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# Limitations of Measurements

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There are some physical limitations in measurements, which are linked to the instrument used or to changes in the measured quantity. The variation in the value of the measured quantity, which can be small or large, is called uncertainty.

## Limitations due to the instrument’s accuracy

Some limitations in measurements are the result of the measuring instruments. The limitations of the instrument can produce results that differ from the true values. There are two sources for these errors, instrumental accuracy and instrumental functioning, as in the examples below:

Instrumental accuracy: the variation of the property you want to measure is smaller than the scale of your instrument. An example of this is when you measure the length of an object whose total length is 19.5mm, but your ruler only has centimetre marks. In this case, your measurement will only be approximate, i.e., you might get a reading of 2cm, which is a close approximation but not the true value.

Instrumental functioning: your instrument has a defect or has become inaccurate over time. An example of this is using a digital thermometer that differs from the actual temperature by 2 degrees Celsius so that all temperature readings will be off by those 2 degrees.

Figure 1. The scale of the instrument can limit the accuracy of our measurements. Source: Flickr (Public Domain).

## How does data deviate and produce ‘errors’?

Every time we make a measurement or read data, we can introduce errors. The source of the error can be the instrument, the user, or the system. The errors fall into two main categories, systematic errors and random errors. There is a third type of error, known as a gaffe error, which can be a broken sensor or a wrong reading.

### Systematic errors

Systematic errors have their origin in the instruments or the system and do not happen accidentally. Systematic errors appear consistently in every measurement we take. These errors come from using an instrument in the wrong way, from a deviation within the instrument, or from the system that analyses the data. Systematic errors are always present in the system.

There are several sources of systematic errors:

1. Instruments: the measuring instrument adds, subtracts, or modifies the measured data during every measurement. There is, therefore, a consistent deviation in the data measured by the instrument.
2. Systems: this error source is a defect in the system we are using for our measurements.
3. Observations: this error type has its source in the user. Also called observational error, it is a discrepancy between the measured value and how the individual reads that value. One example is when the individual measures length with a ruler and makes a parallax error. In this case, the measurement of an objects length differs from its true length because the experimenter is looking at the markings on the ruler from an angle.

### Random errors

Random errors are the product of chance and present when the data suddenly deviates from the measured values. They can have two sources:

1. Systems: an error may be produced by a defect in the system, such as the sudden malfunction of a sensor. This is not a systematic error but a one-off event as there is no consistent malfunction.
2. Observations: in contrast to a parallax error, these errors are just blunders, such as a wrong reading.

## What are precision and accuracy?

Precision and accuracy are two concepts related to measurements. They determine the quality of our measured values.

### Precision

Precision indicates how repeatable our measured value is. If our measuring instrument is precise, every measurement it makes will be close to every other measurement. So, measuring the weight of an object whose value is 4.3kg, we will always get a value very close to 4.3kg.

Precision does not mean the measurements are correct. An instrument can be precise but consistently deviate far from the true value. In our example of an object weighing 4.3kg, a scale might consistently produce values close to 4.0kg.

### Accuracy

Accuracy means that the instrument delivers a value that is identical or very close to the true one. A highly accurate scale measuring the weight of a 4.3kg object will always produce values very close to 4.3kg, with only very minor variations.

To achieve measurements of high quality, we, therefore, need instruments with high accuracy and high precision.

Figure 2. The object in this image has a length of 4.25cm. Source: Manuel R. Camacho, StudySmarter.

• The closer we are to the white circle at the centre, the more accurate the measurement is.
• The measurements in purple are precise but not accurate.
• The measurements in dark blue are neither precise nor accurate.
• The measurements in light blue are precise and accurate.

## Physical Limitations of Measurements - Key takeaways

• When taking measurements, there will always be limitations. These limitations lead to differences between the measured values and the real ones.
• Limitations in measurements are the product of the instruments or the user.
• Limitations in measurements produce errors in the measured values.
• Any deviation of measured values from the real ones produced by errors or any measurement limitations are uncertainties.
• Errors can be either systematic or random. Systematic errors have their origin in the instruments or the system, while random errors are the results of pure chance.
• Precision and accuracy describe the quality of measurements. Accuracy is the property of measuring a value close to the real one, while precision is the property of consistently repeating the same value.

Physical measurements are measurements of an object’s physical properties, such as its length, mass, luminous intensity, electrical charge, temperature, particle quantity, and time. Also, any combination of the seven elemental physical properties can be measured.

When measuring any property, limitations are present in the instruments or in how the user reads the measured values. Other limitations can come from the theory or the system used to measure physical properties.

## Final Limitations of Measurements Quiz

Question

What are two sources of limitations encountered in measurements?

Instruments or variations in the quantity measured.

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What is a systematic error?

A consistent error in the measurement.

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What is a random error?

An error produced by chance.

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Can a random error be introduced by an instrument?

Yes, it can.

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Can a systematic error be introduced by an instrument?

Yes, it can.

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What is precision?

The ability to reproduce a measured value.

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What is accuracy?

The ability to produce a measurement that is close to the real value.

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Question

A sensor suddenly failing and producing the wrong measurement is an example of which type of error?

Random error.

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Question

If most of our measurements are close to each other, is our instrument accurate?

No, it isn’t.

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If the results of a measurement are close to each other and close to the real value, can you describe them as precise and accurate?

Yes, you can.

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If we make some measurements and our results are close to the real value but not close to each other, can you describe them as accurate?

Yes, we can.

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Can instruments introduce limitations to your measurements?

Yes, they can.

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How are random errors produced?

Random errors are the product of changes in the measurements.

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What is the name given to the deviation introduced in the data while doing measurements?

Errors.

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What is the name of the variation in the value of a measured quantity?

Uncertainty.

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Can the size of an object introduce limitations while measuring it?

Yes, it can.

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Question

If your instrument has a scale in cm and you measure an object with a length of 22mm, which kind of error are you introducing?

Systematic error.

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If you have a ruler whose length is 99cm instead of 1m and you use it to measure, which kind of error are you introducing?

Systematic error.

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If you have an experiment where no vibration happens while measuring something, and suddenly something drops, causing a vibration, which kind of error was introduced?

Random error.

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Question

What is the difference between systematic errors and random errors?

One is produced by chance, the other is always in the system.

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