What Is a Measurement?

A measurement compares a physical quantity with an agreed unit.

In physics, we commonly measure quantities such as:

• Length
• Time
• Mass
• Temperature
• Force
• Voltage
• Current
• Pressure
• Speed
• Light intensity

Measurements can be obtained using either analog or digital instruments.

The main difference is how the measurement is represented and read.

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Analog Measurements

An analog instrument represents a measurement using a continuously changing scale or indicator.

Examples include:

• Metre rulers
• Liquid-in-glass thermometers
• Analog voltmeters
• Analog ammeters
• Spring balances
• Dial pressure gauges
• Analog stopwatches

With many analog instruments, the user determines the measurement by observing where a pointer, liquid level, or object lies relative to a scale.

For example, an analog thermometer might have scale divisions every:

1°C

If the liquid level lies between:

22°C and 23°C

the observer must judge the value between the markings.

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Digital Measurements

A digital instrument displays a measurement as a numerical value.

Examples include:

• Digital thermometer
• Digital balance
• Digital stopwatch
• Digital multimeter
• Electronic force sensor
• Digital pressure sensor
• Photogate timer
• Motion sensor

A digital thermometer might display:

23.7°C

rather than requiring the observer to estimate the position of a liquid column.

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Analog vs Digital

Both types can produce useful scientific measurements.

Digital does not automatically mean better in every situation.

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Continuous and Discrete Values

An analog display can usually vary continuously across its scale.

For example, the pointer on an analog voltmeter can move smoothly between:

2 V and 3 V

A digital display represents measurements using discrete numerical steps.

For example:

2.41 V

then:

2.42 V

then:

2.43 V

The physical voltage may vary continuously even though the display changes in steps.

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Reading an Analog Instrument

Suppose a thermometer has markings every:

2°C

The liquid lies halfway between:

20°C and 22°C

A reasonable reading would be approximately:

21°C

The observer must interpret the position.

Different observers might estimate slightly different values.

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Reading a Digital Instrument

Suppose a digital thermometer displays:

21.4°C

The observer records:

21.4°C

There is no need to estimate the position between scale markings.

This reduces one source of human reading error.

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Parallax Error

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

Imagine reading a pointer on an analog meter.

Viewed directly from the front, it might indicate:

4.2 V

Viewed from an angle, it might appear to indicate:

4.4 V

The pointer itself has not changed. The apparent position changes because of the viewing angle.

Digital displays greatly reduce this particular problem because the numerical reading is displayed directly.

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Digital Does Not Mean Exact

Suppose a digital thermometer displays:

25.6°C

This does not mean that the actual temperature is known perfectly.

The instrument still has limitations involving:

• Resolution.
• Accuracy.
• Calibration.
• Range.
• Response time.
• Electronic noise.

Every measurement has some degree of uncertainty.

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Resolution

Resolution is the smallest change that an instrument can detect or display.

Suppose a digital balance displays:

125.4 g

The smallest displayed increment is:

0.1 g

Its display resolution is therefore:

0.1 g

Another balance might display:

125.437 g

with a resolution of:

0.001 g

The second balance has finer resolution.

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Resolution and Precision

Digital instruments often provide finer resolution than simple analog instruments.

For example:

Instrument A

Displays:

12.4 cm

Instrument B

Displays:

12.437 cm

Instrument B provides more displayed digits.

This may allow smaller differences to be detected.

However, additional digits do not automatically mean that the measurement is more accurate.

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Accuracy

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

Suppose the actual mass of an object is approximately:

100.0 g

A balance repeatedly displays:

105.3 g

The display has fine resolution, but the measurements are inaccurate.

This could result from poor calibration.

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Precision

Precision describes how closely repeated measurements agree with one another.

Suppose repeated measurements are:

105.2 g

105.3 g

105.2 g

105.3 g

These measurements are very close together.

They are precise.

However, if the actual value is 100.0 g, they are not accurate.

Therefore:

precision ≠ accuracy

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More Digits Do Not Guarantee Better Data

Consider two thermometers.

Thermometer A

Displays:

21°C

Thermometer B

Displays:

21.4837°C

It may appear that Thermometer B is automatically superior.

But suppose its uncertainty is:

±2°C

Those extra decimal places are not useful indicators of the actual temperature.

A digital display can create false precision if the user assumes every displayed digit is meaningful.

