Calculating uncertainty and errors

In a nutshell

Uncertainty is the amount of error a measurement has. Uncertainty can be calculated using percentage uncertainty.

Equations

$Percentage \space uncertainty \, = \, \frac{uncertainty}{reading} \, \times \, 100$

Uncertainty

Definition

Uncertainty is the amount of error a measurement has.

The uncertainty will vary for each piece of equipment that is used. They have different scales. Uncertainty is represented by $\pm$ , which indicates a range in error values (margin of error). The uncertainty for a piece of equipment will be half the value that the scale increases by, if this is analogic. For digital equipment, the uncertainty equals the value that the scale increases by.

Example

The uncertainty for a $100 \, cm^3$ beaker that increases by $1 \, cm^3$ will be $\pm \, 0.5\,cm^3$ .

To find the total uncertainty for measurement with the same units, sum together the total uncertainties.

Example

A digital balance measures to the nearest $0.01 \, g$ . To carry out their experiment, a chemist uses the balance for three difference mass readings: $0.45 \, g$ , $1.54 \, g$ and $5.32 \, g$ . What is the total uncertainty for the use of this balance?

Work out the uncertainty for the balance. Because it is digital, this will equal its sensibility:

$\pm 0.01 \ g$

There are three measurements taken, so sum together three lots of uncertainties:

$0.1 \, + \, 0.1 \, + \, 0.1 \, = \, 0.3 \, g$

The total uncertainty for these measurements is $\underline{\pm \, 0.3 \,g}$ .

Percentage uncertainty

Percentage uncertainty can be calculated using:

$Percentage \space uncertainty \, = \, \frac{uncertainty}{reading} \, \times \, 100$

Example

A $10 \, cm^3$ pipette has an uncertainty of $\pm \, 0.25 \, cm^3$ . Calculate the percentage uncertainty.

First, write the equation for percentage uncertainty:

$Percentage \space uncertainty \, = \, \frac{uncertainty}{reading} \, \times \, 100$

Substitute in the values for uncertainty and the reading value to find the percentage uncertainty:

$Percentage \space uncertainty \, = \, \frac{0.25}{10} \, \times \, 100 \, = \, 2.5\%$

The percentage uncertainty for the pipette is $\underline{2.5\%}$ .

Minimising percentage uncertainty

Use precise equipment
Plan your experiment so a large volume of liquid is used

Total uncertainty

During an experiment the total uncertainty for the final result can be found.

procedure

1.	Find the percentage uncertainty for each piece of equipment.
2.	Add the percentage uncertainties together to find the percentage uncertainty of the final result.
3.	Use the percentage uncertainty of the result to work out the actual total uncertainty of the final result.

This method can be used to work out the total uncertainty for the result of a unknown concentration in a titration.

Example

$35 \, cm^3$  of a $NaOH$ solution is titrated with $16.30 \, cm^3$ of $H_2SO_4$ . The uncertainty in the volume of $NaOH$  is $0.025 \, cm^3$ and the uncertainty in the volume of $H_2SO_4$ is $0.10 \, cm^3$ . The concentration of the $NaOH$  solution was calculated to be $1.45 \, mol \, dm^{-3}$ . What is the uncertainty for the concentration calculated?

Use the sum of the percentage uncertainties for each volume to find the percentage uncertainty of the final concentration:

Percentage uncertainty of $NaOH \, + \,$ Percentage uncertainty of $H_2SO_4 \, = \,$ Percentage uncertainty of final concentration

$(\frac{0.025}{35} \, + \, \frac{0.10}{16.30}) \, \times \, 100 \, = \, 0.685\%$

Calculate the uncertainty of the final result:

$0.685\%$ of $1.45 \, mol \, dm^{-3} \, = \, 9.93 \, \times \, 10^{-3} \, mol \, dm^{-3}$

The uncertainty for the concentration is $\underline{9.93 \, \times \, 10^{-3} \, mol \, dm^{-3}}$ .

Errors

There are two types of errors: systematic and random.

Systematic errors

Systematic errors are the same for each experiment, this is caused by the set up or equipment during the experiment itself.

Random errors

Random errors will vary for each repeat of the experiment. Experiments can be repeated to find a mean to help minimise the effect of random errors.