Become a Probability & Statistics Master

Build a data foundation in probability and statistics by visualizing data with bar graphs and pie charts to quickly interpret large datasets.

Explore data concepts and data tables, identifying individuals and variables, and distinguishing quantitative from categorical data. Learn how one-way data differs from two-way data and table orientation.

Learn to visualize one-way data with bar graphs and pie charts by counting continents hosting the summer games, comparing totals, percentages, and using legends and titles.

Explore line graphs and ogives as tools for displaying change over time and accumulated totals, compare with bar graphs, and analyze rate of change in data.

Understand how two-way data extends one-way data, using revenue by month and year to build two-way tables with row and column totals and comparative graphs.

Learn how to visualize two-way data with Venn diagrams by converting a two-way table of frosting and sprinkles into a four-region diagram, showing intersection, single, and neither cases.

Explore frequency tables and dot plots by constructing a frequency table from parking lot car types, then plot dots to visualize category frequencies and compare counts.

Convert two-way data tables to relative frequency tables by column or by row, using blue and white surfboards. Compute percentages or decimals with column or row totals summing to one.

Explore how a joint distribution links hours spent exercising and weight loss to reveal their correlation, like a relative frequency table.

Explore histograms and stem-and-leaf plots as data visualizations, grouping data into equal-sized buckets and encoding values with stems and leaves.

Learn to build a histogram from a raw data set by determining range, choosing bin counts, assigning data to bins, and plotting frequencies.

Explore central tendency and data spread by analyzing mean, median, and mode, and observe how changes affect the center and spread of data.

Explore measures of central tendency, including mean, median, and mode, and learn how to compute and interpret them as balancing points in data.

Explore measures of spread, including range and interquartile range, and learn how data disperses around the center using golf scores as examples and quartile concepts.

Explore how shifting and scaling a data set affects mean, median, mode, range, and interquartile range, and compare outliers' impact on the mean versus the median.

Learn how box-and-whisker plots show the median, minimum, maximum, and interquartile range. Track the five-number summary and quartiles, and remember the number line marks positions, not data.

Explore how data distribution shapes arise from spread around the mean, including outliers. Trace histogram to frequency polygon to density curve, and study the normal distribution, bell-shaped and symmetric.

Distinguish population from sample and apply the correct mean, variance, and standard deviation formulas. Learn how N versus n and mu versus x-bar shape unbiased estimates and interpretation.

Learn how a histogram becomes a density curve by smoothing the relative frequency histogram and frequency polygon, then use the area under the curve to interpret age distribution.

Compare symmetric and skewed distributions, note the effect of outliers on the mean, and apply the 1.5 IQR rule to identify outliers, using the median and IQR for skewed data.

Explore normal distributions, the empirical rule, and z-scores; learn to interpret mean and median at the center, compute area portions, and use z-tables to find percentiles.

Chebyshev's theorem provides a conservative estimate of the data within k standard deviations of the mean for any distribution shape, unlike the empirical rule for normal data.

Learn how covariance measures how two variables move together, with positive, negative, or zero covariance, using population and sample formulas, and note its unit sensitivity prompting the correlation coefficient.

Learn how Pearson's correlation coefficient standardizes covariance into a unit-free measure of linear relation, and how to interpret strength versus causation, with age and net worth example.

Learn to compute weighted means for population and sample data, and extend the method to grouped data using class midpoints to estimate mean, variance, and standard deviation.

Calculate the likelihood of heads on a coin flip, pulling a jack from a deck, or rolling a three on a die, then apply basic probability rules and Bayes theorem.

Explore simple probability by defining the sample space, counting favorable outcomes over all equally likely outcomes, and distinguishing theoretical from experimental probability with the law of large numbers.

Derive the addition rule for probability by combining P(A) and P(B) and subtracting P(A∩B) to avoid double counting, demonstrated with union and intersection, mutually exclusive cases, and two-way table examples.

Distinguish independent and dependent events and apply the multiplication rule and conditional probability to coin flips and card draws, including replacement and nonreplacement scenarios.

Explore Bayes' theorem through a two-dice example to compute the probability we chose the biased die given a roll of six, illustrating conditional probability and the formula.

Distinguish between continuous and discrete random variables by comparing heights to family size, and explore probability applications of discrete variables using whole-number outcomes like the number of children.

Explore discrete probability by modeling a discrete random variable, such as the number of heads in two coin flips, and learn its probability distribution, mean, variance, and standard deviation.

Shifting a data set by k moves mean, median, and mode by k. Scaling by k multiplies mean, median, mode, range, IQR, and standard deviation by k; area remains constant.

Explore linear combinations of random variables, derive mean and variance for sums and differences, and apply to normally distributed components to compute the probability a popcorn tin weighs under £3.25.

