Accuracy, precision and kinds of error

Ten readings, ten numbers

Time one swing of a pendulum with a stopwatch, ten times. You will not get the same number ten times. Your thumb is a little early, then a little late. Is the stopwatch wrong? Is the pendulum changing? No: every measurement has an error.

An error here is not a mistake. It is the uncertainty that every real measurement carries. Our job is to make it small, and then to say how big it is.

Accuracy and precision

These two words mean different things.

• Accuracy: how close a result is to the true value.
• Precision: how close repeated readings are to each other.

A group of shots can be tight but off the centre: precise, not accurate. It can be spread out but centred: accurate on average, not precise. Play with both.

You should find that the shift spoils accuracy and the spread spoils precision. They are separate. Fixing one does not fix the other.

Two kinds of error

Systematic errors push every reading the same way.

• A zero error: a scale that reads $0.2\ \text{mm}$ when it should read $0\ \text{mm}$.
• A faulty instrument: a stretched tape, a slow clock.
• Personal habits: always reading a scale from one side (parallax), always starting the watch late.

More readings do not help, because the mean is shifted too. You must find the cause and correct it.

Random errors push readings up sometimes and down sometimes, by chance: a shaky hand, a draught, the moment you press the button. They average out: the more readings you take, the closer the mean comes to the true value.

There is also the least count error: no instrument can show anything finer than its smallest division. A metre rule reads to $1\ \text{mm}$; you cannot do better with it.

Stating a result

Take $n$ readings $a_1, a_2, \ldots, a_n$.

1. The best value is the mean: $\bar{a} = \dfrac{a_1 + a_2 + \cdots + a_n}{n}$.
2. The absolute error of each reading is its distance from the mean, without a sign: $|\Delta a_i| = |a_i - \bar{a}|$.
3. The mean absolute error is the mean of these: $\Delta \bar{a}$.

We write the result as $\bar{a} \pm \Delta \bar{a}$.

To compare errors, divide by the value:

An error of $1\ \text{mm}$ is huge on a $5\ \text{mm}$ screw ($20\%$) and tiny on a $5\ \text{m}$ room ($0.02\%$).

A worked example

A marble rolls down a ramp five times: $1.84\ \text{s}$, $1.78\ \text{s}$, $1.87\ \text{s}$, $1.81\ \text{s}$ and $1.85\ \text{s}$.

1. Mean. $\dfrac{9.15\ \text{s}}{5} = 1.83\ \text{s}$.
2. Absolute errors. $0.01\ \text{s}$, $0.05\ \text{s}$, $0.04\ \text{s}$, $0.02\ \text{s}$, $0.02\ \text{s}$.
3. Mean absolute error. $\dfrac{0.14\ \text{s}}{5} = 0.028\ \text{s} \approx 0.03\ \text{s}$.
4. Result. $(1.83 \pm 0.03)\ \text{s}$; percentage error $\dfrac{0.028\ \text{s}}{1.83\ \text{s}} \times 100\% \approx 1.5\%$.

Watch out. If you keep the signs, the differences from the mean always add to zero. That is why we drop the signs.

Think first. What if all five readings were $1.83\ \text{s}$? The mean absolute error would be $0\ \text{s}$, but the watch still cannot read finer than its least count. Then the error to state is the least count.

Practise with new numbers

Try it in Python

The program does the four steps of the worked example.

Remember these

• Accuracy: close to the true value. Precision: close to each other.
• Systematic errors shift every reading; find and remove the cause. Random errors scatter readings; more readings help.
• Result $=$ mean $\pm$ mean absolute error; percentage error $= \dfrac{\Delta \bar{a}}{\bar{a}} \times 100\%$.
• When every reading agrees, state the least count as the error.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A clock runs $2$ minutes fast every day. Is that a systematic or a random error? Will timing many events help?

2. Three readings of a length are $14.2\ \text{cm}$, $14.4\ \text{cm}$ and $14.3\ \text{cm}$. Write the result with its mean absolute error.

3. Which is measured better: $(2.0 \pm 0.1)\ \text{m}$ or $(50.0 \pm 0.5)\ \text{m}$?

4. Can a set of readings be precise but not accurate? Give an example.