Name: measurement uncertainty and significant figures
Uploaded: 2020-09-03T13:38:25.000Z
Duration: 2 min 46 s
Channel: JoVE

K12 Standards

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Ohio’s Learning Standards and Model Curriculum

9-12

Chemistry

Structure And Properties Of Matter

Quantifying matter

uncertainty-in-measurement-reading-instruments

Uncertainty in Measurement: Reading Instruments

Measurements from defined quantities or the direct counting of distinct objects give exact numbers. For example, a dozen bananas is exactly 12 bananas. Similarly, apples in a carton can be directly counted.

However, other measurements are not exact, and have uncertainty associated with them due to limitations in the measurement process.

Consider a runner competing in a 5 kilometer race. The judges record the finish time using a simple analog watch and a digital watch.&nbsp;

The analog watch reads the finish time as between 25 and 30 minutes, so, the judges estimate the time to be 28 minutes. Only the first digit, &lsquo;2&rsquo;, is certain. The last digit, &lsquo;8&rsquo; is an estimate between 25 and 30, and is thus uncertain.

The digital watch, on the other hand, read 26.25 minutes, with full certainty in the first three digits. The finish time could vary between 26.24 and 26.26 minutes, but&nbsp;is certainly not 28 minutes.

In a laboratory setting, a digital balance may display a reading of up to six digits. A reading of 10.1241 grams indicates that the first 5 digits are certain, but the last digit is uncertain. However, all the numbers displayed on a digital instrument are to be reported.

In contrast, when reading analog instruments, the last digit must be estimated. A 50-mL graduated cylinder may have markings every one milliliter. The measurement read is dependent on estimating the volume of liquid at the meniscus, the lowest point on the curved surface of the liquid in the graduated cylinder.&nbsp;

If the meniscus lies between the 42 and 43 markings, the quantity of liquid is more than 42 mL&nbsp;but less than 43 mL. The measurement should be estimated between the two markings; therefore, the volume should be read to the nearest 0.1 mL.

In this case, the meniscus is closer to the 43-mL&nbsp;mark, therefore the number could be reported as 42.8. Another person could estimate the volume as 42.7 or 42.9 as the last digit is uncertain, but the measurement should always be reported to the last uncertain digit.

uncertainty in measurement: reading instruments

Counting is the type of measurement that is free from uncertainty, provided the number of objects being counted does not change during the process. Such measurements result in exact numbers. By counting the eggs in a carton, for instance, one can determine exactly how many eggs are there in the carton. Similarly, the numbers of defined quantities are also exact. For example, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilograms. Quantities derived from measurements other than counting, however, are uncertain due to practical limitations of the measurement process used.

<h4>Significant Figures</h4>

Every measurement has some uncertainty, which depends on the device used (and the user&rsquo;s ability). For instance, the volume of liquid in a graduated cylinder is measured by reading the bottom of the meniscus &mdash; the lowest point on the curved surface of the liquid. Suppose the bottom of the meniscus lies between the 15 and 16 markings;&nbsp;it means the liquid volume is certainly greater than 15 mL but less than 16 mL. The meniscus appears to be a bit closer to the 16-mL mark, and so a reasonable estimate of the liquid&rsquo;s volume would be 16.6 mL. In this measured value, the digits 1 and 6 are certain, but the last digit at the tenths place, 6, is an estimate. Some people might estimate the meniscus position to be equally distant from each of the markings and estimate the tenth-place digit as 5, while others may think it to be even closer to the 16-mL mark and estimate it to be 7. The numerical scale on this graduated cylinder has 1-mL divisions;&nbsp;thus, volumes may be measured to the nearest 0.1 mL. Similarly, a standard electronic balance may read the mass of a quarter as 5.74 g. The digits 5 and 7 are certain, and the 4 indicates that the mass of the quarter is likely between 5.73 and 5.75 grams. The quarter weighs about 5.74 grams, with a nominal uncertainty in the measurement of + 0.01 grams. If the coin is weighed on a more sensitive balance, the mass might be 5.743 g. This means its mass lies between 5.742 and 5.744 grams, an uncertainty of 0.001 grams.

