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**Errors and Uncertainties © Christopher Talbot and Cesar Reyes 2008**

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**Uncertainty in Measurement**

All measurements have uncertainties Important to report uncertainties when data is shared Uncertainties limit conclusions that can be legitimately drawn

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**Uncertainty in Measurement**

When using measuring apparatus in the laboratory check for the manufacturer’s statement of accuracy. A general rule for analogue scales: uncertainty is plus or minus half the smallest division of the scale. Example, 6.60 ± 0.05 cm (ruler with 0.1 cm graduations) implies a length between 6.55 and 6.65 cm A general rule for digital scales: uncertainty is plus or minus one in the least significant digit displayed. Example, 8.94 ± 0.01s (measured using a computer controlled timer) implies a time between 8.93 and 8.95 s.

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**Other Sources of Uncertainty**

Extra uncertainties should be noted even if they are not actually quantified in experimental work When time measurements are taken the reaction time of the experimenter should be considered judging the point that an indicator changes color judging the temperature at a particular time during and exothermic reaction judging the voltage of an electrochemical cell

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Experimental Error The experimental error is the difference between the recorded value and the generally accepted or literature value. Experimental error results from systematic and/or random errors. Experimental error relates to the degree of confidence in a measurement. Propagation of uncertainties must be calculated and taken into account.

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**Types of Errors - Random Errors**

cannot be avoided and are always present a reading has an equal probability of being too high or too low can be reduced through repeated measurements If the same person duplicates an experiment with the same results the results are repeatable. If several experimenters duplicate the results they are reproducible.

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**Causes of Random Errors**

The readability of the measuring instrument The effects of changes in the surroundings such as temperature variations and air currents Arise from instrumentation ‘noise’ Reading a scale from multiple positions Insufficient data The observer misinterpreting the reading

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**Types of Errors - Systematic Errors**

Systematic errors occur as a result of poor experimental design or procedure Can be identified and corrected. Can be detected by careful experimental design Analyze samples of known composition. Analyze ‘blank’ samples and verify that the instrument will give a zero result. Obtain results for a sample using multiple instruments to verify the accuracy of the instrument.

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**Causes of Systematic Errors**

Reading the top of the meniscus rather than the bottom (parallax) Overshooting the volume of a liquid delivered in a titration Heat losses in an exothermic reaction Use of sodium hydroxide as a primary standard Using a leaky gas syringe Using a voltmeter that reads 0.1 V when not part of a circuit (a zero error)

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**Accuracy and Precision**

Precise measurements have small random errors and are reproducible in repeated trials The smaller the random uncertainties the greater the precision Accurate measurements have small systematic errors and give a result close to the accepted values The smaller the systematic error, the greater the accuracy

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**Uncertainties in Calculated Results**

Uncertainties in the raw data lead to uncertainties in processed data Be consistent when propagating uncertainties Whenever you multiply of divide data, quote answer to the same number of significant figures as the least precise data. Whenever you add or subtract data, quote answer to the same number of decimal places as the least precise value.

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**Percentage Uncertainties and Errors**

Consider measuring a volume of 25.0 cm3 with a 25 cm3 pipette which measures to cm3. The absolute uncertainty is plus or minus 0.1 cm3. This means the volume measured lies between the range 25.1 cm3 and 24.9 cm3. The percentage uncertainty is the absolute uncertainty divided by the measured value multiplied by 100: (0.1/25) x 100=2%. The percentage uncertainty shows the size of the uncertainty with regard to the measurement. Percentage uncertainty is needed in error propagation for multiplication and division. Percentage error is a measure of how close the experimental value is to the literature or accepted value

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**Propagation of Uncertainties**

If we are simply adding or subtracting two measurements (with the same units) the absolute uncertainties can be added. Examples: suppose we add two volumes of 25.0 cm cm3. The final answer should be reported as 50.0 cm cm3. Suppose we have two temperatures: ºC ºC and 31.7 ºC ºC. The temperature difference would be reported as 10.3 ºC ºC

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**Propagation of Uncertainties (continued)**

If you are using multiplication, division or powers, then percentage uncertainties should be used during the calculation and then converted back into an absolute uncertainty when the final result is presented. A balance ( g) weighs the sample of liquid bromine as g. A 50 cm3 burette that measures cm3 measured the sample as 5.00 cm3. Calculate the density. Density is 3.12 g/cm3 ±0.05 g/cm3 value Absolute Uncertainty % Uncertainty Mass (g) 15.613 0.001 (0.001/15.613)x100% = 0.006% Volume (cm3) 5.00 0.08 (0.08/5.00)x100% = 1.6% Density g/cm3 3.12 0.016x3.12=0.0499

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**Discussing Errors and Uncertainties**

Experimental conclusion must address systematic errors and random uncertainties State if the uncertainty of one measurement dominates the effects on the final result Can the difference between the experimental and literature value be explained in terms of the uncertainties of the measurements or were other systematic errors involved? A graph gives a measure of the data reliability; it identifies data points which do not fit the trend

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**Errors and Graphs Systematic error produces a displaced straight line**

Random uncertainties lead to points on both sides of the perfect straight line An polynominal equation which produces a “perfect fit” is not necessarily the best description of the relationship between variables Think carefully about the origin; most accurate or irrelevant

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