In this explainer, we will learn how to define measurement accuracy and precision and explain how different types of measurement errors affect them.
When referring to measurements, the terms accurate and precise have distinctly different meanings.
Let us first explain what is meant by accuracy.
A measurement has a value (often this value has a unit). This is the numerical part of the measurement.
For example, the distance from Earth to the moon has been measured by various professional astronomers in observatories over many years to be 384 400 kilometres.
Suppose that a person whose hobby is stargazing, acting alone, measures this distance themselves to check the value, using their own instruments. The person measures a distance of 404 000 kilometres.
If this happens, the usual assumption made is that the person has made some errors in their measurement and that these errors are responsible for them measuring a noticeably different value to the accepted measured value.
If we accept this assumption, we can compare the value measured by the hobbyist to the accepted value to see how much difference there is between these values.
The less the difference between the values, the greater the accuracy of the measurement made by the hobbyist.
We see then that an accurate measurement is one that gives a value close to the true value of what is measured. The difference between the measured value and the true value can be called the error value of the measurement. For the example with the hobbyist measuring the distance from Earth to the moon, the error value is 19 600 km.
It is important to understand that it is not possible to be certain that a measurement has an error value of zero.
In making a measurement, it is possible to make errors that no one detects, or even considers. If errors go unnoticed by anyone, no one will know that a measurement failed to obtain a true value of what was measured.
For this reason, experimental scientists pay a lot of attention to the possible errors that they might make that could affect the accuracy of a measurement. In a scientific experiment, it is often assumed that a measurement is affected by errors, unless it can be shown by testing that these errors did not occur.
There are many, many possible errors that can occur when making many, many possible measurements. One way of classifying this great multitude of possible errors is to define errors as random or as systematic.
The following table compares error values expected due to random and systematic errors.
| Random Errors | Systematic Errors |
|---|---|
| The error value due to a random error is usually different for each measurement. | The error value due to a systematic error is (in the simplest case) the same for each measurement. |
| The error value for a measurement is apparently unrelated to that of any other measurements; hence, the value is random. | More complex systematic errors could change the error value for successive measurements, but the change in error value would be predictable rather than random. |
| It would be unusual for a random error to produce the same error value for successive measurements. | The error value might, for example, increase by the same amount for each successive measurement. |
One way that random errors can be detected is if repeated measurements are made of something that should not change. If the values recorded in the measurements vary unpredictably, then it must be concluded that a random error affects the measurements made.
Consider a measurement of the time taken for a ball to roll down a slope.
If the ball, the slope, the air around the ball, and the gravity that makes the ball move down the slope do not change in any way, then the time taken for the ball to roll from the top of the slope to the bottom should also not change.
Suppose, however, that the time is repeatedly measured, and the values of time measured vary unpredictably. This would show that a random error in the measurements existed. The random error could be due to many reasons, such as the following.
- The surface of the slope is not perfectly smooth, and some paths down the slope slow the ball down more than others.
- The surface of the ball is not perfectly smooth, and some starting orientations of the ball slow it more than others.
- The ball does not start from exactly the same height for each measurement.
- The air around the ball moves differently for different measurements.
- The instrument that measures time does not behave identically for each measurement.
One way that systematic errors can be detected is to repeatedly make a measurement for which the expected value is easily known.
It is well known that for distilled water at Earth’s surface, the temperature at which the water will start to boil is .
Suppose a thermometer is used to repeatedly measure the temperature at which water starts to boil under these conditions, and the thermometer consistently gives the same reading that is not equal to .
It would be reasonable to suspect that the thermometer was somehow producing a systematic error.
If the same thermometer was used to measure a different well-known value of temperature and it consistently gave the same unexpected reading for that measurement, it would be even more plausible that the thermometer produced a systematic error.
Even if the thermometer measured values other than as expected, the thermometer might nevertheless produce a systematic error at temperatures of or close to .
One particular value that can be a very useful value to attempt to measure to check for errors is the smallest value that it can measure.
For instruments that measure only positive values, the smallest value that can be measured by the instrument is zero. If the instrument reads a nonzero value when it is not being used to measure anything, then this can be a good reason to think that the instrument produces an error.
Suppose that an electronic balance that apparently has nothing placed on its pan gives a nonzero reading. We might think that something on the pan has a mass that is not detectable by sight or touch. It might be that the pan has become covered by a very thin layer of some very dense substance that is not detectable by human senses.
Alternatively, we might think that the balance is introducing an error.
An error that corresponds to a nonzero reading of an instrument for which no measurement is being made is called a zero error.
We have considered random errors and systematic errors separately, but it is of course possible for a measurement to be affected by both random and systematic errors.
For all errors, the effect of the error is to increase the error value of a measurement and so reduce the accuracy of the measurement.
Let us now explain what is meant by precision.
There are two key things to understand about precision:
- Precision is a property of a set of measurements, not of a single measurement.
