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Accuracy and Error

Upper and lower bounds

A table is measured to two decimal places. It is 1.86m tall.

So the smallest this table could have been is 1.855m since this rounds up to 1.86 to two decimal places. Any smaller, and the number rounds down.

The largest the table could have been <1.865 since 1.865 itself rounds UP.

Hence the true length lies between these upper and lower bounds. Mathematicians express this in two ways.

1.855 ≤ L < 1.865                                                                or                                        1.86 + 0.005

Note that the upper bound can never be reached and so we write < instead of ≤, however when a question asks you for the "upper bound" the answer is 1.865 and not <1.865

 

Relative and percentage error

The second way of representing an error in a measurement tells us that the "absolute error" of our measurement is 0.005.

Relative error = absolute error/measured value Percentage error = (absolute error/measured value)*100

These are far better ways of representing errors since they take into account the importance of the error compared to the size of the object you are measuring. For example, a three meter error could be dangerous when building a bridge (especially if it is three meters too short!) but is negligible if measuring the distance from the earth to the sun.

Diagrams

You can represent upper and lower bounds using a diagram like the one below. You should know about this but it is rarely asked at GCSE

Significant figures and decimal places

When you record any long number you can write it down to a chosen number of decimal places or significant figures.

When you are dealing with decimal places it is important to remember that 1.372000 is different (more accurate) to 1.372.

When dealing with significant figures, the examiner will not always be able to tell how many s.f. you have written something to. For example 1200 to one s.f. is "1000" but to two s.f. it is "1200", which is the same as to three and four s.f.

Trick question- complication with significant figures

Note that the upper and lower bounds a measurement of 10cm to 2 s.f. are not 9.5 and 10.5 but 9.95 and 10.5. This is because 9.5 to 2 s.f. is still 9.5! However 9.95 to 2 s.f. rounds to 10.

Calculations involving upper and lower bounds

At every stage in a calculation use the upper and lower bounds as appropriate

Question: I cut 20.7 cm off a piece of wood measuring 32.2 cm. Find the upper and lower bounds for the length of the remaining wood.

Answer

The upper bound of this new piece of wood is the longest it could ever be. Therefore I need to assume the original piece was (32.2 + 0.05 upper bound) and then cut as little as possible off. Cut (20.7 - 0.05 lower bound) off. So the upper bound of the remaining wood is 32.25 - 20.65 = 11.6

The lower bound of this new piece of wood is the shortest it could ever be. Therefore I need to assume the original piece was (32.2 - 0.05 lower bound) and then cut as much as possible off. Cut (20.7 + 0.05 upper bound) off. So the lower bound of the remaining wood is 32.15 - 20.75 = 11.4

Hence the actual value can only be given as "10 to 1s.f." since this is the most accurate value that both 11.4 and 11.6 round to. If we tried to say that the true value can be written as "11 to 2s.f." this would be wrong because 11.6 rounds to 12 to 2 significant figures.

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