Calculating the uncertainty

Measurements made when doing experiments always have uncertainty associated with them. Manufacturers specify the expected uncertainty in measurements made using their equipment. Examples of manufacturer's specifications are given below. These are absolute uncertainties because they have the unit of the measurement.

Each of these contributes to the absolute uncertainty in a number calculated from experimental measurements. The procedure for determining the absolute uncertainty in a concentration (0.04896 mol L^–1) calculated using mass and volume measurements is illustrated below.

First convert the absolute uncertainty in each measurement to a percentage uncertainty,
Divide the absolute uncertainty by the measurement and multiply by 100%.

Measurement

Absolute
uncertainty

% Uncertainty

250.0

volumetric flask
±0.2

0.2	× 100% = 0.08 %
250.0

1.213 g
by difference

balance
2 × 0.004 g

2 × 0.004 g	× 100% = 0.66 %
1.213 g

Then add the percentage uncertainties to determine the total percentage uncertainty in the calculated result.
(0.08 + 0.66)% = ±0.74%

Then convert the percentage uncertainty in the calculated result to an absolute uncertainty.
Divide percentage uncertainty by 100 and multiply by the calculated result.

±0.74	× 0.04896 mol L^–1 = ±0.000362 mol L^–1
100

Round the absolute uncertainty to one sig fig.
When converting the absolute uncertainty to 1 significant figure, round UP UNLESS the next digit is zero (as in 0.00020).

±0.000362 mol L^-1 rounds to 0.0004 mol L^-1

Round the reported concentration so that the last digit is in the same decimal place as the first digit in the absolute uncertainty.
0,048960 mol L^-1 rounds to 0.0490 mol L^-1

Report rounded result with ± uncertainty.
0.0490 ± 0.0004 mol L^–1