Explore permutations and combinations and apply their formulas to answer probability questions. Learn how order matters in permutations and how it does not in combinations, illustrated with ice cream choices.

Explore binomial random variables defined by two outcomes, independent trials, fixed trial count, and constant probability of success, with the binomial coefficient and distribution.

Model counts of events with Poisson distributions and apply the Poisson process to compute exact and cumulative probabilities over non-overlapping intervals.

Learn how to compute binomial probabilities for at least or at most occurrences and determine the mean, variance, and standard deviation of a binomial random variable.

Explore Bernoulli random variables as a single-trial case of binomial variables, with mean p, variance p(1-p), and standard deviation sqrt(p(1-p)).

Explore geometric random variables, where trials are independent and stop at the first success; learn the probability formula p(1-p)^{n-1}, and the mean 1/p and variance (1-p)/p^2.

Explore inferential statistics by learning how to sample: randomly select a group from a population, survey it, and infer population traits while considering sampling risk, sampling distributions, and confidence intervals.

Explore sampling, random selection, study design, and data collection to understand observational studies and experiments, and how they reveal population parameters while addressing bias, correlation versus causation, and replication.

Learn how to obtain a representative sample from a population by using simple random, stratified, or clustered sampling. Minimize bias to enable inferences about population parameters.

Explore the sampling distribution of the sample mean and its link to the population mean; learn that its mean equals mu and apply the central limit theorem and standard error.

Explore conditions for inference with the sampling distribution of the sample mean to infer the population mean: random sampling, normal or large counts, and independence or the 10% rule.

Learn how the sampling distribution of the sample proportion becomes normal for n ≥ 30, and compute its mean, standard error, and variance using p_hat and P.

Meet random sampling and the normal large counts condition, plus the 10% independence rule. Use the sampling distribution of sample proportion with p-hat and p, and apply finite population correction.

Explore the student’s t distribution as a wider, flatter alternative to the z distribution, with degrees of freedom n-1 influencing its shape and when to use t vs z.

Construct confidence intervals for the population mean using sample means, margins of error, and confidence levels; apply z or t as appropriate, and note how sample size affects width.

Construct a confidence interval for a proportion using p-hat, margin of error, and z-values; apply the finite population correction when sampling from a finite population, illustrated with sea turtles.

Formulate a good hypothesis statement, then collect data through sampling to test whether the evidence supports or refutes the claim, following the inferential statistics process from start to finish.

Learn how inferential statistics use sample data to infer population parameters via hypothesis testing, with null and alternative hypotheses, alpha levels, and confidence intervals.

Explain how to choose the level of significance and interpret Type I and Type II errors in hypothesis testing, including alpha, beta, confidence level, and tradeoffs with sample size.

Select one- or two-tailed tests from your hypotheses and alpha, then compute the correct z or t statistic using the appropriate formula for known or unknown sigma and sample size.

Learn how to use p-values and alpha to decide when to reject the null, with one-tailed and two-tailed tests, using z-scores.

Explore hypothesis testing for the population proportion, with null and alternative hypotheses, a one-tailed lower test, normality checks, z-score, and p-value based conclusions.

explore how to build and interpret the confidence interval for the difference of means between two populations, using sampling distributions, standard errors, and t- or z-based methods.

Perform hypothesis testing for the difference of population means, define null and alternative, choose two-tailed or upper/lower tail tests, use t or z statistics, and interpret results at significance level.

Explore matched-pair hypothesis testing for dependent samples with before-and-after data, using a weight loss example to compare mean differences, compute t statistics, and interpret results.

Construct a confidence interval for the difference of proportions from two samples, using p-hat, standard error, and a z critical value, and interpret whether zero lies in the interval.

Learn to perform hypothesis tests for the difference of proportions, including two-tailed, upper tail, and lower tail; compute z from p-hats and interpret results from California vs. Colorado truck data.

Learn how to turn scattered data points into a regression equation to model and estimate future values, using regression analysis and scatter plots to assess fit.

Explore regression with a data set and scatterplots to fit the trend line, and learn to compute slope and intercept to estimate Y from X.

Learn how to evaluate linear fit with the correlation coefficient r, interpret residuals, and understand the least squares line as the line of best fit that minimizes squared residuals.

explain how the coefficient of determination r squared measures error reduction by the regression line compared to the mean of y, and how rmse describes residual spread around the line.

Explore chi-square tests, including homogeneity, independence, and goodness of fit, learning how to verify conditions, compute expected values and degrees of freedom, and apply the chi-square statistic.

Celebrate your completion of the course with a solid grasp of probability and statistics, and explore more math courses to continue your journey.