Uncertainty and error in measurement

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.

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.

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

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%

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%

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.

Example of random errors Suppose the mass of a piece of a Mg ribbon is measured several times. 0.1234 g Average value: 0.1234 g 0.1232 g Range 0.1232 to 0.1236 g 0.1233 g mass : 0.1234 ± 0.0002 g 0.1234 g 0.1235 g 0.1236 g

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.

Example of systematic errors Suppose the top pan balance was incorrectly zeroed in the previous example: 0.1236 g Average mass : 0.1236 g 0.1234 g Range 0.1234 to 0.1238 g 0.1235 g All values are too high by 0.0002 g 0.1236 g 0.1237 g 0.1238 g

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 10.000 g known value accuracy or uncertainty: (9.999 / 10.000) x 100 % = 99.99 % 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.

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.

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 4.99-5.01 g volume 2.2-2.4 cmᶟ find the range of density: a) density w/maximum value of mass and minimum value of volume d=5.01/2.2 = 2.277273 g/cmᶟ = 2.3 g/cmᶟ b) density w/minimum value of mass and maximum value of volume d= 4.99/2.4=2.079167 g/cmᶟ = 2.1 g/cmᶟ The precision of density is limited by volume measurement as is the least precise.

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.

Propagation of uncertainties Example: measuring volume with a burette with a % uncertainty of ±0.05 cmᶟ. What is the volume delivered? initial reading 15.05 final reading 37.20 Range of initial reading 15.00-15.10 Range of final reading 37.15-37.25 Maximum volume 37.25 – 15.00 = 22.25 cmᶟ Minimum volume 37.15 – 15.10 = 22.05 cmᶟ The total volume delivered is 22.15 ± 0.1 cmᶟ the uncertainty is the sum of the two absolute uncertainties when adding or subtracting measurements.

Propagation of uncertainties (cont.) Example: mass = 24.0 ± 0.5 g volume 2.0 ± 0.1 cmᶟ absolute uncertainty ± 0.5 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 = 11.19 g/cmᶟ Maximum 24.5/1.9 = 12.89 g/cmᶟ Absolute uncertainty = 12.89 – 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.