- Precision is independent of accuracy, and, therefore,
- measurements that are accurate can either be precise or imprecise,
- measurements that are inaccurate can either be precise or imprecise.
Suppose that two successive measurements are made of the length of an object. The actual length of the object is 15 millimetres, and any change in this length during measurements is by much less than 1 millimetre.
The measurements made do not though have the value of 15 mm but values of 14 mm and 16 mm.
Now suppose that two successive measurements are made of the length of the same object using a different method or instrument. The measurements made have values of 13 mm and 11 mm.
The following table summarizes these results.
| Measurement Set A Length (mm) | Measurement Set B Length (mm) |
|---|---|
| 14 | 13 |
| 16 | 11 |
For measurement set A, we can see that the error value of each measurement is 1 mm, as each measurement is different from the true value by 1 mm.
For measurement set B, we can see that the error value of the first measurement is 2 mm and the error value of the second measurement is 4 mm.
Clearly, the measurements in measurement set B are less accurate than those in measurement set A.
It is important to notice that in both measurement sets, the value of each measurement made is different. It was stated previously that any change in the length of the object was by much less than 1 mm, so the differences between the values measured in either set are due to error in the measurements.
The precision of each set of measurements is defined by how close the values of the measurements in the set are to each other, regardless of how accurate the values are.
For measurement set A, the difference between the greatest and the least value is
For measurement set B, the difference between the greatest and the least value is
The precision of each set of measurements is the same.
Suppose another set of measurements, measurement set C, consisted of two values that were both 50 mm.
| Measurement Set C Length (mm) |
|---|
| 50 |
| 50 |
Measurement set C would be very inaccurate indeed but would be more precise than either measurement set A or B.
Suppose also there is another set of measurements, measurement set D, that consists of two values that were 0 mm and 30 mm.
| Measurement Set D Length (mm) |
|---|
| 0 |
| 30 |
Measurement set D is very imprecise. Also, each measurement in measurement set D is very inaccurate.
It is worth noting, however, that the average value of measurement set D is and the average value of measurement set A is
From this, we see that the average values of measurement sets A and D have the same accuracy.
Let us now look at an example in which accuracy and precision are compared.
Example 1: Distinguishing between the Accuracy and the Precision of a Set of Values
The diagram shows a target board and four sets of hits on it, A, B, C, and D. The shots were all aimed at the bull’s-eye of the target.
- Which set of hits is both accurate and precise?
- Which set of hits is neither accurate nor precise?
- Which set of hits is accurate but not precise?
- Which set of hits is precise but not accurate?
Answer
We can determine the accuracy of a set of hits by how far from the bull’s-eye the hits are. This can either be for each individual hit in a set or for the average of all the hits in a set.
We can see immediately that for sets B and D, all the hits are either in the blue or white rings, so not close to the bull’s-eye. These hits are not accurate and so sets B and D are not accurate.
We can see immediately that for set A all the hits are close to the bull’s-eye. Set A is accurate.
For set C, two out of three hits are in the red ring and the other hit is in the blue ring. The hits in set C are less accurate than those in set A but more accurate than those in sets B and D. Also, we can consider the average of the hits in set C by taking the point that is equidistant from all the hits in set C, as shown in the following figure.
We can see that the point corresponding to the average of the hits in set C is over the bull’s-eye. We can say that the accuracy of set C is closer to that of set A than of set B or D, and so we call set C accurate.
We can determine the precision of a set of hits by how close the hits are to each other. The sets where the hits are close to each other are sets A and B. In sets C and D, the hits are further from each other.
We can summarize what we have deduced in a table.
| Set | Accuracy | Precision |
|---|---|---|
| (A) | High | High |
| (B) | Low | High |
| (C) | High | Low |
| (D) | Low | Low |
We see then that
- the set that is both accurate and precise is set A,
- the set that is neither accurate nor precise is set D,
- the set that is accurate but not precise is set C,
- the set that is precise but not accurate is set B.
Now let us look at an example in which the effect on accuracy and precision of measurements errors is considered.
Example 2: Distinguishing between the Effect of Systematic Errors on Accuracy and Precision of Measurements
Which of the following statements most correctly describes how systematic measurement errors affect the accuracy and the precision of measurements?
- Systematic errors decrease measurement accuracy.
- Systematic errors decrease both the accuracy and the precision of measurements.
- Systematic errors decrease measurement precision.
- Systematic errors do not affect measurement accuracy or measurement precision.
Answer
In answering this question, we will assume that we are only considering the simplest type of systematic error, for which the error value of a measurement is changed by the same amount for all measurements made.
We can use the analogy of aiming at a target for making a measurement, where the closer to the center of the target a hit is, the more accurate the measurement is. The following figure shows three hits on a target that represent measurements.
An error in a measurement is represented by an increase in the distance between the center of the target and a hit. For systematic errors, the error value is the same for each measurement. This is represented by the hits all moving the same distance in the same direction, as shown in the following figure.
