Statistics Fundamentals

Explore descriptive statistics to summarize data with central tendency and variability, and learn how data visualization complements numeric metrics to reveal relative positions, with Python basics.

Explore how a frequency table summarizes data by class intervals, showing class width, class value, frequency, relative frequency, and cumulative relative frequency for a dataset of 50 students.

Create a frequency table for exam scores in Python using pandas and numpy. Compute class intervals with arange, frequencies with histogram, and cumulative relative frequencies displayed in a data frame.

Explore stem and leaf diagrams that split data into stems and leaves to visualize dataset of 20 observations. Learn how a stem-and-leaf diagram clarifies values and why histograms suit data.

Create a stem and leaf plot of exam scores using pandas, numpy, and stemgraphic; load the dataset, preview rows, and set scale to 10 with asc False for ascending order.

Use a frequency histogram to visualize data distributions and compare to the frequency table. Learn about class width, bins, skewness, and multimodal patterns in a 50-student score dataset.

Create a histogram from a pandas series of exam scores using seaborn and matplotlib, set 0 to 100 axes with arange, and compare frequency histogram using norm_hist and Sturges’ formula.

Display data with dot plots by stacking dots on category values along the horizontal axis. Show distribution, skewness, and outliers; histograms suit large datasets.

Focus on the center of data with mean, median, and mode as representative values. Understand how the sample mean (x̄) and population mean (μ) arise from sums divided by n.

Compare the mean and median as measures of center, showing how outliers distort the mean and how the median better represents the dataset’s center using examples and a dot plot.

Discover that the mode is the most frequent value for categorical and numerical data, including two modes, while dot plots show distributions can differ despite the same median.

Compute mean, median, and mode from a student exam dataset using pandas, handling missing values with dropna and exploring group differences via groupby.

Explore the geometric mean as the square root of the product and see why it better represents average growth rates in profits than the arithmetic mean.

Explore the harmonic mean, its formula, and how it corrects average speed calculations by using total distance over total time, illustrated with a two-way trip.

Compute the trimmed mean by discarding high and low values and averaging the remainder, reducing outlier influence and illustrating the Olympic average in figure skating.

Use simple moving average, calculated as the arithmetic mean over a centered five-day window, to smooth stock price or temperature data and reveal the underlying trend.

Compute the expected value by multiplying each outcome of a random variable by its probability. Then sum these values to obtain the mean, illustrated with dice and a card game.

Explore binary data with outcomes 1 or 0 and use proportion as the center to measure how often churn occurs, dividing churners by total customers and interpreting with domain knowledge.

Learn to compute geometric, harmonic, arithmetic, and trimmed means using Python tools like pandas, NumPy, and SciPy, and apply moving averages on stock price data.

Compare two datasets with equal means, medians, and modes but different variability to show how dispersion matters, and use dot plots to illustrate the higher spread in dataset B.

Analyze the range as max minus min and compare datasets A and B. Note how an outlier can mislead, and define residuals as observed minus mean or model predictions.

Compare two datasets by calculating each observation's deviation from the mean, then averaging the absolute deviations to obtain the mean absolute deviation (MAD) for dispersion.

Compute variance by averaging squared residuals with n, xi, and x-bar. Datasets A and B show variances 0.8 and 1.8, illustrating squaring weights deviance over mean absolute deviation.

Explore how variance measures data variability across datasets and why the square root forms the standard deviation, returning the same unit as the original value for meaningful comparisons.

Use the coefficient of variation to compare data variability across differently scaled datasets, since it is the ratio of the standard deviation to the mean and is unitless.

Compute variability metrics in Python using pandas, including range, mean absolute deviation, variance, standard deviation, and coefficient of variation on exam scores, by loading data and formatting results.

Understand how a data point's relative position is measured by percentile and interquartile range. Percentile shows what percent of data are at or below the focal value.

Explore quartiles and hinges to compute the interquartile range, the difference between q3 and q1, illustrated by a 12-point dataset with q1=5.5, q3=13.5, and iqr=8.

Apply the empirical rule to normally distributed data, showing that about 68% lie within one standard deviation, 95% within two, and 99.7% within three of the mean.

Chebyshev's theorem applies to any dataset and states that at least 75% lie within two standard deviations and at least 88.888% within three, with the true value possibly higher.

