statistics for ApplicationsMIT 18.650 (formerly 18.443) · Open on OCW
How these courses are chosen

Listed as Statistics courses:

  • those whose title contains “Statistics”, “statistical method”, “statistical methods”, “statistical thinking” or “statistical learning”, or
  • those tagged “Statistics” as one of at most 2 topic tags, or as the first of 3.

Free Statistics textbooks and books

Show all 60 listed booksShow fewer books

Search all 70 Statistics books on the Books page →

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