Measurement and Uncertainty

Measurement is an essential part of science. Scientists measure quantities such as length, mass, time, temperature, and volume so that observations can be compared and analysed.

However, no measurement is perfectly exact. Every measurement has some degree of uncertainty. Good scientists choose appropriate equipment, record measurements correctly, and consider how uncertainty may affect their conclusions.

Choosing the Correct Measuring Instrument

Different scientific quantities require different measuring instruments.

The instrument chosen should match both the quantity being measured and the precision required.

Recording Measurements Correctly

A scientific measurement should normally contain two pieces of information:

numerical value + unit

For example:

12.4 cm

The number tells us the size of the measurement, while the unit tells us what measurement scale was used.

Writing simply 12.4 is incomplete because we do not know whether this means 12.4 cm, 12.4 g, 12.4 s, or something else.

SI Units

Scientists use the International System of Units (SI) so that measurements can be understood and compared around the world.

Some commonly encountered SI units include:

• length → metre (m)
• mass → kilogram (kg)
• time → second (s)
• temperature → kelvin (K)

In school laboratories, other convenient units such as centimetres, millimetres, grams, millilitres, and degrees Celsius are also commonly used.

Units should always be recorded using the correct symbols.

For example:

25 cm, not 25 cms

4.2 kg, not 4.2 kgs

What Is Measurement Uncertainty?

No measuring instrument can give an infinitely precise measurement.

Suppose you measure the length of an object and record:

Length = 12.4 cm

This does not mean its length is known perfectly. The true value could be slightly higher or lower.

Measurement uncertainty describes the range within which we reasonably expect the true value to lie.

A measurement might therefore be written as:

12.4 ± 0.1 cm

The ± symbol means "plus or minus."

This tells us that the measured value is 12.4 cm, with an estimated uncertainty of 0.1 cm.

The likely range is therefore:

12.3 cm to 12.5 cm

Estimating Uncertainty

A useful introductory rule for analogue instruments is to estimate uncertainty as approximately:

± half of the smallest scale division

Suppose a ruler has markings every 1 mm.

The estimated uncertainty might be:

±0.5 mm

If a measuring cylinder has scale divisions every 2 mL, an estimated reading uncertainty might be:

±1 mL

The exact method used to report uncertainty can depend on the instrument and the conventions used in the investigation, so scientists should state the method they have used.

Analogue Instruments

An analogue instrument has a continuous scale that must be read by the observer.

Examples include:

• rulers
• analogue thermometers
• measuring cylinders
• burettes
• analogue force meters

When using an analogue instrument, the observer usually estimates between the smallest markings.

This introduces some uncertainty into the measurement.

Digital Instruments

A digital instrument displays a numerical value directly.

For example, an electronic balance might display:

24.36 g

The smallest displayed increment is 0.01 g.

For introductory laboratory work, the resolution of a digital instrument is often used when considering its measurement uncertainty. However, the manufacturer's stated uncertainty should be used when it is available.

Digital instruments reduce some reading errors, but they do not eliminate uncertainty.

Resolution

The resolution of an instrument is the smallest change that the instrument can display or detect.

Consider two balances:

Balance A: measures to the nearest 1 g

Balance B: measures to the nearest 0.01 g

If we need to measure a small sample of salt accurately, Balance B would usually be more appropriate because it has a finer resolution.

Similarly, a ruler marked every millimetre provides finer measurements than one marked only every centimetre.

Choosing Appropriate Equipment

The most appropriate instrument is not always simply the largest or most complicated one.

Suppose you need to measure 23 mL of water.

You could use:

• a 500 mL beaker
• a 100 mL measuring cylinder
• a suitable pipette

A beaker would provide only a rough measurement.

A measuring cylinder would provide a more precise measurement.

A suitable volumetric pipette could provide even greater precision if the required volume matched its calibrated volume.

Equipment should therefore be selected based on:

• the quantity being measured
• the expected size of the measurement
• the required precision
• the instrument's range
• the instrument's resolution

Reading a Measuring Cylinder

When measuring liquid volume, the measuring cylinder should be placed on a flat, level surface.

The observer's eye should be level with the liquid surface.

For many liquids such as water, the surface curves downward. This curved surface is called the meniscus.

The measurement is taken from the bottom of the meniscus.

Reading the scale from above or below can produce a parallax error.