Explore computing relative positions in Python by using percentile and quartile functions to derive the IQR and related metrics, including mean, median, and standard deviation, with pandas and scipy.

Explore why data visualization complements quantitative metrics by contrasting distributions with dot plots and learning basic visualization techniques to explain data intuitively.

Explore box plots to visualize distribution using a box for quartiles and a median line, with whiskers denoting min and max, and learn how iqr, skewness, and outliers shape interpretation.

Create a boxplot of salary data using pandas and seaborn, compare salary distributions by gender, and adjust outliers and whiskers with sym and whis.

Visualize categorical data with a bar chart by sorting bars to reveal category rank. Keep bars separated, show categories on the x-axis, and use the mode to indicate central location.

Illustrate bar plots with pandas, seaborn, and matplotlib to explore salary data: use countplot by gender and location, and barplot with hue for location, including optional 95% confidence intervals.

Visualize categorical data with a pie chart by dividing a circle into slices and labeling each category’s percentage, noting when it’s best for few categories or widely different proportions.

create a donut pie chart of employees' work locations using pandas and matplotlib, building a relative frequency table and customizing labels, colors, and donut hole for clarity.

Use line plots for time series to track changes, with the x-axis as the timeline and the y-axis as the value. Avoid too-short periods, as longer ranges reveal true trends.

Import and visualize time-series stock data with pandas, seaborn, and matplotlib to plot price and its 5-day moving average, then refine the x-axis to month/day and tick every five days.

Explore cross-tabulation tables to visualize relationships between two variables. Classify data into rows and columns, display frequencies or percentages, and interpret row by column (m by n) structures.

Compare stacked bar charts and mosaic plots (Marimekko diagrams) for cross-tab data, displaying proportions on both axes and illustrating sales by region across companies.

Create cross-tabulation tables and stacked bar charts from location and gender data using pandas crosstab and matplotlib, displaying both frequencies and 100% stacked percentages.

Create a mosaic plot of salary and work location data using Python, pandas, and Matplotlib, plus statsmodels mosaicplot, customizing colors with a props dictionary and setting gap and title.

Plot three age-group percentages on a ternary plot by assigning each class to a vertex and drawing lines to locate each company's intersection point.

Create a ternary plot of country population age composition data (0-14, 15-64, 65+) using pandas and plotly.express, cleaning spaces, handling missing values, and converting strings to numeric.

Explore probability fundamentals, from objective and subjective probability to Bayes' theorem, using data to analyze events and predict outcomes, including employee churn, and apply to multivariate analyses and machine learning.

Explore factorial fundamentals, including n!, its role in permutations and combinations, and why 0! = 1, with examples using four distinct objects.

Learn to compute permutations of r objects from n distinct objects in a line using the factorial formula n!/(n−r)!, with examples like 5P3 and 4P3×4P2.

This lecture contrasts combinations with permutations, shows that order does not matter, derives the combination formula by dividing permutations by r!, and uses 4 choose 3 and 4 choose 1.

Learn to compute factorial, permutation, and combination with Python using math.factorial and scipy's perm/comb, create permutation and combination functions, and display integer results with exact True.

Explore the basics of set theory as the foundation of probability, defining experiments, trials, deterministic and random outcomes, events, and sample spaces, with a die example illustrating outcomes 1–6.

Explore the concept of a set in set theory as a collection of elements treated as one object, with examples like die pips and continents.

Differentiate between an event and an element: an event is the outcome of an infinitely repeatable experiment, while an element is a constituent of a set.

Explore how a Venn diagram visualizes the universal set and relationships among subsets, including the overlap, superset-subset relationship, and that greater than 4 is a subset of greater than 3.

Understand the complement and complementary event as the remaining part of the universal set after removing A, with A and its complement mutually exclusive, and P(A) + P(A^c) = 1.

Explore the intersection of sets, denoted by cap, with shared elements like 5 in A and B, and the empty set phi with probability 0 for mutually exclusive events.

Use the cup symbol to compute the union of two sets and its probability. P(A or B) = P(A) + P(B) − P(A ∩ B); disjoint sets yield zero.

Learn set difference, where A minus B contains elements in A not in B, and the symmetric difference is the union of A minus B and B minus A.

Define sets in Python, compare them with lists, and perform intersection, union, and difference. Add or discard elements, and test disjoint, subset, and superset relations, noting sets have unique elements.

