Library Guides: Chemistry Textbook: Measurements and Uncertainty in Measurement

Measurement and Uncertainty in Measurement

Measurements
Measurement Uncertainty, Accuracy, and Precision

Measurements

By the end of this section, you will be able to:

Explain the process of measurement
Identify the three basic parts of a quantity
Describe the properties and units of length, mass, volume, density, temperature, and time
Perform basic unit calculations and conversions in the metric and other unit systems

Measurements provide much of the information that informs the hypotheses, theories, and laws describing the behavior of matter and energy in both the macroscopic and microscopic domains of chemistry. Every measurement provides three kinds of information: the size or magnitude of the measurement (a number); a standard of comparison for the measurement (a unit); and an indication of the uncertainty of the measurement. While the number and unit are explicitly represented when a quantity is written, the uncertainty is an aspect of the measurement result that is more implicitly represented and will be discussed later.

The number in the measurement can be represented in different ways, including decimal form and scientific notation. (Scientific notation is also known as exponential notation; a review of this topic can be found in Appendix B.) For example, the maximum takeoff weight of a Boeing 777-200ER airliner is 298,000 kilograms, which can also be written as 2.98 $\times$ 10⁵ kg. The mass of the average mosquito is about 0.0000025 kilograms, which can be written as 2.5 $\times$ 10⁻⁶ kg.

Units, such as liters, pounds, and centimeters, are standards of comparison for measurements. A 2-liter bottle of a soft drink contains a volume of beverage that is twice that of the accepted volume of 1 liter. The meat used to prepare a 0.25-pound hamburger weighs one-fourth as much as the accepted weight of 1 pound. Without units, a number can be meaningless, confusing, or possibly life threatening. Suppose a doctor prescribes phenobarbital to control a patient’s seizures and states a dosage of “100” without specifying units. Not only will this be confusing to the medical professional giving the dose, but the consequences can be dire: 100 mg given three times per day can be effective as an anticonvulsant, but a single dose of 100 g is more than 10 times the lethal amount.

The measurement units for seven fundamental properties (“base units”) are listed in Table 1.2. The standards for these units are fixed by international agreement, and they are called the International System of Units or SI Units (from the French, Le Système International d’Unités). SI units have been used by the United States National Institute of Standards and Technology (NIST) since 1964. Units for other properties may be derived from these seven base units.

Base Units of the SI System
Property Measured	Name of Unit	Symbol of Unit
length	meter	m
mass	kilogram	kg
time	second	s
temperature	kelvin	K
electric current	ampere	A
amount of substance	mole	mol
luminous intensity	candela	cd

Table 1.2

Everyday measurement units are often defined as fractions or multiples of other units. Milk is commonly packaged in containers of 1 gallon (4 quarts), 1 quart (0.25 gallon), and one pint (0.5 quart). This same approach is used with SI units, but these fractions or multiples are always powers of 10. Fractional or multiple SI units are named using a prefix and the name of the base unit. For example, a length of 1000 meters is also called a kilometer because the prefix kilo means “one thousand,” which in scientific notation is 10³ (1 kilometer = 1000 m = 10³ m). The prefixes used and the powers to which 10 are raised are listed in Table 1.3.

Common Unit Prefixes
Prefix	Symbol	Factor	Example
femto	f	10⁻¹⁵	1 femtosecond (fs) = 1 $\times$ 10⁻¹⁵ s (0.000000000000001 s)
pico	p	10⁻¹²	1 picometer (pm) = 1 $\times$ 10⁻¹² m (0.000000000001 m)
nano	n	10⁻⁹	4 nanograms (ng) = 4 $\times$ 10⁻⁹ g (0.000000004 g)
micro	µ	10⁻⁶	1 microliter (μL) = 1 $\times$ 10⁻⁶ L (0.000001 L)
milli	m	10⁻³	2 millimoles (mmol) = 2 $\times$ 10⁻³ mol (0.002 mol)
centi	c	10⁻²	7 centimeters (cm) = 7 $\times$ 10⁻² m (0.07 m)
deci	d	10⁻¹	1 deciliter (dL) = 1 $\times$ 10⁻¹ L (0.1 L )
kilo	k	10³	1 kilometer (km) = 1 $\times$ 10³ m (1000 m)
mega	M	10⁶	3 megahertz (MHz) = 3 $\times$ 10⁶ Hz (3,000,000 Hz)
giga	G	10⁹	8 gigayears (Gyr) = 8 $\times$ 10⁹ yr (8,000,000,000 yr)
tera	T	10¹²	5 terawatts (TW) = 5 $\times$ 10¹² W (5,000,000,000,000 W)

