Not formulas for their own sake, but a tool to tell signal from noise. The core concepts — each with a "why, deeper" and an interactive you can play with.
The core
Descriptive statistics
Level 0
The first skill: boil a pile of numbers down to a picture you can hold in your head — the center (mean, median, mode), the spread (std, IQR), the shape, and any outliers. Every analysis starts here: before you prove anything, you just need to see what's going on.
Common mistake: Trusting the mean on skewed data — one Bezos walks into a bar and the 'average' guest is a billionaire. For salaries, prices and durations, reach for the median and look at the whole distribution, not a single number.
Mean and median split apart exactly when the shape is lopsided — and that gap is itself a signal. Two groups with the same average can be worlds apart in spread, so a center without a spread is nearly useless. And an outlier isn't always trash: sometimes it's a typo, sometimes it's the most interesting thing in the data — so find out where it came from before you decide to delete it.
Probability & distributions
Level 1
The language of chance. A distribution is a map of which values are how likely — the normal 'bell', the binomial, the Poisson. Without this language you can't even say what 'it happened by chance' really means.
Common mistake: Assuming everything is normal. It usually isn't — incomes, waiting times, click counts follow very different laws. Before you apply a method that assumes a bell curve, just look at the histogram first.
The 68-95-99.7 rule tells you how much data sits within 1, 2 and 3 standard deviations of the mean for a normal curve — your quick ruler for 'is this value weird?'. Why is the normal everywhere? The Central Limit Theorem: the sum or average of many small independent effects almost always lands on a bell, no matter how twisted the original distribution was. The CLT is the bridge from probability to statistics, and the best way to feel it is to simulate it.
Sampling & uncertainty
Level 2 · the heart
The heart of the whole thing: use a slice of data (a sample) to speak about the whole (the population) — honestly, with a margin of error. This is where standard error, confidence intervals and the bootstrap live — that lovely trick where you resample your own data thousands of times and watch how much the estimate wobbles.
Common mistake: Reading a '95% CI' as 'there's a 95% chance the true value is in here.' The honest version: if you repeated the experiment many times, the method would capture the truth 95% of the time. And always report an estimate with its interval — '12% ± 2%' is far more honest than a bare '12%'.
Standard error shrinks as the sample grows — but by the square root: to get twice as precise you need four times the data, which is why endlessly collecting more stops paying off. The bootstrap is beautiful because it hands you a confidence interval for almost any statistic without a single formula: resample with replacement, compute the statistic 10,000 times, take the 2.5th and 97.5th percentiles. Here the computer replaces half a calculus course.
Hypothesis testing
Level 3 · crucial
The formal way to answer: is this difference real, or just noise? You set a null hypothesis — 'no effect' — and compute a p-value: the chance of seeing a result this extreme (or more) if there really were no effect. A small p means the data fit badly with 'nothing is going on.'
Common mistake: The million-dollar mistake: a p-value is NOT 'the probability the hypothesis is true' and NOT 'the probability the result is chance.' p = 0.03 does not mean 'the effect is 97% real.' The cure is simple: say the correct definition out loud until it's automatic.
Two errors hide here: Type I — seeing an effect that isn't there (a false alarm, whose rate you set with the 0.05 threshold), and Type II — missing a real effect. Power is your chance of catching an effect when it exists; it grows with sample size, and it's exactly power that drives the 'how much data do I need?' calculation for A/B tests. A t-test compares two group means, chi-square compares proportions and categories — the two everyday workhorses, both resting on the same p-value foundation.
Relationships & regression
Level 4
How one thing relates to another. Correlation is a number from −1 to +1 for how much two quantities move together (Pearson for linear, Spearman for monotonic). Regression goes further: it fits a line Y = a + b·X, and the slope b tells you how much Y changes when X goes up by one — your first step toward prediction.
Common mistake: Confusing correlation with causation. Ice-cream sales correlate with drownings, but the culprit is the heat, not the ice cream — there's almost always a lurking third factor or pure coincidence. Only a randomized experiment proves causation, never an observed association.
R² tells you what fraction of Y's variation the model explains (0 is nothing, 1 is perfect), but a high R² alone doesn't mean 'the model is right' — always look at the residual plot, not just the number. Spearman is sturdier than Pearson against outliers and catches curved-but-monotonic links. And remember: regression describes the relationship in the data you saw; it makes no promise that things stay the same outside that range.
Experiments & A/B tests
Level 5 · capstone
The thing a product analyst actually gets paid for. You split users randomly into groups A and B, change one thing, compare a metric — and it's the randomization that lets you talk about causation instead of mere correlation. From there it's a straight line: hypothesis to a 'ship it or not' decision.
Common mistake: Chasing 'p < 0.05' and forgetting the effect size. On a huge sample even a pathetic 0.1% lift in conversion turns 'significant' — statistically significant ≠ worth doing. Always ask: is this effect actually big enough to ship?
Sample size and power are computed before the test: skip that and you'll either miss the effect or run the test forever. Watch out for multiple comparisons — check 20 metrics at a 0.05 threshold and one will 'fire' by chance; fix your hypothesis up front and correct for it (Bonferroni and friends). And the Bayesian approach is a great 'what to learn next': it gives you what your gut actually wants — the probability of the hypothesis given the data, not the other way around.
See it live
Interactive visualizations and videos to "see" statistics by hand. Start with Seeing Theory.
A visual textbook of probability and statistics from Brown University, where every animation is yours to play with. It walks the whole path — probability, distributions, inference, regression — beautifully and step by step. The best place to start if you want to feel statistics rather than memorize it.
Tip: Don't read it front to back: grab a concept from the core, find it here, drag the sliders, then apply it to your own data right away.
Questions
How does the bell shift when you drag the mean and the spread?
What happens to the sample mean as the sample size grows?
Kristoffer Magnusson's set of interactive visualizations: confidence intervals, correlation, p-values, effect size, statistical power. Move a slider and watch it change instantly. The best cure for the most abstract ideas.
Tip: Open the power visualization and drag the sample size — you'll instantly get why A/B tests compute it up front.
Visual explainers of key ideas: conditional probability, ordinary least squares (regression), Markov chains. Each one comes with an interactive picture you steer yourself.
A tool where bootstrapping and randomization tests happen with clicks — you watch the distribution and the confidence interval build up on real data. It's the pure embodiment of 'simulate instead of memorizing formulas.'
The classic teaching applets: sampling distributions, the Central Limit Theorem, hypothesis testing. Perfect for watching the CLT emerge before your eyes and seeing how a sample behaves run to run.
Josh Starmer explains any concept in statistics and ML as simply as humanly possible — with little songs and his trademark 'BAM!'. If something won't click from text, look it up on his channel and it will.
Visual math at its finest. For statistics, the videos on the binomial distribution, Bayes' theorem and the Central Limit Theorem are the ones that matter — after them it truly clicks.
A free, structured course with exercises and instant checking. Great as a skeleton: go topic by topic in order and lock it in with practice when you need the discipline.