Explore the foundations of probability by defining random experiments, distinguishing deterministic and random outcomes, and applying basic theorems and the 0 to 1 probability scale with examples.

Learn how to calculate probability using the classical definition (k/n) for equally likely elementary events, illustrated with a six-sided die and concepts like complement and sample space.

Compute probabilities using combination formulas for drawing 3 balls from 7, including 3 red and 1 red with 2 blue, and apply complement method to find at least one blue.

Compare coin flips, where outcomes are independent, with exam scores that are dependent, and show how independence leads to multiplying probabilities for intersections.

Compute the expected value of a random variable using the probability-weighted average, illustrated with a five-card game where even cards yield 12 and odd cards yield -10, giving -1.2.

Explore conditional probability with green ball examples, and learn that the probability of A given B equals the probability of A and B divided by the probability of B.

Explore how conditional probability changes with statistical independence using ball-drawing examples. Distinguish dependent cases where P(A|B) differs from P(A) and independent cases where they match.

Apply the conditional probability framework to derive the multiplication theorem, switch A and B, and show P(A∩B)=P(A)P(B) for independence.

Explore Simpson's paradox and conditional probability through the UC Berkeley admission data, showing how department-level trends can reverse when data are aggregated.

Calculate conditional probability of universities with international students above 30% using Python and pandas, analyzing a Times Higher Education 2020 top 20 dataset by country.

Bayes' theorem links prior, likelihood, and evidence to update posterior probabilities in conditional probability problems, with practical examples from churn prediction, A+ training outcomes, and confusion matrices.

Learn to implement Bayes' theorem in Python to compute conditional probability A given B, including calculating B's probability from A and not A, and apply to churn prediction example.

Explore probability distributions, from symmetric to asymmetric, and learn how they describe data, estimate probability, expected value, and variance for hypothesis testing. Start with random variables and the normal distribution.

Define a random variable and its link to a random experiment. Explain its capital and lowercase notation and the discrete versus continuous distinction, with coin flips and dice pips.

Explore statistics fundamentals by learning discrete probability distribution, probability mass function, and how to compute mean, variance, and standard deviation from a probability distribution.

Explore continuous probability distributions using BMI as an example, explaining why a continuous random variable has probability zero for any exact value and how infinite values sum to one.

Understand how the probability density function (pdf) defines probabilities for continuous distributions, use the area under the curve to compute interval probabilities, and apply the integral.

Explore the cumulative distribution function, or CDF, and how it gives probabilities for discrete and continuous variables using PMF, PDF, and integration. Visualize CDF behavior, non-decreasing properties, and interval probabilities.

Learn to compute the expected value of discrete and continuous random variables from probability distributions using sums and integrals, while applying constants, additivity, and key properties.

Explore the variance of random variables, covering discrete and continuous cases, using expectation and pdf, and learn that constants do not change variance while scaling by c yields c^2 variance.

Explore how to compute variance from the expected value, using Var(X) = E[X^2] - (E[X])^2, with discrete proofs and notes on continuous cases.

Explore the additivity of variance for random variables and how covariance affects var(X+Y); learn that independence makes var(X+Y) = var(X) + var(Y).

Explore the normal distribution, a Gaussian, continuous and symmetric about the mean, where mean equals median equals mode, with mu and sigma shaping the bell curve.

Explore the standard normal distribution with mean zero and variance one, and learn to standardize data to a z-value for using the probability density function and the standard normal table.

Explain how the standard normal distribution table, or z-table, shows the percentage to the left of a z-score and how to use it for probabilities in a normal distribution.

Explore skewness as a measure of data asymmetry and kurtosis as a trait of the normal distribution shape, including the Fisher-Pearson coefficient of skewness and sample-size adjustments.

visualize a normal distribution with Python by using SciPy.stats.norm, setting mean 50 and sd 10, generating x from 0 to 100, computing pdf, and plotting a blue, symmetrical bell curve.

Explore binomial distribution by analyzing Bernoulli trials and binary outcomes, deriving the formula with combinations, and applying it to sequences of independent trials like coin flips.

Learn to compute the expected value of a binomial distribution using E[X] = np, with n trials and success probability p, illustrated by a 0.6 coin flip.

Learn how to compute the variance of a binomial distribution using n and p, with the key result var(x) = np(1-p) and practical coin-flip examples.

