An overview of statistics and probability: collecting and summarising data, correlation and regression, probability rules, and the binomial and normal distributions, with what HL adds.
Each one has its own page once something is published for it.
Statistics and probability are the mathematics of uncertainty and variation. Statistics starts from data that have already been collected and asks what they show: what is typical, how much the values vary, and whether two quantities are related. Probability works in the other direction, starting from a model of a random process and predicting how often each outcome will happen. The two meet in the probability distributions, which serve as models for real data and connect the patterns seen in a sample with the population it came from.
The topic falls into three strands that build on each other. The first, descriptive statistics, covers how data are collected, presented and summarised (4.1 to 4.3), and how relationships between two variables are measured (4.4 and 4.10). The second, probability, starts with the basic language and rules (4.5 and 4.6) and makes them precise with conditional probability and independence (4.11). The third, probability distributions, models random quantities with discrete random variables, the binomial distribution and the normal distribution (4.7 to 4.9 and 4.12).
Subtopic | What it covers |
|---|---|
4.1 Populations, samples and sampling | Types of data, bias, missing data and outliers, and the main sampling methods |
4.2 Presenting data | Frequency tables, histograms, cumulative frequency graphs and box-and-whisker diagrams |
4.3 Measures of central tendency and dispersion | Mean, median, mode, quartiles, IQR and standard deviation, and the effect of changing units |
4.4 Linear correlation and regression | Scatter diagrams, Pearson's correlation coefficient and the regression line of on |
4.5 Concepts of probability | Sample spaces, events, complementary events and expected numbers of occurrences |
4.6 Venn diagrams, tree diagrams and combined events | Combined, mutually exclusive and independent events, worked from diagrams and tables |
4.7 Discrete random variables | Probability distributions, expected value and fair games |
4.8 The binomial distribution | Repeated independent trials with two outcomes; its mean and variance |
4.9 The normal distribution | The bell-shaped model for continuous data, and normal probability calculations |
4.10 Regression of x on y | Predicting from , and why the two regression lines differ |
4.11 Conditional probability and independence | The formal conditional probability formula and tests for independence |
4.12 Standardising normal variables | -values, and finding an unknown mean or standard deviation |
4.13 Bayes' theorem (HL) | Reversing conditional probabilities, with up to three events; medical testing |
4.14 Continuous random variables (HL) | Probability density functions, their mean, variance, median and mode, and linear transformations |
Each subtopic depends on earlier ones. Quartiles and outliers from 4.2 feed the choice of summary measures in 4.3; the mean and standard deviation of 4.3 become the parameters and of the normal distribution in 4.9. The probability rules of 4.5 and 4.6 give the distributions of 4.7, and the binomial distribution in 4.8 uses the binomial coefficients of 1.9. At HL, 4.14 brings in integration from Topic 5 to describe continuous distributions.
Modelling: a probability distribution is a model of a real random process. A binomial model for the number of defective items in a batch, or a normal model for adult heights, is useful only as long as its assumptions hold, and checking those assumptions is part of the work.
Validity: a conclusion is valid only if the data behind it were collected without bias and analysed with appropriate methods. A strong correlation does not show that one variable causes the other.
Representation: the same data can be shown as a table, a histogram, a box plot or a single number, and each representation shows some features while hiding others.
Approximation: a sample statistic approximates a population value, the mean of grouped data is an estimate, and the relative frequency of an event approaches its probability as the number of trials grows.
Relationships: correlation and regression describe how two variables vary together, and conditional probability describes how knowing one event changes the probability of another.
Both SL and HL students study subtopics 4.1 to 4.12, which make up the SL course for this topic. HL students also study 4.13 and 4.14: Bayes' theorem for up to three events, and continuous random variables with probability density functions, which need integration. HL students also meet the variance of a discrete random variable, , and the effect of linear transformations on and .
The entries of the mathematics: analysis and approaches formula booklet that 4 uses. Each opens in the explained booklet.
And 13 more in the formula booklet.
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