A-Z Maths for Data Science.

Explore how linear equations model unknowns in real problems, from bank robbery chases to house pricing, and see how linear models underpin data science with simple, explainable equations.

Explore how linear algebra underpins mathematical streams and fuels data science, machine learning, and optimization, with probability, statistics, calculus, and dimensionality reduction via principal component analysis and singular value decomposition.

Explore vectors in 1d, 2d, and 3d, and see how higher-dimensional equations arise in data science. Define a scalar as magnitude only, while a vector carries magnitude and direction.

A point is defined by coordinates equal to the dimension, for example x, y or x1, x2, ..., and distance uses the generalized Pythagoras formula in n dimensions.

Learn that vectors have magnitude and direction and represent points with coordinates like (2,3), and perform operations on expanded components x i plus y j rather than magnitude and angle.

Explore row and column vectors, their shapes and compatibility in vector and matrix operations, including dot products, with emphasis on consistent definition, dimensionality, and the data point versus feature interpretation.

Understand that the transpose swaps rows and columns, reversing the shape (2x3 to 3x2; 3x3 example), and note the matrix-transpose relationship to explore after learning basic operations.

Compute a vector’s magnitude using the L2 norm or Euclidean distance, and convert any vector to a unit vector by dividing by its magnitude, with 2D and 3D examples.

Master vector addition and subtraction through element-by-element operations on vectors with matching lengths and formats, yielding a new vector with the same dimensions; explore the geometric view and dot product.

Explore the inverse of a vector as another vector with the same magnitude but opposite direction, like (2,3) and (-2,-3), whose sum is the zero vector and magnitude sqrt(13).

Explore the dot product of two vectors, its compatibility rules (columns of v1 equal to rows of v2), and how it yields a scalar, unlike the cross product.

Multiply a vector by a scalar by applying the scalar to each element, scaling the magnitude while preserving direction; a negative scalar reverses direction.

Explore the distributive properties of vectors and scalars, and define the angle between two vectors as theta, with examples on the x and y axes, often using the smallest angle.

Compute the angle between two vectors using the dot product and magnitudes, illustrated with v1=(1,0) and v2=(0,1); cos theta = (v1 dot v2)/(|v1||v2|), theta=90 degrees.

Understand orthogonal vectors, defined as perpendicular with a 90-degree angle, where the dot product is zero; see examples with perpendicular vectors.

Master orthonormal vectors. Divide by the magnitude to obtain a unit vector from orthogonal vectors with unit magnitude, and use the dot product to relate magnitudes to cosine of angle.

Learn how a line is represented as a vector, from y = mx + c to n-dimensional form w^T x + w0 = 0 via dot products in linear models.

Explore how a line is represented in vector form as w^T x + w0 = 0, and how w0 indicates whether the line passes through the origin.

Express the line as w^T x = 0, the dot product with w is zero, showing w is perpendicular to the line and x lies on it.

Represent lines through origin as w^T x = 0, where w is a normal vector; shifting non-origin lines to origin preserves normal direction, and angle depends on direction, not magnitude.

Learn to project a vector onto a line, using coordinates or angles, with a projection equal to the vector magnitude times the cosine of the angle between them.

Discover how to compute a point’s distance from a line using the vector form w^T x + w0 = 0, via projection and perpendicular distance, including origin and shifted lines.

Learn how line equations use W^T x and W^T x + w0 to define points via signed distance, and how these ideas extend to circles, spheres, and higher dimensions.

Explore matrices as 2d arrays and distinguish vectors as 1d arrays, explain rows, columns, and matrix size m x n, with element notation A11.

Master matrix operations: element-wise addition and subtraction, Hadamard product, and dot-product multiplication with shape compatibility. Apply these ideas to linear equations and matrix representations.

discover how transpose reveals symmetry in square matrices, define diagonal and identity matrices, and compare scalar multiplication with dot products, including compatibility rules and non-square diagonal cases.

Learn how an orthonormal matrix has every row and column as an orthonormal vector, so A^T A equals the identity; examine identity, diagonal, and symmetric matrices as orthonormal examples.

Explore how to compute determinants using minors and cofactors, and derive a matrix inverse by applying these concepts to 2x2 and 3x3 cases.