Table 1.3

LINK TO LEARNING

Need a refresher or more practice with scientific notation? Visit this site to go over the basics of scientific notation. Also check this video for a better understanding.

SI Base Units

The initial units of the metric system, which eventually evolved into the SI system, were established in France during the French Revolution. The original standards for the meter and the kilogram were adopted there in 1799 and eventually by other countries. This section introduces four of the SI base units commonly used in chemistry. Other SI units will be introduced in subsequent chapters.

Length

The standard unit of length in both the SI and original metric systems is the meter (m). A meter was originally specified as 1/10,000,000 of the distance from the North Pole to the equator. It is now defined as the distance light in a vacuum travels in 1/299,792,458 of a second. A meter is about 3 inches longer than a yard (Figure 1.14); one meter is about 39.37 inches or 1.094 yards. Longer distances are often reported in kilometers (1 km = 1000 m = 10³ m), whereas shorter distances can be reported in centimeters (1 cm = 0.01 m = 10⁻² m) or millimeters (1 mm = 0.001 m = 10⁻³ m).

One meter is slightly larger than a yard and one centimeter is less than half the size of one inch. 1 inch is equal to 2.54 cm. 1 m is equal to 1.094 yards which is equal to 39.36 inches. — Figure 1.14 The relative lengths of 1 m, 1 yd, 1 cm, and 1 in. are shown (not actual size), as well as comparisons of 2.54 cm and 1 in., and of 1 m and 1.094 yd.

Mass

The standard unit of mass in the SI system is the kilogram (kg). A kilogram was originally defined as the mass of a liter of water (a cube of water with an edge length of exactly 0.1 meter). It is now defined by a certain cylinder of platinum-iridium alloy, which is kept in France (Figure 1.15). Any object with the same mass as this cylinder is said to have a mass of 1 kilogram. One kilogram is about 2.2 pounds. The gram (g) is exactly equal to 1/1000 of the mass of the kilogram (10⁻³ kg).

The photo shows a small metal cylinder on a stand. The cylinder is covered with 2 glass lids, with the smaller glass lid encased within the larger glass lid. — Figure 1.15 This replica prototype kilogram is housed at the National Institute of Standards and Technology (NIST) in Maryland. (credit: National Institutes of Standards and Technology)

Temperature

We use the word temperature to refer to the hotness or coldness of a substance. It is a measure of average kinetic energy of molecules in a substance. Temperature is an intensive property as it does not depend on the amount of substance. If one scooped a spoon of water from a cup of water, both would have the same temperature. One way we measure a change in temperature is to use the fact that most substances expand when their temperature increases and contract when their temperature decreases. The mercury or alcohol in a common glass thermometer changes its volume as the temperature changes, and the position of the trapped liquid along a printed scale may be used as a measure of temperature.

Temperature scales are defined relative to selected reference temperatures: Two of the most commonly used are the freezing and boiling temperatures of water at a specified atmospheric pressure. On the Celsius scale, 0 °C is defined as the freezing temperature of water and 100 °C as the boiling temperature of water. The space between the two temperatures is divided into 100 equal intervals, which we call degrees. On the Fahrenheit scale, the freezing point of water is defined as 32 °F and the boiling temperature as 212 °F. The space between these two points on a Fahrenheit thermometer is divided into 180 equal parts (degrees).

