Measurements

What measurements covers

This unit sets the rules every other measurement on the course follows: which units are allowed, how to check an equation is dimensionally sensible, and how much doubt attaches to a number you have measured. It is examined directly in the practical and data-analysis questions, and it reappears in every other unit as soon as you handle repeats or a gradient.

The main ideas

• The six SI base quantities used at A-level, and unpacking any derived unit back into them.
• Checking an equation for homogeneity, and what a failed check proves against what a passed one does not.
• Prefixes with standard form, and conversions within one quantity, such as joules to electronvolts or to kilowatt hours.
• Random against systematic error, and which treatment reduces each.
• The vocabulary mark schemes use: precision, accuracy, resolution, repeatability, reproducibility.
• Putting a number on the doubt: half the resolution for one reading, half the range for a set of repeats.
• Combining uncertainties through sums, products and powers, and taking a gradient's uncertainty from the steepest and shallowest lines.

Key equations

percentage uncertainty = (uncertainty/value) × 100$\text{percentage uncertainty} = (\text{uncertainty}/\text{value}) \times 100$percentage uncertainty = (uncertainty/ value) × 100

turning an absolute uncertainty into a percentage

add the absolute uncertainties

for a sum or a difference

add the percentage uncertainties

for a product or a quotient

multiply the percentage uncertainty by n

for a quantity to the power n

uncertainty = 12 the range$\text{uncertainty} = \tfrac{1}{2}\,\text{the range}$uncertainty = (1/2) the range

from a set of repeated readings

Every equation the measurements notes display is collected, with its conditions, on the measurements equations page.

Common errors

• Precision and accuracy are independent. Repeating readings reduces random scatter but leaves a systematic offset where it was, so readings can be tightly clustered and wrong by the same amount every time.
• Subtracting two nearly equal quantities adds the absolute uncertainties but shrinks the value, so the percentage uncertainty can grow large enough to make the result worthless.
• The power rule applies to the percentage uncertainty, and the power is easy to miss inside a rearrangement such as g = 2h/t^2, where t counts twice.

Suggested order

Take SI units and prefixes first, since the rest of the course assumes them. Uncertainty and error next, because the practical work in every later unit relies on it. Estimation is short and self-contained, so it can go last.