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Digital Instruments and Measurement Precision

Digital instruments can improve measurement precision in several ways.

They can:

• Provide finer resolution.
• Reduce observer estimation.
• Reduce parallax error.
• Record measurements consistently.
• Detect rapid changes.
• Store measurements automatically.
• Connect to data-logging systems.

However, the improvement depends on the quality and suitability of the instrument.

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Example: Timing a Moving Cart

Suppose a cart travels down a short ramp.

Manual Method

A student uses a stopwatch.

Measured time:

1.3 s

The result may be affected by reaction time.

Digital Photogate

A photogate records:

1.247 s

The electronic system can detect the event without the student manually starting and stopping the timer.

For a short event, this can greatly improve the measurement.

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Reaction Time

Human reaction time can significantly affect manual timing.

Suppose an event lasts:

0.50 s

If the person starting and stopping the stopwatch introduces an error of around a few tenths of a second, the percentage effect can be large.

Electronic timing can reduce this problem.

This is why photogates are often useful for investigating:

• Falling objects.
• Carts.
• Pendulums.
• Collisions.

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Digital Balances

A digital balance converts the effect of an object's mass into an electronic signal and displays a numerical value.

For example:

73.26 g

Before measuring, the balance may need to be:

• Switched on.
• Zeroed.
• Tared.
• Checked for calibration.
• Placed on a stable surface.

A precise-looking display does not compensate for poor experimental technique.

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Taring a Balance

Taring allows the mass of a container to be removed from the measurement.

Suppose an empty beaker has a mass of:

82.4 g

You place it on the balance and press tare.

The display becomes:

0.0 g

After adding water, the balance displays:

125.6 g

This value represents the mass of the water rather than the combined mass of the beaker and water.

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Digital Multimeters

A digital multimeter can measure electrical quantities such as:

• Voltage.
• Current.
• Resistance.

Depending on the device, the user selects:

• Measurement type.
• Appropriate range.
• Correct terminals.

The display then gives a numerical reading.

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Selecting the Correct Range

Many digital instruments have different measurement ranges.

Suppose you expect a voltage of approximately:

5 V

A range suitable for values around 5 V should be selected.

Choosing an inappropriate range may result in:

• Poor resolution.
• An overload warning.
• No useful reading.
• Possible equipment damage in some situations.

Some modern instruments automatically select the range. This is called autoranging.

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Measuring Voltage

A digital voltmeter might display:

4.87 V

This means the instrument has produced a numerical estimate of the potential difference.

The final digit should not be treated as perfectly exact.

The manufacturer's specifications determine the actual measurement uncertainty.

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Measuring Current

Suppose a digital ammeter displays:

0.326 A

This can also be expressed as:

326 mA

Students should pay careful attention to the displayed unit.

A reading of:

326 mA

is not the same as:

326 A

Digital displays reduce some reading errors but do not prevent unit errors.

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Digital Thermometers

Digital thermometers can be useful because they:

• Provide direct numerical readings.
• Can respond quickly.
• Can sometimes record continuously.
• May connect to data loggers.

However, temperature probes may have a response time.

If the temperature changes rapidly, the sensor may not immediately display the new temperature.

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Response Time

Response time describes how quickly an instrument responds to a change in the measured quantity.

Suppose a temperature probe is moved from cold water into hot water.

The actual water temperature may already be:

70°C

but the display might change gradually:

25°C → 38°C → 51°C → 63°C → 69°C → 70°C

The sensor needs time to reach the new reading.

This can be important when measuring rapidly changing systems.

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Digital Force Sensors

A digital force sensor can measure forces continuously.

It may display values such as:

3.42 N

and can sometimes send data directly to a computer.

This is useful for investigations involving:

• Springs.
• Friction.
• Tension.
• Collisions.
• Newton's laws.

A force sensor should normally be zeroed before collecting data.

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Zero Error

Suppose a force sensor has no force applied but displays:

0.37 N

This indicates a possible zero error.

If the sensor is not corrected, every measurement may be shifted.

For example, an actual force of:

5.00 N

might appear as approximately:

5.37 N

Zeroing the sensor can correct this problem.

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Calibration

Calibration involves checking an instrument against known reference values.

Suppose a digital thermometer is placed under a known reference condition.

If it consistently gives a value different from the expected value, calibration may be required.

Calibration is important because:

digital ≠ automatically accurate

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Measurement Range

Every digital instrument has a limited range.