The equations relating Celsius and Fahrenheit temperatures is:

T_{°F} = (\frac{9}{5} \times T_{°C}) + 32

Rearrangement of this equation yields the form useful for converting from Fahrenheit to Celsius:

T_{°C} = \frac{5}{9} (T_{°F} - 32)

The SI unit of temperature is the kelvin (K). Unlike the Celsius and Fahrenheit scales, the kelvin scale is an absolute temperature scale in which 0 (zero) K corresponds to the lowest temperature that can theoretically be achieved. Since the kelvin temperature scale is absolute, a degree symbol is not included in the unit abbreviation, K. The early 19th-century discovery of the relationship between a gas’s volume and temperature suggested that the volume of a gas would be zero at −273.15 °C. In 1848, British physicist William Thompson, who later adopted the title of Lord Kelvin, proposed an absolute temperature scale based on this concept (further treatment of this topic is provided in this text’s chapter on gases).

The freezing temperature of water on this scale is 273.15 K and its boiling temperature is 373.15 K. Notice the numerical difference in these two reference temperatures is 100, the same as for the Celsius scale, and so the linear relation between these two temperature scales will exhibit a slope of $1 \frac{K}{°C}$ . The equations for converting between the kelvin and Celsius temperature scales are derived to be:

T_{K} = T_{°C} + 273.15

T_{°C} = T_{K} - 273.15

The 273.15 in these equations has been determined experimentally, so it is not exact. Figure 1.19 shows the relationship among the three temperature scales.

A thermometer is shown for the Fahrenheit, Celsius and Kelvin scales. Under the Fahrenheit scale, the boiling point of water is 212 degrees while the freezing point of water is 32 degrees. Therefore, there are 180 Fahrenheit degrees between the boiling point of water and the freezing point of water. Under the Celsius scale, the boiling point of water is 100 degrees while the freezing point of water is 0 degrees. Therefore, there are 100 Celsius degrees between the boiling point and freezing point of water. Under the kelvin scale, the boiling point of water is 373.15 K, while the freezing point of water is 273.15 K. 233.15 K is equal to negative 40 degrees Celsius, which is also equal to negative 40 degrees Fahrenheit. — Figure 1.19 The Fahrenheit, Celsius, and kelvin temperature scales are compared.

Although the kelvin (absolute) temperature scale is the official SI temperature scale, Celsius is commonly used in many scientific contexts and is the scale of choice for nonscience contexts in almost all areas of the world. Very few countries (the U.S. and its territories, the Bahamas, Belize, Cayman Islands, and Palau) still use Fahrenheit for weather, medicine, and cooking.

EXAMPLE 1.11

Conversion from Celsius Normal body temperature has been commonly accepted as 37.0 °C (although it varies depending on time of day and method of measurement, as well as among individuals). What is this temperature on the kelvin scale and on the Fahrenheit scale?

Solution:

K = °C + 273.15 = 37.0 + 273.2 = 310.2 K

°F = \frac{9}{5} °C + 32.0 = (\frac{9}{5} \times 37.0) + 32.0 = 66.6 + 32.0 = 98.6 °F

Check Your Learning:

Convert 80.92 °C to K and °F.

Answer:

354.07 K, 177.7 °F

EXAMPLE 1.12

Conversion from Fahrenheit Baking a ready-made pizza calls for an oven temperature of 450 °F. If you are in Europe, and your oven thermometer uses the Celsius scale, what is the setting? What is the kelvin temperature?

Solution:

°C = \frac{5}{9} (°F - 32) = \frac{5}{9} (450 - 32) = \frac{5}{9} \times 418 = 232 °C ⟶ set oven to 230 °C (two significant figures)

K = °C + 273.15 = 230 + 273 = 503 K ⟶ 5.0 \times 10^{2} K (two significant figures)

Check Your Learning:

Convert 50 °F to °C and K.

Answer:

10 °C, 280 K

Time

The SI base unit of time is the second (s). Small and large time intervals can be expressed with the appropriate prefixes; for example, 3 microseconds = 0.000003 s = 3 $\times$ 10⁻⁶ and 5 megaseconds = 5,000,000 s = 5 $\times$ 10⁶ s. Alternatively, hours, days, and years can be used.

Derived SI Units

We can derive many units from the seven SI base units. For example, we can use the base unit of length to define a unit of volume, and the base units of mass and length to define a unit of density.

