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.
Every ball on the Galton board makes the same simple decision at each peg: bounce left or right, with a coin-flip's worth of randomness. No ball "knows" about a bell curve, and no two runs ever drop in exactly the same pattern — the shape only emerges once you add up thousands of independent random choices. That's the Central Limit Theorem in physical form, and it's exactly the same mathematics behind why so many unrelated things in nature — heights, measurement errors, exam scores — end up roughly normally distributed.
An IQ score of 130, with a population mean of 100 and standard deviation of 15, has a z-score of exactly 2.0 — and by the 68-95-99.7 rule, only about 2.3% of people score that high or higher.
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.