The scientific method.

Step 1 of 6

A hypothesis and a prediction

Everyone has hunches about how the world works. Science starts from them too, and adds one habit: before trusting an idea, turn it into a hypothesis, a claim definite enough to predict the result of a measurement, and go and check whether the prediction comes true.

Five pendulums hang from one bar on the bench. Each has a 1 m string and is released from $15^\circ$⁠, but the bobs differ: from 50 g to 800 g. The Hypothesis is fixed on Heavier swings faster: four times the mass, half the period. At Prediction only the blue ghosts move, each at the beat the hypothesis predicts for its bob. Predict what the real bobs will do, then look.

Step 2 of 6

One clear miss

At Test each bob is timed with a hand-held stopwatch. That is not exact: your thumb starts and stops the watch a little early or a little late, so each timed swing is off by about 0.12 s. Each timing is a trial. The mean of several trials is surer, and the chart beside the Scene draws an error bar around it, two standard errors each way.

At Verdict every pendulum gets a tag: mint where the mean is within its bar of the prediction, pink where it clearly misses. A general hypothesis claims something about every case, so one clear miss is enough to refute it.

Step 3 of 6

Survives, not proven

Now the Hypothesis is the textbook formula, $T = 2\pi\sqrt{L/g}$⁠, and the pendulums differ in string length: from 0.2 m to 1.0 m. Trials per pendulum is fixed at $20$⁠, and the error bars are narrow.

What can you conclude when every measurement agrees? The philosopher Karl Popper pointed out an asymmetry: no number of agreements proves a general claim, while one clear miss refutes it. Scientific ideas are kept because they have survived hard tests, and they stay open to the next one.

Step 4 of 6

A sharper test

Width does not matter says a pendulum keeps the same beat whether it swings wide or narrow. Galileo claimed it around $1600$⁠, and for small swings it is very nearly right. The pendulums are released from $10^\circ$⁠ up to $60^\circ$⁠.

With Trials per pendulum at 1, every tag is mint, and the hypothesis survives. But the standard error of one trial is the whole 0.12 s. Predict what will happen when we raise it to $20$⁠ trials.

Step 5 of 6

How many trials it takes

The standard error of a mean of $n$⁠ trials is $\dfrac{0.12}{\sqrt{n}}$⁠ seconds: the more trials, the narrower the bar, and the smaller the effects the test can see. Width does not matter is fixed, and so is Run the test, at Verdict. Raise Trials per pendulum one at a time, and find when the first pink tag appears.

Try it: after checking, carry on up to $20$⁠ trials. After the $60^\circ$⁠ tag, the $45^\circ$⁠ one turns pink too, while the narrow swings stay mint. Galileo was not simply wrong: his rule holds for small swings, and the refutation marked where it stops working.

Back to the experiment ↑

Step 6 of 6

A surprising result

A miss is only as solid as the measurement behind it. A two-standard-error bar holds a true value in about $95$⁠ of every $100$⁠ measurements, so even a true hypothesis misses now and then, by chance. The whole cycle runs: notice something, form a hypothesis, draw a prediction from it, test it against measurements with honest error bars, and accept the verdict. A failed prediction sends you to a new hypothesis, and a passed one to a harder test.