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 uncertainty=readinguncertainty×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 ±, 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 100cm3 beaker that increases by 1cm3 will be ±0.5cm3.
To find the total uncertainty for measurement with the same units, sum together the total uncertainties.
Example
A digital balance measures to the nearest 0.01g. To carry out their experiment, a chemist uses the balance for three difference mass readings: 0.45g, 1.54g and 5.32g. 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:
±0.01 g
There are three measurements taken, so sum together three lots of uncertainties:
0.1+0.1+0.1=0.3g
The total uncertainty for these measurements is ±0.3g.
Percentage uncertainty
Percentage uncertainty can be calculated using:
Percentage uncertainty=readinguncertainty×100
Example
A 10cm3 pipette has an uncertainty of ±0.25cm3. Calculate the percentage uncertainty.
First, write the equation for percentage uncertainty:
Percentage uncertainty=readinguncertainty×100
Substitute in the values for uncertainty and the reading value to find the percentage uncertainty:
Percentage uncertainty=100.25×100=2.5%
The percentage uncertainty for the pipette is 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
35cm3 of a NaOH solution is titrated with 16.30cm3 of H2SO4. The uncertainty in the volume of NaOH is 0.025cm3 and the uncertainty in the volume of H2SO4 is 0.10cm3. The concentration of the NaOH solution was calculated to be 1.45moldm−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 H2SO4=Percentage uncertainty of final concentration
(350.025+16.300.10)×100=0.685%
Calculate the uncertainty of the final result:
0.685% of 1.45moldm−3=9.93×10−3moldm−3
The uncertainty for the concentration is 9.93×10−3moldm−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.