How to learn statistics

This stage builds the core intuition that a sample statistic is a random quantity. You simulate sampling from a known population, estimate standard error, and construct bootstrap intervals before any formal test. It sits first because every later method is a way of managing the variability you see here.

• population vs sample
• random sampling
• sampling bias
• sample mean distribution
• standard error
• bootstrap resampling
• confidence interval interpretation

You can now: Take a dataset of 100 values, draw 1,000 bootstrap samples, and state a 95% interval for the mean, explaining what the interval does and does not say.

Once you know a sample is uncertain, you can summarise it without overclaiming. This stage covers measures of centre and spread, skew, outliers, and the normal distribution as a model, not a fact. It sits after sampling because summaries are only useful when attached to the sample that produced them.

• mean vs median
• variance and standard deviation
• quartiles and IQR
• histograms and box plots
• skewness
• outliers
• z-scores
• normal distribution

You can now: Given a skewed dataset, choose median and IQR over mean and standard deviation, and justify the choice with a plot and one sentence.

Probability supplies the language for uncertainty. You learn rules for combining events, conditional probability, Bayes' theorem, and the common distributions that generate data. It sits here, not first, because probability only matters once you have a question about a sample and a population.

• probability rules
• conditional probability
• independence
• Bayes' theorem
• random variables
• expectation and variance
• binomial and Poisson distributions

You can now: Given disease prevalence 1%, test sensitivity 90%, specificity 95%, calculate the probability that a positive result is a true positive.

Here the sampling intuition becomes formal method. You construct confidence intervals for means and proportions, then meet null hypotheses, p-values, and the errors a test can make. It sits after probability because a p-value is a conditional probability, and after sampling because a test is a claim about sampling variability.

• point estimation
• confidence intervals for a mean
• confidence intervals for a proportion
• null and alternative hypotheses
• p-value
• type I and type II errors
• statistical power
• two-sample t-test

You can now: Run a two-sample t-test on two small datasets and report the difference in means with a 95% confidence interval, not just a p-value.

Regression connects variables and lets you estimate effects while holding others constant. You learn to fit a line, interpret coefficients, check residuals, and recognise confounding and interaction. It sits after testing because a regression coefficient is an estimate with a standard error, and every test you ran applies here.

• simple linear regression
• least squares
• interpretation of slope and intercept
• R-squared
• residual plots
• multiple regression
• confounding
• logistic regression

You can now: Fit a simple linear regression to a dataset, check the residuals for curvature or changing spread, and interpret one coefficient in plain language.

The final stage forces you to work with messy, unfamiliar data and report what you can and cannot conclude. You choose methods, check assumptions, and write results with uncertainty intervals. It sits last because everything before it is a tool; here you decide which tool fits and defend that choice.

• data cleaning
• exploratory data analysis
• assumption checking
• method selection
• effect sizes
• confidence intervals in reporting
• visualisation
• limitations and caveats

You can now: Take an unfamiliar dataset, choose an appropriate analysis, and write a one-page summary that includes uncertainty intervals and states what the data cannot tell you.