Simple statistics for traders

After this lesson you'll know the three statistical facts about market returns that break naive intuition — tails, outliers, and what "average" hides.

The bell curve markets don't follow

Most statistics education starts from the normal distribution, the bell curve, where extreme events are vanishingly rare and the average describes the typical case. Daily stock returns look bell-ish in the middle and then part company with the model in the tails: extreme days occur hundreds to thousands of times more often than a bell curve predicts. The textbook example is 19 October 1987, when the US market fell over 20% in one session. A normal model built on the volatility of the time rates that as effectively impossible, and the astronomical "25-standard-deviation" figures sometimes quoted are a measure of how badly the model fits, since the day happened regardless. Smaller versions of it come along every few years.

This one fact sits underneath machinery you've already built, which is why it leads the course. Fat tails are why gap risk gets a course of its own (School II) and why the scenario drill prices the correlated day (School VI, course 5). They're also why the bad-year lesson declines to trust statistics gathered in calm years. If you size risk for normal conditions, the tail days are the ones you haven't sized for.

The outlier's weight

The second fact is that your own results are outlier-dominated in the same way, because trend-following is a fat-tailed business. Take a realistic 30-trade campaign: eighteen losses near −1R, ten modest winners around +1R, and two big ones at +5R and +7R. Sum: −18 + 10 + 12 = +4R. Now delete the +7R trade, one trade out of thirty, and the campaign is negative. The average was mostly those two trades all along, spread thin across the other twenty-eight.

A few consequences, all of them pointing back into the curriculum. The median trade of a healthy trend-following book is a small loss, so the day-to-day experience of running one feels worse than the results are (School VII, course 2). Letting winners run (School V) is where the expectancy lives, and taking profits impatiently caps the outliers, which quietly removes most of what the system earns. And when you review a track record, your own included, ask for the distribution as well as the average. "+0.13R per trade" and "two trades made all of it" can both be true of the same book, and the second one is what tells you how next quarter might feel.

Reading any statistic like a trader

The general skill comes down to a few questions you can put to any number. What's the distribution behind this average? (Means hide outliers; the median tells you the typical experience.) How big was the sample, and across how many regimes? (Course 1.) What got selected before this reached me? (Course 5 makes that one a full lesson, since the results being shown were chosen by whoever is showing them.)

Check yourself

1. Why does 1987 justify the scenario drill better than any argument? (The tail event that models called impossible arrived anyway — the drill prices what the bell curve refuses to.)
2. In the worked campaign, what's the median trade, and why must you know it? (About −1R — the typical day of a profitable trend book feels like losing, and expecting that is what makes holding the discipline possible.)
3. A service advertises +0.3R average per trade over 200 trades. What two follow-ups does this course demand? (The distribution — how much is outliers? — and the regime coverage of the sample.)

The idea this lesson installs

Ask every average what its outliers are hiding.

Next: Course 5 — "How to evaluate any guru, service or course."