Mathematics: Analysis and Approaches HL

Statistics and probability

Higher-level probability reasoning, Bayes theorem, and continuous random variables.

Use Bayes theorem and conditional probability Apply conditional probability and Bayes theorem in structured probability contexts. Use continuous random variables Use density functions, cumulative probability, expectation, and intervals for continuous variables. Use expected value and variance of discrete variables Calculate and interpret expectation, variance, and standard deviation for discrete variables. Use probability generating functions Use generating functions to extract probabilities, expectation, and distribution information. Use moment generating functions Extract moments and interpret distributions using moment generating functions. Use covariance and correlation of random variables Calculate and interpret covariance, correlation, and linear association for random variables. Use bivariate distributions Calculate and interpret probabilities from joint distributions. Read values from a bar chart Read the height of a named bar directly off a bar chart on a value grid. Builds the data-handling skill of extracting a value from a categorical chart, not being given it in the text. Use conditional expectation Calculate and interpret conditional expectation in discrete or continuous contexts. Find a difference from a bar chart Read the heights of two bars off a bar chart and find the difference. Extends bar-chart reading into "read then compute" by comparing two categories. Use transformations of random variables Find distributions, means, or probabilities after transforming random variables. Predict from a line of best fit Use the line of best fit on a scatter graph to predict a y-value at a given x. Reads the trend line, not an individual data point - a core skill for correlation and regression. Use normal approximations Apply and interpret normal approximations with appropriate conditions and corrections. Read a value from a pie chart Read a sector angle off a pie chart and calculate the number in that category as (angle / 360) times the total. Combines chart reading with proportion. Use joint probability density functions Calculate and interpret probabilities from joint density functions. Read a frequency from a histogram Read the frequency of a class interval off a histogram. Reinforces that a histogram has touching bars over a continuous scale, unlike a bar chart of categories. Find marginal and conditional distributions Derive marginal and conditional distributions from a joint model. Estimate the median from a cumulative frequency curve Use a cumulative frequency curve (ogive) to estimate the median or quartiles: read across from the cumulative frequency axis to the curve, then down to the data axis. Use linear combinations of random variables Calculate means and variances for linear combinations of random variables. Read values from a box-and-whisker plot Read the five-number summary (minimum, quartiles, median, maximum) off a box-and-whisker plot, and find the interquartile range and range. Apply the central limit theorem Use the central limit theorem to approximate sampling distributions. Use covariance matrices Interpret covariance matrices and relationships between random variables. Analyse correlation under linear transformations Determine how linear transformations affect correlation, mean, and variance. Use moment relationships for distributions Use raw and central moments to interpret distribution behaviour. Approximate distributions with continuity corrections Apply continuity corrections when approximating discrete distributions. Use conditional variance Calculate and interpret conditional variance in probability models. Apply law of total expectation Use total expectation to combine conditional components. Use transformations of continuous variables Transform continuous random variables and interpret resulting densities or probabilities. Analyse sampling distribution approximations Use sampling distribution approximations and interpret assumptions.