Discover how to compute the inverse of a square matrix using determinant, cofactors, adjoint, and transpose, yielding the identity matrix; learn the orthogonal matrix property where transpose equals inverse.

Learn how the probability mass function represents discrete distributions and how the probability distribution function applies to continuous cases, illustrated with a fair dice example.

Explore continuous random variables and the probability density function, height distributions, that the area under the curve equals one and probability between two values; exact heights have zero probability.

Explore the Bernoulli distribution, a discrete two-outcome model where a success occurs with probability p and a failure with one minus p, with examples like heads or tails.

Explore the binomial distribution, an extension of the Bernoulli distribution with multiple independent trials. Compute the probability of k successes in n trials using the binomial formula, nCk p^k (1-p)^(n-k).

Explore the expected value of a random variable, a weighted average, the arithmetic mean of outcomes, guiding decisions under risk with a gambling example and the law of large numbers.

Compute the expected value of sales changes using a three-outcome distribution (2000, 0, -1000) with probabilities 6/50, 35/50, and 9/50, then relate to variance and standard deviation.

Learn how to compute the expected value of a Bernoulli distribution by weighing outcomes 0 and 1 with their probabilities, showing that the mean equals the success probability p.

Explore the expected value of the binomial distribution, deriving E[K] = np from nCk p^k (1-p)^{n-k} applied to six balls in an over with p = 0.3.

Explore the law of large numbers and how the average of many trials converges to the expected value, illustrated with dice rolls.

Explore the normal (Gaussian) distribution, a symmetric bell curve where the area under the curve equals one; learn how the mean and standard deviation shape it and the 68–95–99.7% rules.

Explore how standard deviation shapes the pdf of a normal distribution, showing that larger deviation flattens the curve and lowers its peak while preserving the area under the curve.

Understand the cumulative distribution function and relation to histograms and PDFs, focusing on how the CDF represents the area under the curve of the PDF up to a given point.

Explore the normal distribution by examining its formula, parameters mu and standard deviation, and how changing these values reshapes the curve for each x using real data in Excel.

Explore normal distribution with an interactive Excel utility, adjusting mu and sigma to see pdf and cdf changes, visualize 68-95-99.7 rules, and learn by working numeric examples.

Standardize any normal distribution by subtracting its mean and dividing by its standard deviation. Achieve zero mean and one standard deviation, then apply the z-table for area calculations.

Explore extreme values in the normal distribution, showing the asymptotic curve assigns a positive, though tiny, probability to every x from minus to plus infinity, even for plus 100.

Learn how the z-score standardizes any normal distribution to a unit normal, enabling probability calculations on the standard normal curve by subtracting mu and dividing by sigma.

Understand the z score, defined as (x - mu)/sigma, and how many standard deviations a value lies from the mean in both original and unit normal distributions.

Learn to read z score table by understanding z = (x - mu) / sigma, and recognizing positive and negative values and left and right areas.

Compute the probability of a free pizza using a normal distribution with mean 16.3 and standard deviation 0.2 for pizza sizes less than 16 inches, via the z-score -1.5.

Apply z scores to a normal distribution with mean 16.3 and standard deviation 0.2 to compute probabilities for sizes above 16.5 and between 15.95 and 16.63, using symmetry and z-table.

Use z scores and the z score table to estimate how many students score below 65, above 75, and between 65 and 75 with mean 70 and sd 5.

Solve a normally distributed battery lifespan problem using z-scores. Derive mu and sigma from 15% below 30 hours and 10% above 50 hours by two linear equations based on z-values.

Explore symmetric distributions and skewness, show how in a normal distribution the mean equals the mode equals the median, and describe how positive or negative skew affects tails and moments.

Explore the central limit theorem, showing that for any population distribution, the distribution of sample means becomes normal with the same mean and with standard deviation sigma over sqrt(n).

Revisits the central limit theorem, showing sample means become normal as sample size exceeds 30, with mean equal to the population mean and standard deviation divided by sqrt(n).

Central limit theorem states that sample means are normally distributed with mean mu and standard deviation sigma/√n; for population, use n≥30, but if the population is normal, any n works.

Apply the central limit theorem with z-score methods to solve sampling distribution problems. Find the probability that a 40-customer sample has mean at least 87, given μ=85 and σ=9.

