Interactive Math & Science Lessons
Statistics is the study of how much you are allowed to conclude. Every tool here answers a version of one question: given this much data, what could still be a coincidence?
Short sequences rather than single tools. Each step hands the next one something specific, so they are worth doing in order.
L1: foundations
MATHFour tools that build one lesson: a summary number is a lossy compression, and how you compress determines what the summary is honest about. Each step hands the next one a specific limitation of single-number summaries.
L2: school
MATHFour tools that build one idea: money is worth different amounts at different times, and a repeated payment can be collapsed into a single formula. Each step here hands the next one something specific.
L2: school
MATHThe birthday problem looks like a party trick and is actually the standard tool for reasoning about collisions. The second step is where computer scientists meet it again, under a different name.
L2: school
STATSix tools and one distribution. Everything here is the binomial in a different limit, or the binomial read backwards: count the successes, let the count get large, and watch a bell curve arrive that nobody put there.
L2: school
STATFour tools asking one question with progressively less to work with: two groups, then any number of them, then counts with no average in them at all. The last tool asks it before the data exist.
L2: school
STATNobody objects to the arithmetic in Bayes' theorem. They object to the prior — where the first number came from, and who chose it. These three tools take that question in the order it gets harder: a prior you can look up, a prior the apparatus hands you, and a prior nobody has.
L2: school
STATA p-value answers one question about one planned test. Real studies run twenty tests, look at the data as it comes in, and publish the results that came out significant, and each of those changes what p < 0.05 is worth. Every page here takes a procedure that sounds harmless and works out exactly what it costs.
L2: school
STATAn expected value is a promise about the average over a great many players, and a gambler is one player with one bankroll. A game can be fair, or even in your favour, and still ruin almost everyone who plays it. Each page draws the average as a dashed ghost and lets you watch what happens to the players who make it up.
L3: undergraduate
MATHSeven tools that turn a picture of scattered points into a number you can defend, then ask how the number is found, how much of it to believe, what happens when nobody tells you the answer — and finally how to grade a prediction that was only ever a probability. The route runs through geometry rather than statistics, because the geometry is what makes the statistics honest.
L3: undergraduate
MATHOne question, asked four times with less and less to go on: what does this matrix actually do to space? The columns answer it outright. The eigenvectors answer it for the directions that survive. The SVD answers it when there are none, and PCA points the same machinery at a cloud of measurements.
L3: undergraduate
STATSix tools on the same 98 years of returns, each handing the next a specific quantity. The sequence answers one question in six steps: what is actually risky about an investment, and how much of what you are told about it can be checked?
L3: undergraduate
COMPA signal is a sum of spinning arrows, and every practical way of reading them off costs something the textbook formula leaves out. A finite window smears a tone that does not fit it across every bin; a shorter window buys time and pays in pitch; an average is a filter with a shape of its own; and the last page is the reason any of it is affordable.
Fitting the mean to your data costs exactly one observation
Everyone is told that the sample variance divides by n − 1 because a degree of freedom was consumed. That names the bookkeeping. The mechanism is that the mean was fitted to the data, and a fitted mean always makes the data look tighter — by one observation’s worth, at every sample size.
You do not pay off a national debt, you outgrow it
For fourteen straight years the US debt got bigger and smaller at the same time. Which of those two things you notice depends entirely on what you divide by.
The average return is the one number your money never earns
An average is a summary of years that already happened. Your money experiences the order they happened in, the risk that produced them, and the uncertainty around the average itself — and each of those costs something the average does not show.
"Cuts your risk in half" can mean one person in a hundred
Relative risk sounds enormous and can be tiny; absolute risk sounds tiny and is what actually happens to you. The gap between the two is where most health reporting lives, and the fix is one subtraction.
Run twenty tests on nothing at all, and one of them will be significant
p < 0.05 promises a 5% error rate on a single test. Run twenty and the chance of at least one false alarm is 64%. The fix is arithmetic; the hard part is that nobody counts the tests they did not report.
Every group in the data trends down. The total trends up.
The pooled slope is +0.0928. Split the same rows by one extra column and the two groups slope at −0.5842 and −0.6109. Both descriptions are arithmetically correct, and only one of them answers your question.
There are two regression lines, and R² is the gap between them
Fitting y against x and fitting x against y give two different lines through the same points. The ratio of their slopes is exactly R squared, which turns that familiar goodness-of-fit number into something you can see.
Why a 99% busy system is not slightly worse than a 75% busy one
Queue length grows as one over the headroom you have left, so the last few percent of utilisation cost more than all the rest combined. This is why systems seem fine and then fail without an intervening warning.
Randomness buys accuracy at a terrible exchange rate
A Monte Carlo estimate improves as the square root of the sample count, so every extra correct digit costs a hundred times the work. The strange part is that this rate holds whatever you happen to be simulating.
Twenty-three people, and the coincidence you should expect
With 23 people in a room the odds of a shared birthday pass half. A room that size holds 253 pairs, and it is pairs that matter — the count grows with the square of the guest list.
A 99% accurate test is usually wrong
Screen a rare condition with a 99% accurate test and 495 of every 594 positives are false alarms. How rare the condition is decides that, and it is the figure nobody quotes beside the accuracy.
