Probability and statistics

Randomness has structure, but you often need to run the experiment.

This prototype gives learners two linked ways into statistics: changing a binomial distribution and exploring what happens when many sample means are collected.

The aim is to make probability feel less like isolated formulae and more like a way of investigating patterns, uncertainty and evidence.

Distribution playground

Change a binomial distribution and watch the shape move

This first panel focuses on X following a binomial distribution: the number of successes in a fixed number of independent trials.

Expected value

4.5

Standard deviation

1.57

Largest probability

0.24

Sampling simulator

See sample means become more predictable

Draw many samples from a population and plot the mean of each sample. Increasing the sample size usually makes the sample means less spread out.

Mean of means

2.19

Spread of means

0.92

Samples drawn

240

Controls

Binomial trials: 10

Success probability: 0.45

Population

Sample size: 5

Number of samples: 240

Skewed population

Most values are small, with occasional larger values. This is useful for seeing how averaging reduces skew.

What this shows

Probability distributions describe long-run patterns, not what must happen next.

Increasing the sample size usually makes the mean of a sample less variable.

The central limit theorem is powerful because sample means often become more predictable even when the population is not normal.

Guided tasks

Move the success probability. When is the binomial distribution most symmetric?

Keep the population fixed and increase the sample size. What happens to the spread of sample means?

Use the skewed population. Why do the sample means look less skewed than the original population?

Joy in the process

The point is not just to finish. It is to notice, test and return.

These tools are invitations to explore. A good mistake, a surprising pattern or a question you cannot yet answer is part of the work, not a failure of it.

The challenge is deliberate: the site should support thinking, not remove the need for it.

Before changing a setting, pause and predict what you think will happen.

Change one thing at a time. What stayed the same, and what changed?

Try to create a surprising case, a broken case, or a beautiful pattern.

Ask what this connects to outside the page: maps, movement, nature, systems or decisions.

Reset, then try again with a new question in mind.

Future extensions

This can become a wider statistics playground.

Add normal distribution shading and inverse probability questions.

Add hypothesis testing simulations with rejection regions.

Add correlation and regression with editable data points.

Add Monte Carlo simulations for estimating areas and probabilities.