Uncertainty and Equipment Error

Absolute uncertainty and recording data When you record measurements you should also record the absolute uncertainty (mm) is no longer sufficient it should (+/-0.5 mm) or (±0.5 mm) The number of significant figures used should be such that the uncertainty is in the last significant figure. It is illogical to report values with more significant figures than indicated by the uncertainty or error. E.g. in 9.63 ±0.6 the ‘3’ has no meaning and the number should be reported as 9.6 ±0.6.

Systematic Error Analysis of systematic error looks at how rigorous your method of controlling variables was Analysis focuses heavily on choice of equipment and the use of equipment Therefore % equipment errors are key to analysing systematic error Equipment errors ideally should be below 5%

Deciding on the level of uncertainty Precision/uncertainty may well be written on a piece of equipment or in it’s manual – use that The precision of measurements made with analogue (e.g. a ruler) instruments are equal to ½ of the smallest unit of measurement The precision of measurements made with digital instruments (e.g. an electronic balance) are equal to the smallest unit of measurement Stopclocks and stopwatches are different as they depend on the reaction time of the user and how they are used: It takes us 0.1 – 0.3 seconds to start stop a watch therefore to both start and stop a stopwatch the uncertainty is in the region of ±0.5s, or ±1s conservatively If you are taking measurements on a timed basis, e.g. 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 1/100 th of a gram e.g 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 27.5 mm – 28.5 mm This can be written as 28 mm (±0.5 mm), where the ±0.5 mm is the absolute uncertainty For us it is enough to put (±0.01 g) or (±0.5 mm) under the column label in a data table

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 mm mm) Hence the uncertainty in my measurements is 3 times as big (0.5 mm mm mm) Therefore my measurement is 970 mm (±1.5 mm)

What about uncertainties in processed data? Means and standard deviations are calculated from raw data and are measure in the same units and therefore are subject to the same uncertainty 10 mm (±0.5 mm) 12 mm (±0.5 mm) and 14 mm (±0.5 mm) have a mean = (10 mm + 12 mm + 14 mm) / 3 = 12 mm (±0.5 mm) Standard deviation = 2 mm (±0.5 mm)

You need to calculate % equipment errors You need calculate % equipment errors for each piece of equipment If the equipment was reused in a different way you need to calculate the error again The smallest measurement will give the biggest error – you only need to check on the smallest measurement repeated calculations can be useful to show cases where only a couple of measurements break the 5% error rule Put your equipment errors in a table – much easier to read and less possibility of mistakes

Calculating % equipment errors Put all your measuring equipment into a table such as the one below this The table can be part of the results section or an appendix, either way will won’t use it until the conclusion and evaluation of the investigation (it doesn’t contribute to DCP) Measuring Instrument and use UncertaintySmallest amount measured % Error (= uncertainty x 100 / amount measured)

Example Calculations 13 mm was the smallest length measured 0.5 mm the uncertainty % Equipment Error = uncertainty x 100 / amount measured = 0.5 mm x 100 / 13 = 3.85 %

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 all below 5%? Still comment on it to show that you’ve 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