--- title: Variance & Covariance description: Variance measures the spread of a single dataset, while covariance measures how two variables change together. Learners will understand how to calculate these metrics, interpret joint variability, and apply them to risk assessment and statistical modeling. category: mathematics subcategory: probability-theory difficulty: beginner, intermediate, advanced url: /subject/variance-and-covariance --- # Variance & Covariance Variance measures the spread of a single dataset, while covariance measures how two variables change together. Learners will understand how to calculate these metrics, interpret joint variability, and apply them to risk assessment and statistical modeling. ## Available Resources 1 Videos • 1 Courses • 4 Websites • 1 Papers ## Websites ### 1. Khan Academy - Variance Khan Academy's free video lessons and practice exercises on measuring spread, covering population and sample variance, standard deviation, and mean absolute deviation, so learners can compute these measures by hand and interpret what they say about a dataset's variability. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population **Tags:** variance, standard deviation, descriptive statistics, measures of spread ### 2. Seeing Theory An interactive, D3.js-based walkthrough of probability and statistics in six chapters, from basic probability and distributions through frequentist inference, Bayesian inference and regression analysis. Parameters are manipulable, so sampling behaviour and fit are visible directly. **Difficulty:** Beginner | **Price:** Free **Link:** https://seeing-theory.brown.edu/basic-probability/index.html **Tags:** probability, statistics, data-visualization, regression, bayesian-inference ### 3. probabilitycourse.com ProbabilityCourse.com is an online learning resource that provides a structured probability course with clear explanations, worked examples, and practice problems covering topics from basics to advanced theory. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://probabilitycourse.com **Tags:** websites, technology-computer-science, data-science--ai ### 4. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Papers ### 1. Stanford EE278 Lecture Notes 3: Random Vectors, Mean and Covariance Matrix **Author:** Abbas El Gamal Graduate lecture notes generalizing variance to the covariance matrix of a random vector, proving symmetry and positive semidefiniteness, then deriving coloring and whitening transforms and the Gaussian random vector. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://isl.stanford.edu/~abbas/ee278/lect03.pdf **Tags:** covariance matrix, random vectors, whitening transform, gaussian random vectors, probability theory ## Youtubes ### 1. Covariance, Clearly Explained **Author:** Josh Starmer Visual walkthrough starting from variance, then building covariance from paired scatter data, showing positive, negative and near-zero cases and explaining why the raw covariance value resists interpretation without correlation. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/watch?v=qtaqvPAeEJY **Tags:** covariance, correlation, variance, descriptive statistics ### 2. Statistics 110 Lecture 21: Covariance and Correlation **Author:** Joe Blitzstein Harvard lecture introducing covariance and correlation, then using bilinearity to compute the variance of a sum of dependent variables, worked through to the variance of a Hypergeometric random variable. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/watch?v=IujCYxtpszU **Tags:** covariance, correlation, probability theory, hypergeometric distribution, variance of sums ## Videos ### 1. Lecture 11: Derived Distributions, Convolution, Covariance and Correlation (MIT 6.041) **Author:** John Tsitsiklis Finishes derived distributions, finds the density of a sum of independent random variables by convolution, then defines covariance and the correlation coefficient. The lecture video shows how the variance of a sum decomposes and why correlation lies between minus one and one. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/6-041-probabilistic-systems-analysis-and-applied-probability-fall-2010/resources/lecture-11-derived-distributions-convolution-correlation/ **Tags:** covariance, correlation, convolution, derived-distributions, variance ## Courses ### 1. An Intuitive Introduction to Probability **Author:** Karl Schmedders University of Zurich course taught by Karl Schmedders that builds probability from everyday examples: basic rules, conditional probability, applications, discrete random variables, and the normal distribution. Learners finish able to compute and interpret probabilities in practical decisions. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/introductiontoprobability **Tags:** probability, conditional-probability, discrete-random-variables, normal-distribution --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/variance-and-covariance