One pendulum, one stopwatch, and patience. Every time you time twenty swings your error bar gets a little narrower. After two hundred repeats you have a number stable to three decimal places, and it is still wrong. Not because you did too little, but precisely because all you did was work.
Why this experiment is different
The thirteen experiments before this one were about the instrument, or about the model, or about the choices the experimenter makes. This one is about the statistics the student writes down – the plus or minus they put after the number with such confidence.
The rule everyone learns is that the error on a mean shrinks as the square root of N. That is true, and incomplete. The square root of N only knows about the part of the error that changes between repeats. Anything that stays the same across every repeat is invisible to the formula, and the more you repeat, the more firmly the error bar closes around the wrong answer.
Here the constant thing is an innocent mistake: you read the ruler to the top of the bob rather than to its centre. Every length in your notebook is short by the bob radius. That is not one mistake; it is the same mistake two hundred times. Which is why, when you swap the hand stopwatch for a photogate and the bar gets a hundred times narrower, the answer stays exactly where it was and only its claim gets louder.
In the speed of sound experiment the error bar shrank because you discarded data; there you had a hand in it. Here you did nothing except work hard. And as in the flywheel experiment, the number you arrive at carries no sign of its own that it is wrong.
The way out is to vary the thing you were holding fixed. Across several pendulums, T squared against the length you wrote down is a straight line whose slope gives g and whose intercept gives the hidden offset. What never changed could never be seen.
This is exactly what happened to the OPERA experiment in 2011: neutrinos apparently faster than light, with sixteen thousand events and a very narrow error bar. The cause was a loose fibre optic connector, a constant 73 nanosecond offset that no number of events was going to average away.
The Python section makes it quantitative. It puts the statistical bar you would have quoted next to the offset you never knew about and takes their ratio. The number that comes out is usually around twenty – twenty times what you were reporting, and no amount of extra timing was ever going to change it.