Visualize and solve binomial distribution problems in Python using SciPy and NumPy, calculating pmf for n=10, k=4 with p=0.6, and examining mean and variance.

Explore the Poisson distribution, a discrete model for the number of events in a fixed interval, defined by lambda and Napier's constant e, often used for rare events.

Learn how to determine the expected value of a Poisson distribution, which equals lambda, using Maclaurin series and the exponential function.

Demonstrate how to derive the variance of a Poisson distribution, showing that it equals lambda by linking the expected value and the expected value of X squared.

Explore Poisson and binomial distributions through defect-rate examples, compute lambda as n times p, and compare probabilities for quality-control decisions.

Visualize a Poisson distribution with Python and solve related problems. Use lambda=3 to find P(X=4)=0.168 via manual formula and scipy's poisson pmf, and visualize with mean and variance both 3.

Explore the geometric distribution as a discrete model for the first success in independent Bernoulli trials, and learn to compute the first-success probability at trial k with probability p.

Compute the expected value of a geometric distribution as the average number of trials to first success, using p and the 1/p formula with an example.

Compute the variance of a geometric distribution using (1-p)/p^2, and see why lower p yields greater variance, with the expected value 1/p and a brief proof outline.

Visualize and solve geometric distribution problems with Python and SciPy, using the 0.6 probability example, simulate trials, plot the distribution, and compute mean and variance.

Explore the exponential distribution as a continuous time model for intervals between random events, using lambda, pdf, and cdf to compute probabilities such as a next customer within 3 minutes.

Learn how to calculate the expected value of an exponential distribution, showing that it equals 1 over lambda by applying integration by parts and the exponential property.

Derive the variance of an exponential distribution as var equals one over lambda squared by evaluating E[X] and E[X^2].

Explore memorylessness, a core property of geometric and exponential distributions, where prior outcomes do not affect the next event, as with a fair coin with a head probability of 0.5.

visualize exponential distribution with python and solve problems using lambda 20 and 3 minutes; compute probability density for a 3-minute arrival and cumulative probability, then plot with numpy and matplotlib.

Explore the discrete uniform distribution, where each discrete outcome has equal probability; derive its expected value (N+1)/2 and variance (N^2-1)/12, illustrated by a fair six-sided die.

Explore the continuous uniform distribution, its pdf 1/(b-a) on [a,b], the mean (a+b)/2, variance (b-a)^2/12, and the cdf, with practical examples.

Compute the expected value and variance of a discrete uniform distribution from 1 to 6 using Python and SciPy, by applying scipy.stats.randint with min=1 and max=7.

Explore joint probability distributions for two random variables x and y, discrete and continuous cases, with marginal distributions and a crosstab example of employee churn by department.

this lecture introduces sampling, population versus sample, and inferential statistics, and covers sampling methods, the law of large numbers, the central limit theorem, and study design.

Define population and sample. Contrast descriptive and inferential statistics, and explain how a representative sample and sampling strategy enable inferring population parameters for finite and infinite populations.

Explain the difference between complete survey and sampling survey, and define sample size and the number of samples with concrete examples such as a national census.

Explore probability sampling and non-probability sampling, compare statistics and parameters, and learn why probability methods support reliable inferences from samples.

Explore probability sampling methods, including simple random, stratified, cluster, systematic, and multi-stage sampling, and discuss replacement strategies, proportional allocation, and biases in achieving representative samples.

Explore random sampling in Python using pandas to create a sample from dataset, including with and without replacement, reproducibility with random_state, and viewing a sample from the Boston data.

Discover how the law of large numbers makes the sample mean converge to the population mean as sample size grows, illustrated by rolling a fair die.

Demonstrate the law of large numbers by simulating dice rolls with NumPy, random, and Matplotlib, showing the sample mean converges toward the population mean of 3.5 as trials increase.

Explore the central limit theorem, showing that as sample sizes grow, the distribution of the sample mean becomes normal with mean μ and variance σ^2/n, enabling population mean inference.

implement a 1000-trial dice-rolling simulation to illustrate the central limit theorem, compute trial means, and show convergence to normal and to the population mean with line plots and histograms.

Learn how experimental and observational studies differ, including randomized controlled trials and crossover designs, cross-sectional and longitudinal cohorts, and case-control methods to explore causality and bias.

