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AustraliaNSW
Stage 5.1-3

7.02 Accuracy and measurement

Lesson

Scales

Which of these three numbers is larger? 0.0025,\, 0.002\,82, \, 0.009 The answer is 0.009.

Now let's try again with these numbers:25, \, 28.2, \, 90 The answer is 90.

The second list is the same as the first, just multiplied by 10\,000. Still, it is much easier to tell by sight that the last number of the second list is the largest, compared with the first list.

We have a better intuitive sense of the size of numbers if they are between 1 and 1000, ideally less than 100. When selecting the correct scale to report a measurement, we want to lie somewhere in this range to compare it to others quickly and easily.

Examples

Example 1

It is most appropriate to use centimetres to measure the length of:

A
a cup
B
a horse
C
a country
D
a truck
Worked Solution
Create a strategy

1 centimetre is quite small so choose a small object.

Apply the idea

The smallest object is a cup, option A.

Idea summary

It is important to choose the most appropriate scale (unit) to measure an object.

We want the measurements to lie somewhere in between 1 and 1000 to be able to read and compare them easily.

Absolute error

The aim of a measurement is to obtain the "true" value of a quantity: the height of a tree, the temperature of a room, the mass of a rock, or whatever we want to know.

But is a tree ever exactly 5 metres tall? Is a room ever exactly 22\degree ? Is a rock ever exactly 2 kg in weight?

We can carefully design a measurement procedure to make more and more precise measurements, which makes the number of significant figures in our measurement increase. But we can only ever report the closest marking, and at some point the object we are measuring will fall between the markings.

For any measurement tool, we say its absolute error is equal to half the distance of its smallest unit. Any measurement we make with that tool must be given as plus or minus the absolute error.

Any subsequent measurement that is more precise will fall within this range, but we can't know exactly where until we try with a better tool.

An image of the measurement of an elephant using a metres and centimetres. Ask your teacher for more information.
An image of the measurement of an elephant centimetres. Ask your teacher for more information.

The appropriate unit of measurement to use makes the numerical value lie between 1 and 1000, ideally less than 100.

The absolute error of a measuring tool is equal to half its smallest unit.

Measurements with any tool should always be reported as: \text{Closest mark} \pm \text{Absolute error}

Examples

Example 2

A measuring tape has markings every 20 cm.

a

What is the absolute error of the measuring tape?

A
10 cm
B
20 cm
C
19.5 cm
D
1 cm
Worked Solution
Create a strategy

Multiply the smallest measurement of the tool by \dfrac{1}{2}.

Apply the idea

The measuring tape has markings every 20 cm, so the smallest measurement of the tape will be 20 cm.

The absolute error will be equal to half this amount which is 10 cm, so option A is correct.

b

The length of an object is measured as 120 cm by the measuring tape. A second measurement is then taken, measuring its length to the nearest centimetre. What is the range we should expect this second measurement to lie within?

A
110 cm to 130 cm
B
120 cm to 130 cm
C
110 cm to 120 cm
D
100 cm to 140 cm
Worked Solution
Create a strategy

We can find the range of measurements using: \text{Range of values} = \text{Closest mark} \pm \text{Absolute error}

Apply the idea

For the lower value:

\displaystyle \text{Range of values}\displaystyle =\displaystyle 120 - 10Subtract the error from the measurement
\displaystyle =\displaystyle 110 \text{ cm}Evaluate

For the higher value:

\displaystyle \text{Range of values}\displaystyle =\displaystyle 120 + 10Add the error and the measurement
\displaystyle =\displaystyle 130 \text{ cm}Evaluate

The second measurement should be from 110 cm to 130 cm. Option A is correct.

Example 3

Pauline knows that her bedroom is roughly 3 metres long. To what precision must she measure if she wants to know the length to:

a

2 significant figures?

A
To the nearest 1 millimetre
B
To the nearest 10 centimetres
C
To the nearest 1 metre
D
To the nearest 1 centimetre
Worked Solution
Create a strategy

To measure the length of the bedroom to two significant figures, we would need to include metres and tenths of a metre.

Apply the idea

So we need to measure to the nearest tenth of a metre which is equal to 10 centimetres. The answer is option B.

b

4 significant figures?

A
To the nearest 1 metre
B
To the nearest 10 centimetres
C
To the nearest 1 centimetre
D
To the nearest 1 millimetre
Worked Solution
Create a strategy

To measure the length of the bedroom to hour significant figures, we would need to include metres, tenths, hundreths and thousandths of a metre.

Apply the idea

So we need to measure to the nearest thousandth of a metre. A hundredth of a metre is 1 centimetre, so a thousandth of a metre is 0.1 centimetres or 1 millimetre. The answer is option D.

Idea summary

The absolute error of a measuring tool is equal to half its smallest unit.

We can find the range of values that a measurement could lie between using: \text{Range of values} = \text{Closest mark} \pm \text{Absolute error}

Outcomes

MA5.1-9MG

interprets very small and very large units of measurement, uses scientific notation, and rounds to significant figures

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