A lab note

A measurement is only as good as its uncertainty: a density of 2.48 g/mL means little until you know whether it could be 2.3 or 2.6. This note writes a measurement up the way a lab book does, carrying the uncertainty through the arithmetic, rounding the result to the figures the readings support, and checking it against the value it should be.

# Density of the sample
Mass in g: :mass = 12.4 +/- 0.1              // 12.4 ± 0.1
Volume in mL: :volume = 5.0 +/- 0.2          // 5 ± 0.2
Density in g/mL: mass / volume               // 2.48 ± 0.1
prev to 2 sf                                 // 2.5
check mass / volume ≈ 2.5 within 5%          // ✓ (differs by 0.8%)

# Units around it
12.4 g / 5 mL                                // 2.48 g/mL
12.4 g / 5 cm^3                              // 2.48 g/cm³
2.5 kg * 9.81 m/s^2                          // 24.53 N
21.5 °C in K                                 // 294.65 K
101.3 kPa in atm                             // 1.00 atm
0.5 mm in µm                                 // 500.00 µm

What each part is doing

12.4 +/- 0.1 is a reading and its uncertainty: 12.4, give or take 0.1. The engine carries the spread through the arithmetic, so mass / volume is 2.48 give or take 0.1, the spread worked out from both readings’ spreads rather than guessed. See uncertainty.

The units are in the labels. A value with an uncertainty is a plain number: the engine drops a unit written with it, because it cannot yet carry both through the arithmetic, and refuses to mix a value with an uncertainty and one with a unit rather than quietly losing one of them. So the grams and millilitres are in the text before each colon, and the unit conversions sit in a part of their own below.

prev to 2 sf rounds to two significant figures, the figures counted from the first one that is not zero. Two is what a volume known to 0.2 in 5.0 supports, so 2.5 is the honest way to report the density. See rounding.

check mass / volume ≈ 2.5 within 5% compares the result with the value it should be, allowing a margin. It passes and says how close it came; a result outside the margin would fail and show both sides. See checks.

The units part works in quantities that carry their unit: a mass over a volume is a density, a mass times an acceleration is a force (the engine writes it in newtons, N), and each converts to another unit of the same kind with in. See unit arithmetic and converting units.

What the note leaves out

The spread is combined the standard way for readings that vary independently, which is right for a mass and a volume taken with different instruments and not for two readings that share a source of error. The mole is not a unit the engine reads yet, so an amount of substance is written as a plain number.