Statistics you can watch happen
ProbViz is a free set of statistics and probability calculators that draw what they compute. Every answer comes with its curve, its shaded area or its simulation, so you learn why the number is what it is.
Why the balls make a bell
That board is a Galton board, invented by the Victorian scientist Francis Galton in the 1870s. Each ball hits a peg and goes left or right, a coin flip at every row. To land at the far edge a ball has to go the same way every time, and there is only one path that does that. To land in the middle it needs about as many lefts as rights, and there are hundreds of paths that do that. With 12 rows, 924 of the 4,096 possible paths end in the centre bin and just 1 ends in each corner.
Counting paths is exactly the binomial distribution, and with enough rows the binomial is almost indistinguishable from the normal curve. That is the same reason heights, measurement errors and exam totals tend to be bell-shaped: each is the sum of many small independent nudges. Drag the "chance of bouncing right" slider and the pile leans over, which is a skewed binomial. Then see the same idea with real numbers in the binomial calculator and the central limit theorem simulator.
Probability isn’t the only thing people misread as a straight line: ahaboo untangles the classic tax-bracket myth by letting you pour income into each band and compare marginal with effective rates.
Every tool
Normal curve
Areas under the bell curve, z-scores and the tables that go with them.
Describe data
Centre, spread and position of a data set, with every step shown.
Graph data
Turn a list of numbers into a picture you can read.
Probability
Chance, counting and the distributions of discrete and continuous events.
Inference
From a sample to a claim about the population.
See it happen
The ideas that surprise people, run as experiments.
How to learn statistics online for free with these tools
- Start with a picture of the data. Paste a list into the histogram maker or box plot maker before calculating anything. Shape decides which summary makes sense.
- Summarise it. Mean, median and mode for the centre, standard deviation for the spread. Each page shows the full working, so you can check your homework line by line.
- Meet the normal curve. Use the bell curve page to feel what σ means, then the z table and normal calculator to find areas.
- Break your intuition, then rebuild it. Play the Monty Hall game and the birthday paradox. Being surprised is the fastest way to remember a rule.
- Move to inference. The confidence interval page runs 100 samples so that "95% confident" stops being a slogan.
Everything is calculated in your browser with standard formulas (the normal and t areas use the same numerical methods as statistical software), so answers match your textbook and TI-84 to four decimal places.