Distributions: The Shape of Data

You already know the centre (mean, median) and the spread (variance, standard deviation) of your data. But imagine two datasets with the same mean of 50 and the same standard deviation of 10 — one looks like a bell, the other has a long tail to the right. The shape changes everything.

The shape of a distribution determines which summaries you can trust, which models will work, and whether rare events are truly rare — or just hidden in the tail. This article introduces three distribution shapes that cover most of what you will encounter in data science.

The Normal Distribution — The Bell Curve

The dart-board analogy breaks at three points: first, a dart board is two-dimensional while a normal distribution is typically one-dimensional. Second, real players have systematic bias (they don’t perfectly centre). Third — and most importantly: not all data follows a bell curve.

Example: Men in Germany average around 180 cm tall with σ ≈ 7 cm. 68% are between 173 and 187 cm. 95% are between 166 and 194 cm. A 210 cm man is over 4σ above the mean — extremely rare. If σ were 2 cm instead of 7, 95% would be between 176 and 184 cm — a much narrower range with the same mean.

The bell curve is symmetric — equally distributed on both sides. But what happens when the bell hangs crooked?

Skewness & Long-Tail — When the Average Lies

In our thought experiment there are only two extremes (40,000 and 10 million). In the real world, the long tail is continuous — there are also people earning 100,000, 500,000, and 2 million euros. The curve tails off slowly to the right rather than jumping abruptly.

Both companies have a mean salary of 50,000 euros. But reporting "average salary 50,000" for Company B is technically correct yet deeply misleading. The median of 35,000 euros describes the typical salary far more honestly. Long-tail distributions are common in finance ("Black Swan" events), social networks (a few superstar accounts, many with few followers), and machine learning (rare but critical edge cases that live in the tail).

Interactive: Centre, Width and Skewness

Try it yourself. The box starts with the example from the introduction: a bell with mean 50 and standard deviation 10. With μ you move the centre, with σ you make the bell wider or narrower; the arrows show ±1σ, and below you find the ranges of the 68-95-99.7 rule. Then pull the skewness slider: the curve grows a long tail to the right. The median stays at μ while the mean wanders into the tail – just like the CEO in the thought experiment. The skewed curve is a drawn example, not measured data.

So far we have measured continuous data — salaries, heights, temperatures. But what if we are counting events?

Discrete vs. Continuous — Counting vs. Measuring

The analogy breaks at two points: first, real phone calls are not perfectly independent — a call can trigger follow-up calls, violating Poisson assumptions. Second, temperature is measured with finite precision (one decimal place), technically making it discrete in practice — but the continuous model is the better abstraction.

Interactive: The Central Limit Theorem

We just saw that far from all data is bell-shaped — skewed and multi-peaked distributions are everywhere. And yet the bell curve keeps reappearing. The reason is one of the most surprising theorems in statistics: no matter how crooked the source distribution, the distribution of its means turns normal. Pick any source shape below, draw many samples, and watch the histogram of their means form a bell — and grow ever narrower as the sample size n increases.

Takeaways

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