Suppose a balance can measure:

0–500 g

An object with a mass of:

750 g

is outside the instrument's range.

The display might show:

• An error message.
• "OL" for overload.
• A maximum value.
• No reading.

The correct solution is to use a balance with an appropriate range.

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Choosing Between Range and Resolution

Scientists sometimes need to balance range and resolution.

Suppose you need to measure a mass of approximately:

2 kg

A balance with:

0–100 g range

and extremely fine resolution is useless because the object exceeds its range.

A suitable instrument must first be capable of measuring the expected value.

Then consider whether its resolution is sufficient.

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Selecting an Appropriate Instrument

Before choosing a digital measuring device, ask:

• What quantity am I measuring?
• What value do I expect?
• What range is required?
• What resolution is required?
• How quickly does the quantity change?
• Does the measurement need to be logged?
• Is reaction time important?
• Does the instrument need calibration?
• Is the instrument suitable for the environment?

The instrument should match the needs of the experiment.

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Example: Measuring Classroom Length

Suppose you want to measure the length of a classroom.

Possible tools include:

• 15 cm ruler.
• Metre ruler.
• Tape measure.
• Digital laser distance meter.

A 15 cm ruler could technically be used, but it would be inconvenient and could introduce repeated-placement errors.

A tape measure or suitable digital distance meter would be more practical.

The best instrument is not necessarily the one with the most decimal places.

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Example: Measuring a Wire Diameter

Suppose a wire has a diameter of approximately:

0.5 mm

A metre ruler is unsuitable because its scale divisions are too large.

Better options might include:

• Micrometer.
• Digital micrometer.
• Suitable digital calipers.

A digital micrometer might display:

0.47 mm

The finer resolution makes it more suitable for measuring a small diameter.

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Example: Measuring a Race

Suppose students investigate a toy car that travels for approximately:

15 s

A digital stopwatch might be sufficient.

If the car passes through a short section in:

0.03 s

a manually operated stopwatch would be unsuitable.

A photogate or motion sensor would be a better choice.

The duration of the event influences the appropriate instrument.

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Example: Measuring Temperature Over an Hour

Suppose students investigate how hot water cools for:

60 minutes

Manually reading a thermometer every 10 seconds would be tedious.

A digital temperature sensor connected to a data logger could:

• Measure automatically.
• Record at regular intervals.
• Store the values.
• Generate a graph.

Digital measurement is especially useful when many measurements are required.

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Analog Instruments Still Have Advantages

Digital instruments are not always preferable.

Analog instruments can sometimes:

• Be inexpensive.
• Require no electrical power.
• Be easy to use.
• Show trends visually.
• Respond smoothly to changes.

For example, watching a pointer move can make the direction and rate of change easy to see.

The appropriate choice depends on the experiment.

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Seeing Trends on an Analog Display

Imagine a slowly changing electrical current.

An analog pointer may visibly move:

left → right

This makes the trend obvious.

A digital display might show:

1.21 → 1.22 → 1.24 → 1.25 → 1.27

The digital values are easier to record precisely, but the analog pointer can sometimes make the overall trend easier to see immediately.

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Sampling in Digital Measurements

Digital systems often measure a signal at regular intervals.

This process is called sampling.

For example, a sensor might record:

10 measurements per second

or:

1000 measurements per second

The number of measurements collected per second is the sampling rate.

A rapidly changing signal requires a sufficiently high sampling rate.

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Why Sampling Rate Matters

Suppose a rapidly changing event lasts:

0.01 s

If the instrument records only once every second, it may completely miss the event.

A higher sampling rate is required.

This is particularly important when measuring:

• Sound waves.
• Collisions.
• Vibrations.
• Rapid electrical signals.
• Fast motion.

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Digital Data and Data Loggers

Digital instruments can often connect directly to data-logging systems.

The measurement pathway may be:

physical quantity → sensor → digital measurement → data logger → computer

The computer can then:

• Store data.
• Create tables.
• Plot graphs.
• Calculate averages.
• Identify maximum and minimum values.
• Analyze trends.

This can greatly improve the efficiency of experimental data collection.

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Interpreting a Digital Display

Suppose a digital balance shows:

0.084 kg

This is:

84 g

A student must interpret both:

the numerical value

and:

the unit

Ignoring the unit can produce major errors.

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Decimal Places Matter

Compare:

2.5 V

and:

2.50 V

Mathematically, they represent the same numerical value.

