Significant figures and what a balance can promise

A calculation gives you as many digits as you like; an instrument gives you a fixed number of them. Reporting more digits than the instrument can support is the most common way a correct calculation turns into a misleading number.

Readability is not accuracy

A balance that displays 0.001 g has a readability of 1 mg. That is the size of its last step, not a promise about accuracy; the manufacturer’s specification and the balance’s calibration say how close the display is to the truth. A reading of 4.504 g on such a balance means the mass is 4.504 g give or take a few milligrams, at best.

Counting significant figures

• All non-zero digits count: 4.504 has four.
• Zeros between non-zero digits count: 180.156 has six.
• Leading zeros do not: 0.0025 has two.
• Trailing zeros after a decimal point do count: 4.500 has four, and says the balance read to the milligram.

Carrying them through a calculation

• Multiplying or dividing: the result keeps as many significant figures as the least precise input. 0.100 mol/L × 0.250 L × 180.156 g/mol: the first two factors have three, so the honest result is 4.50 g, not 4.5039 g.
• Adding or subtracting: the result keeps the coarsest decimal place of the inputs. 180.156 + 18.015 = 198.171, all to three decimals.
• Round once, at the end. Rounding intermediate results compounds the error.

Constants are not the limit

Standard atomic weights are known to more figures than any bench measurement needs; it is the volume and the concentration you can actually achieve that set the precision. A 250 mL Class A volumetric flask is good to about ±0.15 mL, roughly 0.06%, so three or four significant figures in the final concentration is honest and five is not.