Errors & Uncertainty

Repeat any experiment and you will get slightly different answers. That scatter is not a failure. It is information about how far to trust your result.

Three kinds of error#

Many things can push a result away from the truth:

• Faulty equipment: a scale with a bent zero, a stopwatch that runs fast.
• Carelessness: reading the wrong mark, mis-recording a number.
• Random causes: tiny, unpredictable variations in reaction time, air currents, where exactly your eye lines up.

The first two can be hunted down and removed once you spot the cause. Random errors cannot: there is no single culprit. What saves us is that they are as likely to push a reading up as down. Average many repetitions and they largely cancel, so the mean of many readings sits close to the true value.

Measuring the scatter: standard deviation#

If an experiment repeated $N$ times gives $x_1, x_2, \dots, x_N$, the best estimate is the mean $\bar x = \tfrac1N\sum x_i$. How uncertain is it? We measure how far, typically, the readings stray from that mean using the standard deviation:

We then quote the result as $\bar x \pm \sigma$. Wider intervals buy more confidence: for a large number of readings there is about a 95% chance that the true value lies within $\bar x \pm 1.96\sigma$, and over 99% within $\bar x \pm 3\sigma$.