Accuracy, Errors and Significant Figures: Quick Revision

A brief note on Accuracy, Errors and Significant Figures. Read it before a practical exam or a test. Try each question at the end before you open its answer.

The big ideas

• Accuracy: close to the true value. Precision: close to each other.
• Systematic errors push every reading one way (zero error, a faulty scale, a habit); more readings do not help. Random errors scatter readings and average out.
• Result $=$ mean $\pm$ mean absolute error; percentage error $= \dfrac{\Delta\bar{a}}{\bar{a}} \times 100\%$.
• Least count: vernier $= \dfrac{1\ \text{MSD}}{n}$; screw gauge $= \dfrac{\text{pitch}}{\text{divisions}}$. Correct reading $=$ observed $-$ zero error.
• Combining: sums and differences add absolute errors; products, quotients and powers add relative errors times the powers.
• Significant figures: products keep the fewest figures; sums keep the fewest decimal places.

The rules at a glance

A worked example

Six readings of a wire's diameter with a screw gauge are $0.52\ \text{mm}$, $0.55\ \text{mm}$, $0.51\ \text{mm}$, $0.54\ \text{mm}$, $0.53\ \text{mm}$ and $0.56\ \text{mm}$.

1. Mean. $\dfrac{3.21\ \text{mm}}{6} = 0.535\ \text{mm}$.
2. Mean absolute error. The distances from the mean are $0.015\ \text{mm}$, $0.015\ \text{mm}$, $0.025\ \text{mm}$, $0.005\ \text{mm}$, $0.005\ \text{mm}$, $0.025\ \text{mm}$: their mean is $0.015\ \text{mm}$.
3. Written properly. Keep one significant figure of the error, $0.02\ \text{mm}$, and round the mean to the same place: $(0.54 \pm 0.02)\ \text{mm}$, about $2.8\%$.

Practice with new numbers

A program to try

The program finds the mean and its error, and writes them with sensible digits.

Mistakes to avoid

• Signed differences. The differences from the mean always add to zero. Drop the signs.
• Subtracting errors for a difference. They always add.
• Forgetting the power. In $\dfrac{4\pi^2 l}{T^2}$, the error of $T$ counts twice.
• The wrong sign of a zero error. Correct $=$ observed $-$ zero error, so a negative zero error is added back.
• Too many digits. An answer cannot be more precise than its least precise measurement.

Check yourself

1. A balance always reads $2\ \text{g}$ too high. Will averaging many weighings fix it?

2. A screw gauge has a pitch of $0.5\ \text{mm}$ and $100$ circular divisions. What is its least count?

3. Two times, $(8.4 \pm 0.2)\ \text{s}$ and $(3.1 \pm 0.1)\ \text{s}$, are subtracted. What is the result?

4. How many significant figures are in $0.04020$?

5. Write $2.9 \times 1.135$ with the right number of significant figures.