Apply the central limit theorem to find the probability that the sample mean of 49 shoppers lies between 441 and 446, with mu 448 and sigma 21.

Explore discrete and continuous uniform distributions, with dice illustrating equal likelihood, compare to the normal distribution, and relate area under the curve to probability.

Explore the log normal distribution, where log x is normal and exp of a normal variable yields a log normal, showing right skew and how standard deviation shapes the curve.

Explore common log normal distribution examples across human behavior, online engagement, sports durations, medicine, and economics to see how this pattern appears in real data.

Explore power law distribution, where y scales as x to the k, showing exponential rise or fall; x must be positive and alpha positive, with real-world examples like income.

Explore the Pareto distribution as a power law, shaped by alpha, illustrating the 80/20 rule with wealth, income, city sizes, and other real-world examples, and why distribution choice matters.

Learn how the Pareto distribution uses alpha as the shape parameter and beta as the minimum value, and see how changing these values shapes the curve and its area.

Learn how a Q-Q plot compares an unknown data distribution to a known distribution, using ordered data pairs to assess normal, log-normal, and other distributions with a straight-line criterion.

Explore the Box-Cox transformation to convert data distributions to normal using a lambda parameter; code uses (x^lambda-1)/lambda, or log when lambda is zero, with QQ plots for validation.

Discover how distributions guide real-world decisions, from log-normal and normal sizing of shirts to Pareto-based storage provisioning. Apply these concepts to hypothesis testing and data-science workflows.

Master hypothesis testing with null and alternate hypotheses, alpha and p-values, and the role of confidence levels in two-tailed tests, using normal distribution and central limit theorem concepts through examples.

Describe how to construct a 90% confidence interval for a mean using the sample mean, known sigma, z-scores, and the central limit theorem, with a practical example.

Compute a 98% two-tailed confidence interval for the true mean age of engineers using a 50-sample with x-bar 34.3 and sigma 8, yielding 31.65 to 36.93.

Understand why the z score fails when population standard deviation is unknown and how the t score, with degrees of freedom, uses the sample standard deviation and a t table.

Test the null mu=20 vs alt mu<20 with n=20, mean 19.8, s=3.1 using a t-score; interpret the p-value and fail to reject the null.

Perform a one-tailed hypothesis test to see if the mean height exceeds ten inches, testing null mu=10 with alpha 0.05, using n=15, xbar=11.4, s=2.5, p-value 0.025.

Conduct a one-tailed t-test comparing mu=82 to mu>82 for a 25-student sample; with t=3.65 and alpha=0.05, reject the null and support the alternative.

Explain concluding hypothesis testing by comparing p-values to alpha, interpreting evidence levels, and deciding when to reject or not reject the null hypothesis, with alpha and confidence level notes.

Explore the foundations of probability through everyday examples, define long-term relative frequency and sample space, and analyze events like unbiased coin tosses and dice.

Learn the core laws of probability, including the total probability equals one and the complement rule. Explore conditional probability and Bayes' theorem with sample spaces and intersections.

Explore mutually exclusive and independent events, understand how they affect probability, and apply formulas for union, intersection, and conditional probability with clear examples.

Understand probability by prioritizing question comprehension and using formulas as support when helpful. Explore examples, circular or odd plates and garment shop or second floor, highlighting intuition and union-intersection methods.

Compute the probability of drawing two blue and two green from four blue and three green without replacement, using a six-sequence sample space and mutually exclusive events.

Explore the chain rule of probability by expanding conditional probabilities and intersections into sequential multiplications. Relate Bayes theorem, independence, and mutually exclusive events to simplify complex problems.

explains conditional probability and chain rule using a 52-card deal to four players, calculating the probability that each gets one ace.

Explore the birthday paradox and why intuition misleads. Compute the probability with a 365-day year: denominator 365^10 and numerator 365×364×...×356, yielding no two share a birthday about 0.8803.

Explore probability through practical examples, from three friends sharing a bill to calculating power cut probabilities across days, using independent events and complementary reasoning.

Explore probability through cricket scenarios and reliability problems, applying intersection and conditional probability, and using independence to compute the chance that all ten components work: (0.99)^10.

