1. Uncertainty and error in measurement

2. Uncertainty in measurement
Measurement is an important part of chemistry.
In the laboratory you will have to select the instrument that is the most appropriate for your task from a range of possibilities.
For measuring volume you could choose among cylinders, pipettes, volumetric flask of different sizes.
For measuring mass you could choose analytical balances with different precisions.
All these could be used to measure volume or mass with different levels of uncertainty.

3. Uncertainty in analogue instruments
An uncertainty range applies to any experimental value.
Some apparatus state the degree of uncertainty, but in other cases you will have to make a judgement.
The volume has to be estimated:
66 ± 1 cmᶟ.
The uncertainty of an analogue scale is
± half the smallest division.

4. Uncertainty in digital instruments
The degree of uncertainty of a digital scale is the smallest scale division.

5. Measurement uncertainties
What type of balance is the best to weigh 1.00g and 10.00g in the lab?
a. uncertainty in balance 1 decimal place 0.1g
b. uncertainty in balance 2 decimal places 0.01g
c. uncertainty in balance of 3 decimal places 0.001g
Calculate the % uncertainty or % error:
% uncertainty= (uncertainty equipment/actual measurement) x 100
a. (0.1/1.00) x 100= 10% (0.1/10.00) x 100= 1%
b. (0.01/1.00) x 100= 1% (0.01/10.00) x 100= 0.1%
c. (0.001/1.00) x 100=0.1%

6. Measurement of uncertainties
Which is the best way to measure 25 cmᶟ ?
uncertainties of glassware:
a. graduated cylinder 25 cmᶟ ± 0.5 cmᶟ
b. graduated pipette 25 cmᶟ ± 0.2 cmᶟ
c. graduated burette 50 cmᶟ ± 0.05 cmᶟ
Calculate % of uncertainty:
a. (0.5/25) x 100= 2%
b. (0.2/25) x 100=0.08%
c. (0.05/25) x 100=0.2%

7. Experimental Errors
The experimental error in a result is the difference between the recorded value and the generally accepted or literature value.
Errors can be categorized as random or systematic:
Random errors are cause by :
precision of the measuring instrument
temperature variations and air currents in the surroundings
insufficient data
observer misinterpreting the reading
the way the experiment was set up
Random errors can be reduced through repeated measurements.

8. Example of random errors
Suppose the mass of a piece of a Mg ribbon is measured several times.
g Average value: g
g Range to g
g mass : ± g g g g

9. Systematic errors Systematic errors occur as a result of
poor experimental design or procedure
human error reading instruments
poor calibration of instruments
heat losses in an exothermic reaction will lead to smaller
temperature changes.
Systematic errors can be reduced by careful experimental design.

10. Example of systematic errors
Suppose the top pan balance was incorrectly zeroed in the previous example:
g Average mass : g
g Range to g
g All values are too high by g g g g

11. Accuracy
It is a measure of the degree of closeness of a measured or calculated value to its actual value.
Example:
9.999 g experimental value
g known value
accuracy or uncertainty: (9.999 / ) x 100 % = %
The percent error is the same as the accuracy of the data.
The accuracy of an experimental value is best determined by the average value of multiple measurements.
The smaller the systematic error, the
greater the accuracy.

12. Precision
It is a measurement of how close are all your values or its reproducibility.
The precision of a set of measurements can be determined by calculating the standard deviation for a set of data where n-1 is the degree of freedom of the system.
The relative uncertainty for any
experimental value is dependent
upon the precision of the instrument.
relative standard deviation = S/average x 100
The smaller the random uncertainties, the greater the precision.

13. Uncertainty in calculated results
Consider a sample of NaCl with a mass of 5.00 ± 0.01 g and a volume of 2.3 ± 0.1 cmᶟ. What is its density?
mass g volume cmᶟ
find the range of density:
a) density w/maximum value of mass and minimum value of volume
d=5.01/2.2 = g/cmᶟ = 2.3 g/cmᶟ
b) density w/minimum value of mass and maximum value of volume
d= 4.99/2.4= g/cmᶟ = 2.1 g/cmᶟ
The precision of density is limited by volume measurement as is the least precise.

14. Precision (cont.)
Report the total mass of solution prepared by adding 50 g of water to 1.00 g of sugar. Would the use of a more precise balance for the mass of sugar result in a more precise total mass?
total mass = 51g
the precision of the total mass is limited by the precision of the mass
of water. Using a more precise balance for the mass of sugar have
not improved the precision of the total mass.

15. Propagation of uncertainties
Example: measuring volume with a burette with a % uncertainty of ±0.05 cmᶟ. What is the volume delivered? initial reading final reading Range of initial reading Range of final reading Maximum volume – = cmᶟ Minimum volume – = cmᶟ The total volume delivered is ± 0.1 cmᶟ the uncertainty is the sum of the two absolute uncertainties when adding or subtracting measurements.

16. Propagation of uncertainties (cont.)
Example:
mass = 24.0 ± 0.5 g volume 2.0 ± 0.1 cmᶟ
absolute uncertainty ± absolute uncertainty ± 0.1
% uncertainty of mass measurement= ( 0.5/24.0) 100 = 2%
% uncertainty of volume measurement= ( 0.1/2.0) 100 = 5%
Density= 24.0/2.0=12.00 g/cmᶟ Minimum 23.5/2.1 = g/cmᶟ
Maximum 24.5/1.9 = g/cmᶟ
Absolute uncertainty = – 12.00= ± 0.89
% Uncertainty = (0.89/12.00) 100% = 7.4%
The total % uncertainty is the sum of individual % uncertainties when multiplying or dividing measurements.