Random and systematic error.

Step 1 of 6

The spread of one reading

A steel ball hangs from an electromagnet, 1.000 m above a contact pad. Cut the current, and a timer measures how long the ball takes to fall. The time gives the acceleration of free fall: $h = \tfrac{1}{2}gt^2$$h =$$\tfrac{1}{2}gt^2$⁠, so $g = \dfrac{2h}{t^2}$⁠. Every drop gives a slightly different reading, and each reading becomes a ball on the board, in the column of its value: a histogram lying on its side.

Errors that change from drop to drop, sometimes up and sometimes down, are random errors. The spread $s$⁠ measures their size: roughly how far one reading lies from the mean $\bar g$⁠. At Spread the board fits a bell curve to the pile and prints $\bar g$⁠ and $s$⁠. Timing jitter and Magnet delay are fixed. Predict what will happen to the width of the pile with four times the drops.

Step 2 of 6

The standard error

What the experiment wants is $g$⁠, and its best estimate is the mean. The mean is far steadier than any single reading, because the random errors in it partly cancel: some readings are high, some low, and the more of them there are, the better they balance. The uncertainty of the mean is its standard error:

$\frac{s}{\sqrt{n}}$⁠

The board does not show it yet. Work it out from the spread on the board.

Step 3 of 6

The square root of n

At Error bar a bar of one standard error each way appears beside the pile, with thin whiskers out to two. The amber ticks beside it are the means of $24$⁠ more runs of the same number of drops: the standard error is roughly how far another run’s mean would land from this one. Change Drops and find how many it takes to halve the bar.

Step 4 of 6

The true value

At True value a rose line marks the accepted value of $g$⁠, 9.81 m/s², which a real experimenter would not know in advance. With $25$⁠ drops it is inside the bar. Predict what will happen at $100$⁠ drops, when the bar halves.

Step 5 of 6

Systematic error

The reason is the Magnet delay: leftover magnetism holds the ball for an instant after the current is cut, so every fall is timed too long. An error that pushes every reading the same way is a systematic error. It does not cancel in the mean, because it does not change from drop to drop. Now the delay is 5 ms, and Work out the true value is fixed at Error bar, so the board does not give the answer away.

Try it: after checking, step Work out the true value to True value. The whole pile slides down by the amount you worked out, and the bar misses the line.

Back to the experiment ↑

Step 6 of 6

Precision and accuracy

A careful timer, Timing jitter at 2 ms, shrinks the random error: the pile is half as wide. But the 1.5 ms delay is still there. Random error spoils precision, how close the readings are to one another, and more measurements cure it. Systematic error spoils accuracy, how close they are to the truth, and more measurements do not help. Finding it takes a change of method: drop from several heights, or calibrate against something whose answer is known.