Experimentally, however, the way a measurement is reported can communicate information about measurement resolution or precision.

You should normally record the value as displayed by the instrument unless there is a scientific reason to report it differently.

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Recording Measurements Consistently

Suppose a digital device displays:

4.20 s

Do not automatically record:

4.2 s

if the displayed precision is relevant to the experiment.

Similarly, if repeated measurements are:

4.18 s

4.23 s

4.20 s

record them consistently.

This makes later analysis clearer.

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Digital Measurements and Uncertainty

A digital reading has uncertainty even if the display looks exact.

For example:

12.36 V

does not mean:

exactly 12.360000... V

The actual uncertainty depends on the instrument.

It may be related to:

• Display resolution.
• Calibration.
• Manufacturer specification.
• Measurement conditions.
• Sensor behaviour.

In advanced experimental work, these uncertainties should be considered explicitly.

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Noise

Electronic measurements can contain noise.

Noise is unwanted variation in the signal.

For example, a stationary sensor might display:

5.01 N

4.98 N

5.03 N

4.99 N

even though the physical quantity is approximately constant.

This variation may result from:

• Electronic interference.
• Vibrations.
• Environmental changes.
• Sensor limitations.

Repeated measurements or averaging may help describe the underlying value.

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Digital Measurements Can Drift

Some electronic instruments may experience drift.

This means the reading gradually changes even when the measured quantity should remain constant.

For example:

5.00 N → 5.02 N → 5.05 N → 5.08 N

over a long period.

Possible causes include:

• Temperature changes.
• Electronic instability.
• Sensor behaviour.

Calibration and repeated checks can help identify drift.

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Comparing Digital Tools

Suppose three balances are available:

If you need to measure a:

50 g

sample very precisely, Balance A may be appropriate.

If you need to measure:

750 g

Balance A cannot be used because the sample exceeds its range.

Balance B would be more suitable.

For an object of:

6 kg

Balance C would be required.

The "best" instrument depends on the measurement.

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Selecting a Tool for an Experiment

Selecting equipment is part of good experimental design.

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Evaluating a Digital Measurement

When given a digital measurement, ask:

Is the instrument appropriate?

Is the value within its range?

Is the resolution sufficient?

Was it calibrated?

Was it zeroed if necessary?

Are the units correct?

Is the sampling rate suitable?

Does the value make physical sense?

These questions are more important than simply trusting the number on the screen.

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Worked Example: Choosing a Timer

A student wants to measure the time taken for a cart to travel:

10 cm

The expected time is approximately:

0.08 s

Option A

Handheld stopwatch.

Option B

Photogate.

The photogate is more suitable because the event is too short for reliable manual timing.

This is an example where digital technology can substantially improve the quality of the measurement.

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Worked Example: Choosing a Balance

A student needs to measure approximately:

250 g

Two balances are available.

Balance A

Range:

0–200 g

Resolution:

0.01 g

Balance B

Range:

0–500 g

Resolution:

0.1 g

Balance A has finer resolution but cannot measure 250 g.

Therefore, Balance B is the appropriate choice.

Range must be suitable before resolution becomes useful.

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Worked Example: Identifying False Precision

A sensor measures a distance and displays:

4.738291 m

However, its stated uncertainty is approximately:

±0.1 m

Reporting all six decimal places would give a misleading impression of precision.

A more appropriate result would reflect the actual capability of the instrument.

This demonstrates why:

number of digits ≠ measurement quality

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Digital Tools and Experimental Design

Selecting a digital tool is part of planning a valid experiment.

The tool should allow the dependent variable to be measured:

• Over the required range.
• With sufficient resolution.
• At an appropriate sampling rate.
• With acceptable uncertainty.

Using advanced technology does not automatically make an experiment scientifically strong.

The measurement method must still suit the research question.

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Common Mistakes

Assuming Digital Means Exact

Digital instruments still have uncertainty.

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Assuming More Decimal Places Mean Greater Accuracy

Extra digits indicate display resolution, not necessarily accuracy.

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Ignoring Units

250 mA

is very different from:

250 A

Always record the unit.

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Ignoring Instrument Range

An instrument cannot reliably measure values outside its designed range.

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Choosing Resolution Without Considering Range

A very precise instrument is useless if the expected value exceeds its maximum range.

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Forgetting to Zero an Instrument

Balances and force sensors may require zeroing before use.