Explore Ronald Fisher’s principles for designing experiments—randomization, replication, and local control—to minimize systematic errors and understand complete randomization and the role of replication.

Learn how samples estimate population parameters in statistics, covering point estimation, unbiased variance, and interval estimations used for hypothesis testing.

Learn how point estimation infers the population mean from sample data within inferential statistics, and distinguish estimation from testing, with an introduction to interval estimation.

Explore point estimation of the population mean using sample data, guided by the law of large numbers, with the sample mean as mu hat and unbiased, consistent estimators.

Explain the difference between sample variance and unbiased variance, showing that using n-1 in the denominator makes the estimator unbiased and consistent for the population variance.

Understand standard error as the standard deviation of an estimator, using unbiased standard deviation to estimate population standard deviation, with the sample mean's standard error equal to sigma over sqrt(n).

Load the Boston housing dataset from scikit-learn, sample 10 cases, and compute sample and population means, variances, and standard deviations for the nox variable using Python and the statistics library.

Explore interval estimation and confidence intervals for the population mean, including 95% confidence levels, and when to use the standard normal distribution with known variance or the t-distribution when unknown.

Estimate a population mean via interval estimation with known variance, using a 95% confidence interval from the standard normal distribution and the sample mean.

Explain how to specify and interpret a 95% confidence interval for the population mean, compare it with the 90% interval using z-scores -1.645 and 1.645, and discuss percentiles.

Explore how increasing sample size narrows a 95% confidence interval around the population mean, illustrated by n = 10, 100, 1000 using the z-score 1.96.

Estimate a population mean via interval estimation when variance is unknown, using unbiased variance and the t-distribution with degrees of freedom, and compare to z-based methods.

Compute a 95% confidence interval for the population mean when variance is unknown by using the t-distribution, selecting the t-value from the table and interpreting interval width with sample size.

Learn to construct 95% confidence intervals for the difference between two population means, using dependent (paired) and independent designs, with t-values and pooled variance where appropriate.

Estimate a population proportion via interval estimation using binomial trials and the standard normal z. Form a 95% confidence interval from the sample proportion by replacing p with p'.

Determine the minimum sample size using interval estimation for population mean and proportion, balancing margin of error with sample size, z-score 1.96, and unbiased variance with the t-distribution when needed.

Explore the chi-square distribution, the sum of squares of independent standard normals, how its shape changes with degrees of freedom, and that its mean is k and variance is 2k.

Demonstrate the reproductive property of chi-square distributions. Show sums of independent chi-square variables yield df equal to sum of dfs, and note k-1 degrees of freedom with the sample mean.

Compute a 95% confidence interval for population variance using non-symmetric chi-square distribution with n-1 degrees of freedom, relying on s^2 as unbiased variance and 0.975/0.025 points; illustrated with weight data.

Implement interval estimation in Python for population mean with known and unknown variance using norm and t intervals. Apply binomial and chi-square intervals for population proportion and variance with scipy.stats.

Explore how hypothesis testing uses sample data to infer population characteristics. Statistical hypothesis testing, grounded in theory, provides objective conclusions for data science and machine learning.

Examine statistical hypothesis testing to assess a population parameter hypothesis using sample data, distinguish real differences from chance, and follow the conservative step-by-step process.

Define null and alternative hypotheses (H0 and H1) for comparing two groups, such as left versus right image upvotes, and decide whether to reject the null to claim a difference.

Set significance level to control Type I error in hypothesis testing, using p-values to judge if observed differences are due to chance against null hypothesis; smaller p-values imply stricter criteria.

Compute the test statistic to compare sample data with the null hypothesis. Describe z-values and t-values, their distributions, and how degree of freedom n-1 informs t-tests in hypothesis testing.

Learn the difference between one-tailed and two-tailed tests, how they reject the null hypothesis, and identify rejection regions at a 0.05 significance level.

Apply hypothesis testing for a population mean with a two-tailed t-test, using sample mean 4.8, unbiased variance 0.16, n=10, alpha=0.05 to test mu=5.

Test a population mean hypothesis with Python using a 10‑employee motivation survey (mean 4.8, unbiased variance 0.16) and a two-tailed test at alpha 0.05; compare t-value to the critical value.

Conduct a hypothesis test for a population mean using stress data on a 7-point Likert scale, with null mean 2.5 and 5% significance, to decide whether to reject the null.

