# Estimating Errors in Chemistry

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1. Random Errors

Random errors or uncertainties cause repeat measurements to vary and be scattered around a mean value.

Averaging a number of readings helps to reduce random uncertainties.

Precision

Precise measurements have a small random uncertainty.

Measurements are more precise if the scale used has smaller intervals. e.g. a 50 cm3 measuring cylinder is more precise than a 50 cm3 beaker. A 50 cm3 burette is even more precise.

It is possible for a measurement to have great precision but be inaccurate.

2. Systematic Errors

Systematic errors affect all the measurements in the same way making them all inaccurate, so they are all lower or higher than the true value.

They do not average out so repeating results does not help to eliminate systematic errors. They can occur due to a design or manufacturing fault, incorrect calibration of equipment or incorrect use of equipment. e.g if the top of the meniscus is read in a pipette or a measuring cylinder.

Identifying and eliminating systematic errors is important for increasing accuracy. They can be reduced by using better equipment or improving practical technique.

Accuracy

Accuracy is how correctly a piece of apparatus measures regardless of precision. If a car is travelling at 55 kph and a police speed detector reads 55kph, it is accurate. If it reads 48 kph or 65 kph it is inaccurate. There is something wrong with the equipment.

Example

The temperature is 20.0OC. Some students obtained the following results when they measured the temperature.

George

Edward

Henry

21.9

21.5

20.0

21.7

19.7

20.1

21.8

20.6

19.8

22.0

18.4

20.0

21.8

22.3

19.9

22.0

19.2

20.1

Determining Uncertainties in Calculated Results

Uncertainties can be quoted in absolute or percentage terms.

Absolute uncertainty = Measurement ï¿½ Uncertainty

Absolute Uncertainty

Percentage uncertainty = ______________________ x 100

Measurement

Mass

1. Consider weighing 1.00g of solid. If you use a two decimal place balance, the mass recorded will be to the nearest 0.01g.

The absolute uncertainty is 1.00g ï¿½ 0.01g

0.01

The percentage error is _____ x 100 = 1%

1.00

2. Consider weighing the same mass of solid on a three decimal place balance. The mass recorded will be to the nearest 0.001g.

The absolute uncertainty is 1.00g ï¿½ 0.001g

0.001

The percentage error is _____ x 100 = 0.1%

1.000

3. Consider weighing 10g of solid on a three decimal place balance.

The absolute uncertainty is 10.000g ï¿½ 0.001 g

0.001

The percentage error is _______ x 100 = 0.01%

10.000

Volume

4. Consider measuring 25cm3 in a 25cm3 measuring cylinder. The measurement will normally be to the nearest 0.5cm3, so in this case

The absolute uncertainty is 25.0 cm3 ï¿½ 0.5cm3

0.5

The percentage error is _____ x 100 = 2.0%

25.0

This is a much larger error than normally occurs when measuring mass. Other equipment has lower percentage errors.

5. If you use a transfer pipette or a standard / volumetric flask, the accuracy is quoted on the apparatus itself, but is normally 0.25% or 0.125%. However, they are appropriate only for specific volumes.

6. If you use a graduated pipette, as with all measuring equipment, specify a capacity closest to the target volume. For example, if you want to measure 22 cm3 of liquid using a 25 cm3 graduated pipette, the reading will usually be to the nearest 0.2cm3.

The absolute uncertainty is 22.00 cm3 ï¿½ 0.2cm3

0.2

The percentage error is _____ x 100 = 0.9%

22.0

7. A burette is more accurate still, and you can obtain readings to the nearest 0.05cm3. However there are two uncertainties in the measurement because an initial and a final reading are taken. If 25cm3 is measured, the % error would be:

0.05

_______ x 2 x 100 = 0.4%

25.00

Used well, the burette is very accurate. However the uncertainty increases dramatically for smaller volumes.

Time

10. Finally, consider a reaction time of 100s measured on a digital stopwatch. These often show times to the nearest 0.01s. The % error would be:

The absolute uncertainty is 100.00 s ï¿½ 0.01s

0.01

The percentage error is _____ x 100 = 0.01%

100.0

This is wonderful, but how quickly can you stop and start it? The error is really more like 0.2%, and operator incompetence becomes important.

Propagating Uncertainty in Equations

The overall percentage uncertainty is the sum of all the percentage uncertainties obtained for the different measurements.