This text is adapted from <a href="https://openstax.org/books/chemistry-2e/pages/1-5-measurement-uncertainty-accuracy-and-precision">Openstax, Chemistry 2e, Section 1.5: Measurement Uncertainty, Accuracy, and Precision.</a>

Counting is the type of measurement that is free from uncertainty, provided the number of objects being counted does not change during the process. Such ...

Education

JoVE Core

Introduction: Matter and Measurement

KEY TERMS AND CONCEPTS

measurement uncertainty and significant figures

Measurement uncertainty and significant figures show how carefully a value can be read from an instrument. Counting is different. If the number of objects does not change during counting, the result is exact. Defined quantities are exact too. For example, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilograms.

Most other measurements are not exact because the measuring process has limits. The result depends on the device used and on the person reading it. Every measurement therefore includes some uncertainty. The last digit is often an estimate, not a fully certain number.

A graduated cylinder shows this idea with liquid volume. The bottom of the meniscus, which is the lowest point on the curved surface of the liquid, is the reading point. If the meniscus sits between 15 mL and 16 mL, the volume is clearly greater than 15 mL and less than 16 mL. A reasonable estimate might be 15.6 mL. In that value, the 1 and 5 are certain, and the 6 is the estimated digit.

Different people may estimate the same meniscus slightly differently. One person may choose 15.5 mL, while another may read 15.7 mL if the liquid looks closer to the 16-mL mark. A cylinder with 1-mL divisions can usually measure volume to the nearest 0.1 mL. The scale marks set the limit on how precise the reading can be.

A balance shows the same idea for mass. A standard electronic balance may measure a quarter as 5.74 g. Here, the 5 and 7 are certain, and the 4 shows the estimated part of the measurement. That value suggests the mass is likely between 5.73 g and 5.75 g, with a nominal uncertainty of plus or minus 0.01 g. A more sensitive balance may read 5.743 g instead, which places the mass between 5.742 g and 5.744 g and reduces the uncertainty to 0.001 g.

Measurement Uncertainty and Significant Figures

measurement-uncertainty-and-significant-figures

scientific laws and theories

How Science Explains Laws and Theories

Scientific laws and scientific theories describe science in two different ways. A law is a concise verbal or mathematical statement that summarizes many experimental observations. It describes or predicts aspects of the natural world that stay the same under the same conditions. A scientific theory is different because it explains why natural events happen. It is a unifying principle that gives a well-supported and testable explanation for a broad set of observations. Strong theories are the...

the scientific method

How Scientists Test a Hypothesis

Chemistry uses the scientific method to turn observations into testable answers. Scientists begin by observing a phenomenon in the physical world and then asking a question about it. This process helps explain everyday chemistry through evidence. A question leads to a hypothesis, which is a tentative explanation for what is being observed. Scientists often use earlier research and published literature to shape that idea. A strong hypothesis must be testable, meaning it can be supported by...

physical and chemical properties of matter

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Distinguishing Physical and Chemical Changes

Matter can be identified by its properties, which are the features that help us tell one substance from another. These properties fall into two main groups: physical properties and chemical properties. Understanding the difference helps explain how matter changes. A physical property is a feature of matter that is not linked to a change in chemical composition. Examples include density, color, hardness, melting point, boiling point, and electrical conductivity. Some physical properties can be...

classifying matter by state

States of Matter and Their Properties

States of matter help describe how matter behaves in chemistry. Matter is anything that has mass and takes up space. It is found everywhere, including air, water, soil, mountains, and even our bodies. The three common states on Earth are solid, liquid, and gas. A solid has a definite shape and a definite volume. Its particles, such as atoms or molecules, are packed very closely together and held in place by strong forces, so they only vibrate slightly. This tight arrangement makes solids...

what is energy?

Energy Forms and Conservation

Energy is a property of matter that appears in many forms. On Earth, most energy starts with the Sun. Plants capture light energy from the Sun and use photosynthesis to turn it into chemical energy. That stored energy can then move through living things and materials in several ways. People get energy from plant foods, and wood or coal, which come from fossilized plants, can be burned to produce heat and electricity. Energy is defined as the ability to do work. Work happens when a force moves...

measurement: standard units

Measurement: SI Units and Conversions

Measurement gives three kinds of information: a number, a unit, and an estimate of uncertainty. The number shows size or magnitude. The unit gives a standard for comparison. The uncertainty reflects possible error in the measurement. Numbers can be written in more than one way. A value may appear in decimal form or in scientific notation, which is also called exponential notation. For example, the maximum take-off weight of a Boeing 777-200ER airliner is 298,000 kilograms, or 2.98 × 10^5 kg. <br...

