A mean and a standard deviation describe data as if it were a bell curve. These four views test that assumption instead of relying on it — and if your data is skewed, has two peaks or carries an outlier, every one of them will show it while the two summary numbers quietly will not.
Take repeated samples of size n from any of these sources — none of them a bell curve — and plot the averages. The averages pile up into a normal distribution anyway, and they do it with a spread of exactly σ∕√n. That is the single most-used fact in statistics: it is why a confidence interval works on data you know nothing about.
Each ball bounces left or right at every peg with a coin-flip’s worth of randomness. The shape emerges only when thousands of these choices add up.
Each added row is one more decision per ball; the discrete binomial converges on the smooth normal curve, with a spread of σ/√n for averages.
How many standard deviations a value sits from the mean. 0 is exactly average, ±1 is one σ away, beyond ±3 is about 1 in 370.
The median and quartiles assume nothing. The z-score probabilities do assume a bell curve: use the Q–Q plot to check.
A measure of how spread out values are around the mean. A small σ means values cluster tightly; a large σ means they are scattered widely. It is in the same units as the data itself, unlike variance.
Add up enough independent random effects (coin flips, measurement noise, genetics) and the result tends toward a normal distribution, regardless of what the individual effects looked like. It is why the bell curve shows up so often in nature.
How many standard deviations a value sits from the mean. A z-score of 0 is exactly average; ±1 is one σ away; beyond ±3 is genuinely rare for a normal distribution.
Every peg-bounce is a fresh call to the browser’s random number generator: no seed, no script, no two runs sharing a history. The bell shape is a statistical tendency of thousands of independent coin flips, not a scripted outcome.
Skew is which side the long tail is on: positive means a few large values stretch the right. Kurtosis is about the tails rather than the peak. Positive excess kurtosis means extreme values are more common than a bell curve predicts, which is precisely the assumption that breaks risk models.
A histogram of a small sample is mostly a statement about where you put the bin edges. A Q–Q plot uses every point exactly once and turns “is this normal?” into “is this line straight?”, which the eye judges far better.
Tukey’s fences sit one and a half interquartile ranges beyond the quartiles. On genuinely normal data they flag about 0.7% of readings: rare enough that a flagged point is worth a look, common enough that finding one proves nothing by itself.
σ describes how spread out the data is, and it does not shrink as you collect more. The standard error describes how uncertain the mean is, and falls as σ/√n, which is why halving your margin of error takes four times the readings.
Each ball's path depends on genuine random numbers generated fresh every time you press play — there's no fixed script or seed behind it. Run it ten times and you'll get ten different exact patterns, even though the overall bell-curve shape stays remarkably consistent — that consistency-despite-randomness is the entire point.
It shifts the probability of bouncing right at every single peg away from 50/50 — push it toward 0.9 and the whole distribution skews hard right, since each ball is now far more likely to drift that way at every one of its many decisions.
Each row adds one more independent random decision. More decisions per ball means more possible final bins and a distribution that sits closer to the smooth, continuous normal curve — the discrete binomial distribution converges to it as the row count grows.
The descriptive numbers — median, quartiles, IQR, MAD — make no assumption at all and are always fair. The ones that do assume a bell curve are the z-score probabilities and, more loosely, the confidence interval. Check with the Q–Q plot above: if the points bend away from the line, quote the median and IQR instead of the mean and σ, and treat any “probability of a value this extreme” with real suspicion.
Not on the strength of the 1.5 × IQR rule alone. On genuinely normal data that rule flags roughly 0.7% of readings, so in a sample of 300 you should expect two with nothing wrong at all. A flagged point is a prompt to go and find out what happened — a transcription slip, a different machine, a real rare event. Deleting points because they are inconvenient is how a dataset comes to support a conclusion it never contained.
That is the whole point of it. A single coin flip has only two outcomes and looks nothing like a bell curve, but the average of thirty of them does — with a spread of exactly σ∕√n. The theorem is a statement about averages, not about data, which is why it applies to almost every measurement anyone ever takes.
Use the population formula (divide by n) when your data is the entire group you care about; use the sample formula (divide by n−1) when your data is a smaller sample standing in for a larger population — dividing by the slightly smaller number corrects for the fact that a sample tends to understate the true spread.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.