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Ignoring Calibration

A digital instrument can produce consistent but inaccurate measurements if incorrectly calibrated.

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Using an Inappropriate Sampling Rate

Slow sampling may miss rapid changes.

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Assuming Digital Is Always Better Than Analog

The most appropriate instrument depends on the investigation.

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Confusing Accuracy and Precision

A set of measurements can be very precise while still being inaccurate.

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Check Your Understanding

1. What is an analog measurement?

2. What is a digital measurement?

3. Give three examples of analog measuring instruments.

4. Give three examples of digital measuring instruments.

5. Explain one important difference between analog and digital displays.

6. What is parallax error?

7. Why are digital instruments less affected by parallax when reading the display?

8. Define resolution.

9. A balance displays mass to the nearest 0.01 g. What is its display resolution?

10. Explain the difference between accuracy and precision.

11. Why do additional decimal places not necessarily mean greater accuracy?

12. A force sensor reads 0.42 N when no force is applied. What problem does this suggest?

13. What is calibration?

14. Why is the measurement range of an instrument important?

15. A balance has a maximum range of 500 g. Should it be used to measure a 750 g object? Explain.

16. Why might a photogate be more appropriate than a stopwatch for measuring an event lasting 0.05 s?

17. What is sampling rate?

18. Why is sampling rate important when measuring rapidly changing quantities?

19. Explain one limitation of a digital thermometer.

20. Why might an analog instrument sometimes be preferable to a digital instrument?

21. A multimeter displays 0.275 A. Express this value in milliamperes.

22. A digital balance displays 0.063 kg. Express this value in grams.

23. Explain what is meant by false precision.

24. A digital sensor produces slightly different readings even when the measured quantity appears constant. Give one possible explanation.

25. A student must measure the diameter of a 0.4 mm wire. Explain why a digital micrometer would generally be more suitable than a metre ruler.

26. A student needs to measure a 350 g object. Instrument A has a range of 0–200 g with resolution 0.001 g. Instrument B has a range of 0–500 g with resolution 0.1 g. Which instrument should be selected, and why?

27. Explain why digital measurements should still be checked for reasonableness.

28. Identify an appropriate digital tool for measuring voltage in a circuit.

29. Identify an appropriate digital tool for measuring the motion of a cart.

30. Describe three factors you should consider when selecting a digital measuring instrument.

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Key Terms

• Analog measurement – measurement represented using a continuously varying scale or indicator.
• Digital measurement – measurement represented numerically using discrete displayed values.
• Digital instrument – electronic measuring device that displays or records measurements numerically.
• Resolution – smallest change an instrument can detect or display.
• Accuracy – closeness of a measurement to the true or accepted value.
• Precision – closeness of repeated measurements to one another.
• Range – interval of values an instrument can measure.
• Calibration – checking an instrument against known reference values.
• Zero error – measurement offset present when the instrument should read zero.
• Parallax error – reading error caused by observing an analog scale from an incorrect angle.
• Sampling rate – number of measurements collected in a given amount of time.
• Response time – time required for an instrument to respond to a change.
• Noise – unwanted variation in an electronic measurement.
• Drift – gradual change in an instrument's reading over time.
• False precision – reporting more precision than the measurement method actually supports.
• Tare – reset a balance to zero while a container or other object is on it.

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Key Takeaways

• Analog instruments usually represent measurements using a scale, pointer, or continuously varying indicator.
• Digital instruments represent measurements numerically.
• Digital instruments can reduce parallax and observer-reading errors.
• Digital tools often provide finer resolution and can collect data automatically.
• Greater resolution does not automatically mean greater accuracy.
• More displayed digits do not guarantee better measurements.
• Digital instruments still have uncertainty.
• Range, resolution, accuracy, calibration, response time, and sampling rate all affect measurement quality.
• Instruments should be zeroed or tared when appropriate.
• Digital measurements can be affected by noise and drift.
• Rapid events may require electronic timing or high sampling rates.
• The units shown on a digital display must always be checked.
• Analog instruments can still be appropriate for many measurements.
• The best instrument is the one that suits the physical quantity, expected range, required resolution, and nature of the experiment.
• Scientists should evaluate measurements rather than automatically trusting a digital display.

A useful rule when selecting equipment is:

Choose the instrument because its range, resolution, and measurement method suit the experiment—not simply because it is digital.