The effect of the systematic error on the hits is to move all the hits away from the center of the target. The accuracy of the measurements represented by the hits is now worse, so we can say that a systematic error decreases measurement accuracy.
We can compare the positions of the hits with and without the systematic error, as shown in the following figure.
We can see that the same triangle that connects the three hits without the error connects the hits with the error. This tells us that the distances of the hits from each other have not been changed by the systematic error. We can say then that a systematic error does not change the precision of a set of measurements.
The correct option is then that systematic errors decrease measurement accuracy.
Now let us look at a similar example.
Example 3: Distinguishing between the Effect of Random Errors on Accuracy and Precision of Measurements
Which of the following statements most correctly describes how random measurement errors affect the accuracy and the precision of measurements?
- Random errors decrease both the accuracy and the precision of measurements.
- Random errors decrease measurement accuracy.
- Random errors decrease measurement precision.
- Random errors do not affect measurement accuracy or measurement precision.
Answer
We can use the analogy of aiming at a target for making a measurement, where the closer to the center of the target a hit is, the more accurate the measurement is. The following figure shows three hits on a target that represent measurements.
An error in a measurement is represented by an increase in the distance between the center of the target and a hit. For random errors, the error value is different for each measurement. This is represented by the hits all moving different distances in different directions, as shown in the following figure.
The effect of the random error on the hits is to move all the hits away from the center of the target. The accuracy of the measurements represented by the hits is now worse, so we can say that a random error decreases measurement accuracy.
We can compare the positions of the hits with and without the random error, as shown in the following figure.
We can see that the triangle that connects the three hits without the error is very different to the triangle that connects the hits with the error. This tells us that the distances of the hits from each other have been changed by the random error. We can say then that a systematic error changes the precision of a set of measurements.
The correct option is then that random errors decrease both the accuracy and the precision of measurements.
Let us now look at an example that explains the usefulness of repeated measurements.
Example 4: Identifying the Usefulness of Repeated Measurements
When taking any measurement, why is it recommended to take the measurement several times and then calculate the average of the readings?
- To reduce the effect of the errors in the individual readings
- Because the average of the readings is the real value of the measurement
- To eliminate any measurement errors in the individual readings
- To increase the precision of the measurement
Answer
Suppose that a measurement is made. It is possible that the measured value includes a random error that produces an error value. The size of the error value is not known.
Suppose instead that the same measurement is repeated many times.
We can use the analogy of aiming at a target for making a measurement, where the closer to the center of the target a hit is, the more accurate the measurement is.
If many measurements are made, the effect of the random error makes each measured value different. This is represented in the following figure.
This representation shows which hits are nearer the bull’s-eye and so makes it seem obvious which measured values are closer to the true value. What is misleading about this representation is that the true value of what is measured is not known, so the same hits could just as plausibly be spread around the target, as shown in the following figure.
We must therefore consider each measurement as equally possibly having the same value as the true value.
If we assume that each measurement is equally likely to give the true value, the following figure shows the distance from each hit to the true value (which we recall is not known).
The following figure shows the errors first as distances from the bull’s-eye and then below that as a set of arrows that all start at different points. In both cases the same set of arrows is shown.
Each of these arrows is equivalent to a vertical and a horizontal arrow, as shown in the following figure.
We can consider just these vertical and horizontal arrows.
These arrows can be separated into vertical arrows and horizontal arrows that point either up, down, left, or right. These arrows can be laid head to tail.
The average length of the arrows pointing up, down, left, and right can be found. This is shown in the following figure.
The horizontal and vertical average arrows can be added together, where the length of a downward arrow is subtracted from the length of an upward arrow, and the length of a leftward arrow is subtracted from the length of a rightward arrow. This gives very small horizontal and vertical lengths, shown in purple.
The purple lines are equivalent to one arrow that points in one direction and has one length.
The resultant arrow is very short. This arrow can now be drawn with its tail at the bull’s-eye. The distance of the head of the arrow from the bull’s-eye represents the error that results from averaging all the measurements made.
We can see that the error due to averaging all the measurements is very small, far smaller than the error of even the measurement that was closest to the true value.
It is important to notice that the average of all the measurements is not exactly equal to the true value. A small error value exists for the average, but this error value is much smaller than the error value for any single measurement.
The option that most closely describes this is that averaging repeated readings is done to reduce the effect of the errors in the individual readings.
Let us now summarize what has been learned in this explainer.
Key Points
- The error value for a measurement is the difference between the measured value and the true value.
- The less the error value of a measurement, the greater the accuracy of the measurement.
- Measurement errors include random errors, systematic errors, and zero errors.
- Random errors change the measured values differently for each measurement made.
- Systematic errors change different measured values equally.
- A nonzero measured value for a true value that is zero is a zero error.
- Precision applies to a set of measurements.
- The smaller the differences of the values of a set of measured values, the more precise the measurements.
- Taking the average of repeated measurements of the same thing can greatly reduce inaccuracy due to random error.