classifying matter by composition

Matter: Elements, Compounds, and Mixtures

Matter can be grouped by composition into pure substances and mixtures. A pure substance has a constant composition throughout and uniform properties. For example, any sample of sucrose has the same composition and the same physical properties, such as melting point, color, and sweetness, no matter where it is isolated from. A mixture contains two or more kinds of matter in varying amounts. Its parts can be separated by physical changes, such as evaporation. The components keep their own...

measurement: derived units

SI Units, Density, and Property Types

SI units and derived units help describe physical properties in chemistry. The International System of Units, or SI system, uses agreed-upon base units for seven fundamental properties: length, mass, time, temperature, electric current, amount of substance, and luminosity. Units made by combining these base units are called SI-derived units. Speed is one example. If distance is measured in meters and time in seconds, the derived unit for speed is meters per second, or m/s. Miles per hour is...

dimensional analysis

Using Conversion Factors to Cancel Units

Dimensional analysis is a method for using conversion factors to cancel units and solve measurement problems. It is also called the factor label method. The main rule is that units must be treated the same way as the numbers attached to them. A conversion factor is a ratio of two equal quantities written in different units. For example, 1 meter is equal to 1.0936 yards. That relationship can be used to build a factor that converts one unit into the other. When a value is multiplied or divided...

uncertainty in measurement: accuracy and precision

uncertainty-in-measurement-accuracy-and-precision

Measurement Error: Reading Results Correctly

Measurement error affects how scientists judge the quality of their results. Repeated measurements help show whether results are precise and accurate. A measurement is precise when repeated values are very similar. A measurement is accurate when it is very close to the true or accepted value. Precise values agree with each other, while accurate values agree with the true value. A pharmaceutical quality control chemist can use this idea to check machines that dispense 500 mL of cough syrup. The...

Measurement uncertainty and significant figures show how carefully a value can be read from an instrument. Counting is different. If the number of objects does not change during counting, the result is exact. Defined quantities are exact too. For example, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilograms. Most other measurements are not exact because the measuring process has limits. The result depends on the device used and on the person...

uncertainty in measurement: significant figures

uncertainty-in-measurement-significant-figures

Significant Figures and Rounding Rules

Significant figures and rounding rules help students report measurements with the right level of certainty. A measurement includes every digit that was actually measured, including the last uncertain digit. That is why a value can look simple but still carry useful information about the tool used to measure it. Zero can also be significant when it is part of a measured value. For example, if a scale reads 140 pounds to the nearest pound, the 1, 4, and 0 are all significant. Reporting...

key terms and concepts

Learn how significant figures show measurement uncertainty with a graduated cylinder, a meniscus, and an electronic balance in high school science.

Measurement Uncertainty and Significant Figures | JoVE K12

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Atomic structure

The Periodic Table

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Periodic Classification of the Elements

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Periodic Table
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• Trends

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	...

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Formula Mass of Covalent Compounds

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Quantifying matter

Content elaboration: In earlier grades, properties of materials were quantified with measurements that were always associated with some error. In this course, scientific protocols for quantifying the properties of matter accurately and precisely are studied. Using the International System of Units (SI), significant digits or figures, scientific notation, error analysis and dimensional analysis are vital to scientific communication. There are three domains of magnitude in size and time: the macroscopic (human) domain, the cosmic domain and the submicroscopic (atomic and subatomic) domain. Measurements in the cosmic domain and submicroscopic domains require complex instruments and/or procedures. Matter can be quantified in a way that macroscopic properties such as mass can reflect the number of particles present. Elemental samples are a mixture of several isotopes with different masses. The atomic mass of an element is calculated given the mass and relative abundance of each isotope of the element as it exists in nature. Because the mass of an atom is very small, the mole is used to translate between the atomic and macroscopic levels. A mole is equal to the number of atoms in exactly 12 grams of the isotope carbon-12. The mass of one mole of a substance is equal to its molar mass in grams. The molar mass for a substance can be used in conjunction with Avogadro's number and the density of a substance to convert between mass, moles, volume and number of particles of a sample.

Measurement Uncertainty and Significant Figures

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