Measurement – Foundation of Science (Advanced)

Measuring means comparing with a standard unit: the CGS, FPS, MKS and SI systems, changing units without changing the quantity, and judging the error in an estimate.

Key ideas

A quantity is its number times its unit; changing the unit changes the number, never the quantity.
The bigger the unit, the smaller the number: a floor is fewer lengths of a long stick than of a short one.
An area unit is a length unit squared, so a stick twice as long makes the area number four times smaller.
CGS uses centimetre, gram, second; FPS uses foot, pound, second; MKS (metre, kilogram, second) grew into SI.
SI has seven base units, including the kelvin, ampere, candela and mole, and works in powers of ten.
A common system lets trade and science share measurements without confusion.

Remember

Same quantity, two units	Q = n₁u₁ = n₂u₂, so n₂ = n₁ × u₁ ÷ u₂
Force and density	1 N = 10⁵ dyne; 1 g/cm³ = 1 kg/L = 1,000 kg/m³
Speed	km/h × 5/18 = m/s; m/s × 18/5 = km/h
Other units	1 kg = 2.2 pounds; 1 inch = 2.54 cm
Percentage error	\|estimate − measured\| ÷ measured × 100

Watch out

Multiplying by the ratio of units the wrong way round: a larger unit must give a smaller number.
Scaling an area by the length ratio instead of its square.
Dividing the error by the estimate instead of by the measured value.

Worked example

In an estimation game: the blackboard: estimated 256 cm, measured 260 cm; a bench: estimated 196 cm, measured 180 cm; a cupboard: estimated 92 cm, measured 120 cm. Find the percentage error for (a) the blackboard, (b) a bench and (c) a cupboard. (d) Which estimate was the most accurate? Name the object. (Give the answer to 2 decimal places where needed.)

Working: percentage error = |estimate − measured| ÷ measured × 100; the smallest error is the most accurate

Answer: (a) 1.54%; (b) 8.89%; (c) 23.33%; (d) blackboard