Explore the two-sample t-test for comparing two group means, including dependent (paired) and independent designs. Apply hypotheses, t-values, and 0.05 significance to evaluate interventions such as before and after motivation.

Demonstrates a dependent-sample t-test in python using numpy and scipy to compare before-and-after motivation data, compute t-values and critical values, and draw conclusions with a two-tailed test.

Perform a dependent two-sample t-test on paired stress data from the same employees using Python to compare time 1 and time 2 and assess significance at alpha 0.05.

Explore the independent sample t-test to compare means of two independent groups, using pooled variance, the null hypothesis mu1 - mu2 = 0, and Welch's t-test when variances differ.

demonstrates performing an independent two-sample t-test in python with numpy and scipy, including manual t-value calculation, pooled variance, two-tailed testing, and interpreting a non-significant result.

Compare stress levels between technology and sales with a two-sample t-test. Use Python to compute means, variances, pooled variance, and p-values at 1% significance, concluding the null is not rejected.

Apply hypothesis testing for population proportion using z statistics with large samples, using p-hat and p0 under the central limit theorem to decide if churn exceeds 5%.

Apply Python to perform hypothesis testing for a population proportion, using a churn-rate example, compute z-score and p-value, and decide at alpha 0.05 whether the churn rate differs from 5%.

Perform a population-proportion hypothesis test on a 100-person sample where 30 prefer working from home, using a 5% level to compute the z-score and p-value and conclude more than 20%.

Learn to implement a chi-square goodness-of-fit test, compare observed and expected frequencies, and decide to reject the null hypothesis at a 0.05 level.

Apply a chi-square goodness-of-fit test in Python using scipy.stats to compare f_obs with f_exp, yielding a p-value of 0.178 and not rejecting the null at 0.05.

Perform a chi-square goodness-of-fit test at 5% to check if the IT sample (counts of software, database, data scientists) matches the population; the p-value shows rejecting the null.

Learn to perform a chi-square test of independence on a crosstab, compute expected values under the null, and apply degrees of freedom m-1 by n-1 at 0.05.

Apply a chi-square test of independence in Python to determine whether city area affects economic sector distribution, using a dataframe and chi2_contingency, with p-value above 0.05 indicating independence.

Solve a test of independence using chi-squared with data from two companies across three job categories; calculate p-value with alpha 0.05 and conclude independence.

Test the difference in population proportions using two samples from each population, compute the pooled proportion and z statistic, and evaluate significance at 0.05 to reject the null.

Apply python to test the difference in female CEO proportions between east and west samples, computing sample and pooled proportions, z-score, p-value (two-tailed 0.05), and null rejection decision.

In this exercise, use a two-tailed z-test for population proportion difference to assess gender differences in leadership preference, concluding we cannot reject the null at 0.05.

Explore correlation and simple regression to analyze relationships between two numerical variables, including positive and negative trends, and learn methods and metrics for multivariate and multiple regression.

Learn how scatter plots visualize the relationship between two numeric variables, using x and y coordinates to identify patterns, trends, or lack of correlation.

Understand the linear relationship between two variables, noting positive and negative correlations and when a straight-line fit shows no relation. Remember that correlation does not imply causation.

Learn how the correlation coefficient r measures the strength and direction of the linear relationship between two variables, spanning -1 to 1, with scatter plots illustrating positive and negative trends.

Explore covariance as the x-y co-movement measure and its link to the correlation coefficient. Compute the average product of their deviances from means, noting positive, negative, or zero covariance.

Explore how the correlation coefficient standardizes covariance via standard deviations to measure the strength and direction of linear relationships. Learn why it overcomes covariance's unit and strength issues.

Learn how correlation coefficient captures linear relationships, but misses non-linear patterns, and how outliers can distort the coefficient.

Assess whether a sample correlation reflects a nonzero population correlation using the test of non-correlation, t-distribution, null hypothesis, and Fisher z transformation, and construct confidence intervals for the population correlation.

Explore how spurious correlation misleads when a third factor drives both variables, distinguish correlation from causation, and learn to use partial and stratified correlation analysis to uncover true relationships.

Explore regression analysis and the relationship between independent and dependent variables. Learn how linear regression estimates the regression line, intercept, and slope to predict continuous outcomes in supervised learning.