Parallax Error

Parallax error occurs when a scale is viewed from the wrong angle.

For example, looking down at the liquid level in a measuring cylinder can make the reading appear different from its actual position.

To reduce parallax error:

• place the instrument correctly
• position your eye level with the measurement
• read the scale directly rather than from an angle

Absolute Uncertainty

Uncertainty can be expressed using the same unit as the measurement.

For example:

50.0 ± 0.5 mL

The 0.5 mL is the absolute uncertainty.

Another example:

15.2 ± 0.1 cm

Here the absolute uncertainty is 0.1 cm.

Percentage Uncertainty

Sometimes it is more useful to compare the uncertainty with the size of the measurement.

This is called percentage uncertainty.

Percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%

Example

A student measures a length as:

20.0 ± 0.5 cm

Percentage uncertainty:

(0.5 ÷ 20.0) × 100 = 2.5%

The measurement therefore has a percentage uncertainty of:

2.5%

Why the Size of the Measurement Matters

Consider using the same instrument with an uncertainty of ±0.5 cm.

Measurement A

2.0 ± 0.5 cm

Percentage uncertainty:

25%

Measurement B

50.0 ± 0.5 cm

Percentage uncertainty:

1%

Although the instrument has the same absolute uncertainty in both cases, the uncertainty is much more significant when measuring the smaller quantity.

This is one reason scientists try to choose equipment that is appropriate for the size of the quantity being measured.

How Uncertainty Affects Results

Measurement uncertainty affects the confidence we have in experimental results.

Imagine two students measure the same object:

Student A: 12.4 ± 0.1 cm

Student B: 12.5 ± 0.1 cm

These measurements are extremely close, and their uncertainty ranges overlap.

This means the small difference between them may simply be caused by measurement uncertainty.

Scientists should therefore avoid claiming that two results are genuinely different when the difference is smaller than, or comparable to, their uncertainties.

Reducing Measurement Uncertainty

Scientists can often reduce the effect of uncertainty by improving their experimental methods.

Strategies include:

• choosing instruments with finer resolution
• using equipment appropriate for the quantity being measured
• measuring larger quantities when practical
• repeating measurements
• calculating averages
• reading scales at eye level
• correctly zeroing equipment before use
• using consistent measurement techniques

Repeating measurements is especially useful because it can help identify unusual results and reveal how much measurements vary.

Accuracy and Precision

These two terms are related to measurement but mean different things.

Accuracy describes how close a measurement is to the true or accepted value.

Precision describes how closely repeated measurements agree with one another.

For example:

Accepted value: 10.0 cm

Measurements:

10.8, 10.8, 10.9, 10.8 cm

These measurements are quite precise because they are close together.

However, they are not particularly accurate because they are all noticeably different from the accepted value.

Did You Know?

Sometimes the easiest way to reduce percentage uncertainty is to measure more of something.

Suppose a stopwatch has a reaction-time uncertainty of approximately 0.2 s.

Timing a single pendulum swing lasting 1 s gives a relatively large percentage uncertainty.

Instead, you could time 20 swings and then divide the total time by 20.

This reduces the effect of reaction-time uncertainty on the calculated period of one swing.

Key Terms

Measurement: A numerical observation made using an instrument.

Uncertainty: An estimate of the range within which the true value may reasonably lie.

Resolution: The smallest change an instrument can detect or display.

Absolute uncertainty: Uncertainty expressed in the same units as the measurement.

Percentage uncertainty: Uncertainty expressed as a percentage of the measured value.

Accuracy: How close a measurement is to the true or accepted value.

Precision: How closely repeated measurements agree.

Parallax error: A reading error caused by viewing a scale from the wrong angle.

Key Takeaways

• Scientists should select appropriate instruments for the quantity being measured.
• Measurements must include a numerical value and correct unit.
• Every measurement has some degree of uncertainty.
• For simple analogue measurements, uncertainty can often be estimated from the instrument's smallest scale division.
• Resolution describes the smallest change an instrument can detect.
• Percentage uncertainty allows the uncertainty to be compared with the size of the measurement.
• Small measurements can have large percentage uncertainties.
• Reading instruments incorrectly can introduce errors such as parallax error.
• Repeating measurements and using appropriate equipment can reduce the effect of uncertainty.
• Accuracy and precision are different concepts.
• Scientists must consider measurement uncertainty when deciding whether differences between results are meaningful.