Uncertainty and Equipment 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 Instrument

and use

Uncertainty Smallest amount

measured

% Error

(= 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