Reveal how ordinary least squares determines the optimal regression line by minimizing the sum of squared residuals, highlighting why squaring favors larger deviations and dampens outliers.

Develops a mathematical understanding of the least squares regression: derive optimal beta0 and beta1 by minimizing the squared error, and express the regression equation using covariance and variance.

Compare correlation and regression by showing how tightly datapoints cluster around a line and how y changes with x, illustrated by two datasets.

Explore multiple regression with two or more independent variables, interpret partial regression coefficients, and assess significance, using a Boston housing case and standardization to compare effects.

Explore the math of multiple regression using ordinary least squares to derive the optimal regression equation with two independent variables, detailing beta0, beta1, beta2 estimates, residuals, and the variance-covariance matrix.

Explore the four linear regression assumptions—linearity, independence, normality, and homoscedasticity—and how to test them using scatter plots, correlation, and Durbin-Watson.

Examine hypotheses in a multiple regression analysis, test null vs alternative hypotheses for crim, zn, and indus on price, and interpret t-values, p-values, and confidence intervals to assess significance.

Explore coefficient of determination, or R-squared, which measures proportion of variance explained by the regression model. Learn how explained and unexplained variation relate and how adjusted R-squared penalizes adding variables.

Explore residual analysis to assess regression performance and R-square, derive the optimal intercept with OLS, show zero sum residuals, uncorrelated residuals with x, and interpret residual plots for outliers.

Explore multicollinearity in multiple regression, how highly correlated predictors undermine coefficient reliability, by illustrating unique information, explained variance, and how near-zero determinants reveal unstable estimates.

Learn to detect multicollinearity with the variance inflation factor, using R-squared from models with each predictor regressed on the others, and remove variables with VIFs of 5 or more.

Assess regression model validity with the F-test by comparing the F-statistic to the F-table, test null and alternative hypotheses that predictors are zero, and gauge model significance.

Convert categorical independent variables to numerical dummy variables for regression, using 0/1 coding. For multi-category variables, choose a reference group and create non-reference dummies to interpret coefficients and avoid multicollinearity.

Explore the concept of effect size as a complement to p-values in hypothesis testing, covering difference and correlation metrics such as Cohen's d, Hedge's g, correlation coefficients, and R-squared.

Analyze statistical power and the trade-off between Type I and Type II errors. Learn how sample size, variability, effect size, and 5% alpha influence power and required samples.

Learn to perform correlation analysis in Python using the Boston housing dataset, creating pandas dataframes, visualizing with pair plots and heatmaps, and computing correlations with corrcoef.

Implement simple and multiple regression analyses using sklearn and statsmodels, visualize with regplot, evaluate R-squared and p-values, assess multicollinearity with VIF, and perform residual analysis.

Learn to create dummy variables for categorical data like country and gender using pandas get dummies, drop first to set a reference group, and apply dummies to all categories.

Explore statistics fundamentals by examining ANOVA, testing whether variance between groups shows different mean values, and comparing one-way and two-way ANOVA.

Discover how ANOVA tests whether the means of three or more groups differ, using variance to compare groups and test hypotheses about interventions.

Explore how the F-test under ANOVA compares two group variances using the F-distribution, defining null and alternative hypotheses, selecting a significance level, calculating the F-statistic, and consulting the F-table.

Apply the F-test to compare variances between two groups. F = 2.25 with df 9,9 shows we cannot reject the null that the variances are not different.

Explore one-way ANOVA to compare population means across three or more groups, using body height data from three countries to illustrate levels, the ANOVA table, and F-value interpretation.

Explore Tukey’s HSD for identifying significant pairwise differences after a one-way ANOVA, using the studentized range q, HSD thresholds, and the Tukey-Kramer method for unequal sample sizes.

Identify the ANOVA assumptions: normality of residuals, homogeneity of variance, and independence of observations. Use a Q-Q plot to assess normality, with points along a 45-degree line indicating normal residuals.

Implement one-way anova in Python with pandas, statsmodels, and bioinfokit using a toy body height dataset, run tukey hsd tests, and validate assumptions with a q-q plot.

Explore how two factors: textbook type and study time shape exam scores, build the two-way ANOVA table, and interpret significant main effects with a non-significant interaction.

Implement a two-way ANOVA in Python to analyze exam scores by textbook and time, visualize interactions with an interaction plot, conduct Tukey HSD, and verify assumptions with a Q-Q plot.