The Sampling Lab

You never see the whole population, only a sample. Watch thousands of samples fall, and see why their means end up on a predictable curve.

Statistics begins with an uncomfortable fact: you only ever see a sample. The mean you compute, the spread you measure and the p-value you report would all have come out differently with a different handful of data. This site treats that fact as an experiment you can run. Draw a sample, draw another, draw ten thousand, and watch what the numbers do.

Six experiments follow. The first two give you the vocabulary: spread, the normal curve and the z-score, each with an exact calculator built into the explanation. The middle two show where the familiar shapes come from. The last two use everything to answer the questions people actually ask of data: how sure can I be, and could this just be chance?

A dot is one measurement. A short tick is a statistic: a number computed from a sample, such as its mean. Ticks from many samples stack up into a histogram, as the yellow ones do above. A dashed line is the truth: a property of the whole population, or a curve that theory predicts from it, which in real life you never get to see.

The experiments

Standard deviation and varianceWhat does a standard deviation actually measure, and why divide by n − 1?Spread drawn as squares, a calculator that shows its work, and why n − 1.
The normal distribution and z-scoresWhat is a z-score, and how do I turn it into a probability or percentile?Areas under the bell curve, an exact z table, and when 68–95–99.7 fails.
Where distributions come from: binomial and PoissonWhere do the binomial, Poisson and normal distributions come from?A Galton board, an exact binomial calculator, and the Prussian horse kicks.
Sampling distributions and the central limit theoremWhy do sample means follow a normal distribution, and how fast?Sculpt a population and watch the sample means turn normal.
Confidence intervalsWhat does a 95 % confidence interval actually mean?Interval rain, t versus z, and the most common misreading.
P-values, significance and powerWhat does a p-value mean, and what is statistical power?The null-world simulator, power curves, and the garden of forking paths.

Tables and calculators

Each one shows its working and draws the answer, and links back to the chapter that explains it.

Sampling distribution simulatorA population, one sample from it, and the means of 4,000 samples, side by side as n changes.
Standard deviation calculatorSample and population standard deviation from your numbers, every step shown.
Confidence interval calculatorFor a mean (Student t) or a proportion (Wilson), with the standard error.
Sample size calculatorHow many you need for a margin of error, for a mean or a proportion.
The z tableEvery Φ(z) from −3.49 to 3.49, exact and clickable, with the critical values.
Z-score calculatorA value to its z-score and percentile, and back again.
P-value calculatorFrom a z score, a t score or a sample mean, one- or two-sided.
Binomial calculatorExact P(X = k), at most, at least, or a range.
Galton boardBalls bounce through pins into the binomial distribution, one at a time.

How this is made and checked

Every figure is driven by a small, tested engine: exact formulas where they exist (the normal and Student t distributions, the binomial and Poisson probabilities), and seeded simulations everywhere else, so the same link always reproduces the same samples. An independent test suite, written from the published equations without reading the engine, checks the numbers. Every equation, number and historical claim on the site is listed with its source in a claims register, and each figure says what it simplifies. More about how the site is checked.