You have learned that exact numbers are numbers with a value that is exactly known, and measured numbers are numbers that are obtained when you measure a quantity such as mass, length, time, or temperature. Measured numbers have a value that is not exactly known. Some uncertainty always exists in the value of a measured number.
Error vs. Uncertainty
It is important not to confuse the terms error and uncertainty.
Error is the difference (or disagreement) between the measured value and the “accepted” value (value that is considered to be true) of the thing being measured. In other words, error tells us how far a measurement is from the “true” value.
Uncertainty is the range of possible values within which the true value of the measurement sits.
The degree of uncertainty is affected in part by the quality of the measuring tool. For example, while some balances are capable of measuring masses only to the nearest 0.1 g, other highly sensitive balances are capable of measuring to the nearest 0.001 g or even better.
Other measuring tools such as rulers and graduated cylinders have small lines which need to be carefully read in order to make a measurement. For example:
This image shows two rulers making the same measurement of a red line. The black dotted line on each ruler indicates where the measurement is on the ruler. With either ruler, it is clear that the length of the red line is between 3 cm and 4 cm. How should the length of the red line be recorded with each ruler?
Two centimeter rulers are labelled A and B. Both rulers are being used to measure a red line. Ruler A is divided into centimeters only. Ruler B is divided into centimeters and each centimeter is further divided into millimeters. A black dotted line through both rulers shows that the red line on Ruler A is between 3 cm and 4 cm, but is closer to 3 cm and it is in the same position on Ruler B.
Ruler A contains no millimeter markings. Using Ruler, A, the tenths digit can be estimated, and the length of the line may be reported as 3.1 cm.
However, another person may judge the measurement as 3.2 cm, or perhaps 3.3 cm. While the 3 is known for certain because of the markings on the ruler, the value of the tenths digit is uncertain.
Ruler B contains marks for tenths of a centimeter (millimeter). Using Ruler B, the same red line may be measured as 3.22 cm. The measurer is capable of estimating the hundredths digit because they can be certain of the tenths digit, indicated by the small lines.
However, another measurer may report the length as 3.21 cm or 3.24 cm. In this ruler, there are two certain values--the 3 and the 2, with the hundredths digit being uncertain.
Uncertainty quantifies the doubt that exists about the result of any measurement and is often discussed using the terms shown below. Click each one to learn about it.
Accuracy is how close a measurement is to an accepted value. This is important because bad equipment, poor data processing, or human error can lead to inaccurate results that are not very close to the truth. When measured values are accurate, it can strengthen the confidence you have in them as measured data.
Consider the following measurements recorded by different people using Ruler A to measure the red line:
3 cm, 3.1 cm, 3.2 cm, 3.3 cm, 3.2 cm
These values would be considered accurate for this ruler because they are close to the ending point of the red line. The following measurements would be determined as “low accuracy”:
2.9 cm, 3.5 cm, 4 cm, 2 cm
This is because these measurements are not close to the actual end of the red line.
Precision is how close a series of measurements are to one another. Precision refers to how close the agreement is between repeated measurements (which are repeated under the same conditions). When measured values are precise, it can strengthen the confidence you have in them as measured data.
Consider the following measurements recorded by different people using Ruler A to measure the red line:
3 cm, 3.1 cm, 3.2 cm, 3.3 cm, 3.2 cm
These values would be considered precise for this ruler because they are close to each other. The range from 3 to 3.3 cm is very small. The following measurements would be determined as “low precision”:
2.9 cm, 3.5 cm, 4 cm, 2 cm
This is because these measurements represent a wide range of values (from 2 to 4 cm).
Let's Practice
Complete this activity to practice using the terms accuracy and precision to discuss uncertainty. The archery targets in this table show marks that represent the results of four sets of measurements. Identify the accuracy (high or low) and precision (high or low) in each one, then click each image to check your answer.
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High Accuracy, High Precision The marks on the target are accurate, (very close to the bullseye) and precise (close to each other). |
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Low Accuracy, High Precision The marks on the target have low accuracy (far from the bullseye) but are precise (close to each other). |
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High Accuracy, Low Precision The marks on the target are accurate, (very close to the bullseye) but have low precision (marks are not clumped together). |
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Low Accuracy, Low Precision The marks on the target have low accuracy (far from the bullseye) and low precision (not clumped together). |
