Uncertainty andEquipment Error by Chris Paine Bioknowledgy
Absolute uncertainty and recording data When you record measurements you should also record the (absolute) uncertainty associated with the measurement The uncertainty reflects greatest precision, i.e. the smallest unit to which a measurement can be made. This method of quoting uncertainty is called least count
Absolute uncertainty and recording data For example measuring a length: We measure a length of 213 mm The smallest unit on a ruler is 1mm therefore the uncertainty is (±1 mm) Therefore as we know the value should be no less than 212 mm and no more than 214 mm Therefore we can quote the value as 213 mm (±1 mm)
Uncertainties and recording data It is illogical to report values with more decimal places than that indicated by the uncertainty for example: 9.63 ± 0.6 the last decimal place has no meaning and the number should be reported as 9.6 ±0.6 If a value 48cm3 is measured to an uncertainty of ±0.5cm3 it should be quoted as 48.0cm3 (±0.5 cm3) – this is an IBO guideline
What about uncertainties in processed data? If means and standard deviations are calculated from a data set. Therefore they should be quoted in the same units and the same uncertainty as the data they are calculated from. For example: 10 mm (±1 mm) 12 mm (±1 mm) 14 mm (±1 mm) Mean = (10 mm + 12 mm + 14 mm) / 3 = 12 mm (±1 mm) Standard deviation = 2 mm (±1 mm)
Deciding on the level of uncertainty Uncertainty may be quoted on a piece of apparatus or in it’s manual – use that If this information is not available use the least count method You may choose to increase your uncertainty to reflect the way a piece of equipment is used. Justify this decision in your lab report. Time is different, stopwatches depend both on the reaction time of the user and how they are used: It takes us 0.1 – 0.3 seconds to start and stop a watch. Therefore the uncertainty is in the region of ±1s If you are taking interval measurements, e.g. you observe an investigation every 2 minutes then your uncertainty is the same as your interval ±2 min
Examples of uncertainty For example, the school electronic balances measure to 1/100th of a gram e.g. 2.86 g The precision of the electronic balance is ±0.01 g Hence the reading on the electronic balance should be reported as 2.86 g (±0.01 g) When using a ruler we can usually be accurate to the nearest mm The implied limits of the measurement 28 mm are 27 mm – 29 mm This can be written as 28 mm (±1 mm), where the ±1 mm is the absolute uncertainty N.B. Quote the uncertainty in the column header (e.g. ±0.01 g or ±1 mm) of your data table rather than against each
Be careful of Repeated equipment use “I use a 300 mm ruler to measure 970 mm, what is the uncertainty?” “To measure the length I must of used the ruler 3 times (300 mm + 300 mm + 270 mm)” “Hence the uncertainty in my measurements is 3 times as big (1 mm + 1 mm + 1 mm)” “Therefore my measurement is 970 mm (±3 mm)” N.B. In reality the investigator should have made a better choice of equipment, e.g. 1m ruler
Be careful of Repeated equipment use “I am carrying out a vitamin C titration in a 50cm3 burette with an uncertainty of ±0.05cm3. My starting volume reads 48cm3. When I finish my titration the volume reads 35.6cm3.” “The volume I used in the titration is 12.4cm3 (48cm3 – 35.6cm3)” “I took two reading from the burette therefore my uncertainty doubled to ±0.1cm3 (±0.05cm3 + ±0.05cm3)” “Therefore my volume is 12.4cm3 (±0.1cm3)”
Systematic Error Analysis of systematic error looks at how rigorously your method controlled, varied and measured the different variables One aspect is equipment error, i.e. was the equipment choice and use appropriate? Ideally equipment errors ideally should be below 5%
Equipment Error Although it is optional to assess equipment errors it is highly recommended For each different use of equipment calculate the % error Calculate the error on the smallest quantity measured. The smallest quantity will generate the largest error. Ideally equipment errors ideally should be below 5% Repeated calculations can be useful to illustrate cases where only a couple of measurements break the 5% error rule. Use your judgment.
Calculating % equipment errors Use a table to organise the calculations The table enhances the evaluation therefore add to the evaluation section of the lab report Measuring Uncertainty Smallest % Error Instrument amount and use measured (= uncertainty x 100 / amount measured)
Example Calculations 13 mm was the smallest length measured 1 mm the uncertainty % Equipment Error = uncertainty x 100 / amount measured = 1 mm x 100 / 13 = 6.7 %
Evaluation - What if an Equipment Error is greater than 5%? Recommend a change to a more accurate named example of measuring equipment Suggest than larger (suggest an amount) amounts are measured/sampled to bring error below 5%
Evaluation – what if equipment errors are below 5%? It’s not necessary to suggest a change, but still comment on it to show that you’ve critically evaluated your equipment use.
Design - make sure you are measuring the right amounts If % Equipment Error = uncertainty x 100 / amount measured Then amount measured = uncertainty x 100 / % Equipment Error Therefore smallest amount measured ≥ uncertainty x 20