Visualize complex probability problems with probability trees, map events, and calculate intersections like P(A ∩ B) using diagrammatic decision trees for ad-skipping scenarios.

Explore the total law of probability, partitioning the sample space into mutually exclusive events, and compute P(A) from P(A∩B) and P(A∩C) using conditional probabilities.

Use the total probability theorem and conditional probability to determine if a tape recorder that died within six months had a flaw, based on flaw and death rates.

Explore the total law of probability with real-world examples, from the elevator scenario to five independent questions using binomial reasoning (at least two, not all five).

Explore how counting underpins probability, distinguishing permutations and combinations, where order matters or not, and learn to apply formulas to complex selection problems.

Count travel options from Delhi to Mumbai using permutation and combination, totaling 22 ways across air, train, and land. Frame probability as favorable over total outcomes.

Combine Delhi–Mumbai options (8 flights, 12 trains, 2 roadways) with Mumbai–Chennai options (6 flights, 10 trains, 2 roadways) as independent segments, yielding 22 and 18 options and 396 total ways.

Explore how constraints and dependencies alter the counting of travel combinations across two legs, illustrating mutual exclusivity, independence, and conditional probability in permutations and combinations.

Learn to decide when order matters or not by solving a one-from-each-discipline selection and counting five-digit ternary sequences, addressing independence and first-digit constraints.

Discover factorial notation, where n! is the product of 1 to n, and n! = n × (n−1)! to simplify calculations.

Explore permutation as an ordered arrangement of k distinct elements from n, derive the formula nPk = n!/(n-k)!, and verify with examples like 5P2.

Explore permutation with repetition where n distinct items fill k positions, yielding n^k outcomes. Compare this to standard permutation nPk and show how A, B, C select two digits.

Explore permutation scenarios: no repetition (n p k) and repetition (n to the power k), dividing by factorials of repeating counts, and note k > n1 + n2.

Explore permutation with repetition using the formula, counting arrangements of probability and Mississippi when letters repeat, and apply factorials to handle repeated letters.

Explore counting permutations when vowels appear together by treating them as a single unit with consonants, and expand the vowel group to account for internal arrangements and repeats.

Explore permutation methods by arranging letters in correction with vowels together as a unit, and in collection with vowels not together, accounting for repeats and factorial terms.

Learn permutations with repetition via a two-digit example using A, B, C to yield 9 ways, and observe the 3^n growth when N and K repeat.

Learn how combinations differ from permutations by exploring unordered selections of k distinct elements from n, using nCk formulas and practical examples like committees.

Differentiate permutations from combinations through practical examples. Observe how distinct gift values create permutation problems, while equal values yield combinations, and how roles or responsibilities add order.

Fix one person to break circular symmetry and count via 4P4 for the men and 3! for the other three women, totaling 144.

Practice a combination problem choosing two fiction and two non-fiction books. Set C(n,2) × C(6,2) = 150 to find n, as the caption states n equals 20.

Compute the expected value of an action by weighting outcomes by their probabilities, and see why repeated trials reveal the law of large numbers.

Compute the expected value of the change in sale from observed changes (2000, 0, -1000) with their proportions, illustrating the long-term mean. Then outline variance and standard deviation.

Explore the first property of expected value: adding a constant to a random variable shifts its expected value by the same constant, illustrated with simple examples to ease intuition.

Learn the second property of variance: adding a constant to all data leaves variance and standard deviation unchanged, because that shift does not alter data spread.

Explore how multiplying a random variable by a constant scales its expected value and variance, showing E[CX]=C E[X], Var(CX)=C^2 Var(X), and SD(CX)=|C| SD(X).

Explore the expected value of A X plus minus B. Note E[X+Y] = E[X] + E[Y], and Var(X+Y) = Var(X) + Var(Y) only if X and Y are independent.

Compare hosting a cricket match in England or India using expected value, variance, and standard deviation, considering ticket prices and probabilities, and choose India for its lower coefficient of variation.

Compute the expected total weight using X and Y and the rule E[X+Y]=E[X]+E[Y]. Evaluate independence by comparing P(Y|X) to P(Y) in the sweet and mixture example.

Draw two balls from a bag of eight white, three black, and two red; black yields ten rupees, white minus two, red zero; compute expected value and probabilities for profitability.