Volume

Volume is the measure of the amount of space occupied by an object. The standard SI unit of volume is defined by the base unit of length (Figure 1.16). The standard volume is a cubic meter (m³), a cube with an edge length of exactly one meter. To dispense a cubic meter of water, we could build a cubic box with edge lengths of exactly one meter. This box would hold a cubic meter of water or any other substance.

A more commonly used unit of volume is derived from the decimeter (0.1 m, or 10 cm). A cube with edge lengths of exactly one decimeter contains a volume of one cubic decimeter (dm³). A liter (L) is the more common name for the cubic decimeter. One liter is about 1.06 quarts.

A cubic centimeter (cm³) is the volume of a cube with an edge length of exactly one centimeter. The abbreviation cc (for cubic centimeter) is often used by health professionals. A cubic centimeter is equivalent to a milliliter (mL) and is 1/1000 of a liter.

1 dm³ = 1 L

1 cm³ = 1 mL

Figure A shows a large cube, which has a volume of 1 meter cubed. This larger cube is made up of many smaller cubes in a 10 by 10 pattern. Each of these smaller cubes has a volume of 1 decimeter cubed, or one liter. Each of these smaller cubes is, in turn, made up of many tiny cubes. Each of these tiny cubes has a volume of 1 centimeter cubed, or one milliliter. A one cubic centimeter cube is about the same width as a dime, which has a width of 1.8 centimeter. — Figure 1.16 (a) The relative volumes are shown for cubes of 1 m³, 1 dm³ (1 L), and 1 cm³ (1 mL) (not to scale). (b) The diameter of a dime is compared relative to the edge length of a 1-cm³ (1-mL) cube.

The video below demonstrates that a cubic decimeter is same as a liter. Notice that the each side of the cube is 1 dm (ie. 10 cm).

Also, the volume of a box whose edge length is 1 meter will hold 1000 smaller boxes each of length 1 decimeter. In other words, 1 m³= 1000 dm³. This is because each dimension is mutiplied three times (volume is length x width x height. Another way to see this is: 1 m³ is 1 m × 1m × 1m which is equivalent to 10 dm × 10 dm × 10 dm.

Therefore, to obtain conversion factors between volume units that are cubes of length, we must cube the conversion factor for the length. This applies for areas as well-- to obtain conversion factors between area units that are squares of length, we must square the conversion factor for length.

Conversion factors
Length	Area	Volume
1 in = 2.54 cm	1 in² = 2.54² cm²	1 in³ = 2.54³ cm³
1 m = 100 cm	1 m² = 100² cm²	1 m³ = 100³ cm³
1 dm = 10^-1 m	1 dm² = (10^-1 )² m²	1 dm³ = (10^-1 )³ m³

Density

We use the mass and volume of a substance to determine its density. Thus, the units of density are defined by the base units of mass and length.

The density of a substance is the ratio of the mass of a sample of the substance to its volume. The SI unit for density is the kilogram per cubic meter (kg/m³). For many situations, however, this as an inconvenient unit, and we often use grams per cubic centimeter (g/cm³) for the densities of solids and liquids, and grams per liter (g/L) for gases. Although there are exceptions, most liquids and solids have densities that range from about 0.7 g/cm³ (the density of gasoline) to 19 g/cm³ (the density of gold). The density of air is about 1.2 g/L. Table 1.4 shows the densities of some common substances.

Densities of Common Substances
Solids	Liquids	Gases (at 25 °C and 1 atm)
ice (at 0 °C) 0.92 g/cm³	water 1.0 g/cm³	dry air 1.20 g/L
oak (wood) 0.60–0.90 g/cm³	ethanol 0.79 g/cm³	oxygen 1.31 g/L
iron 7.9 g/cm³	acetone 0.79 g/cm³	nitrogen 1.14 g/L
copper 9.0 g/cm³	glycerin 1.26 g/cm³	carbon dioxide 1.80 g/L
lead 11.3 g/cm³	olive oil 0.92 g/cm³	helium 0.16 g/L
silver 10.5 g/cm³	gasoline 0.70–0.77 g/cm³	neon 0.83 g/L
gold 19.3 g/cm³	mercury 13.6 g/cm³	radon 9.1 g/L

Table 1.4

While there are many ways to determine the density of an object, perhaps the most straightforward method involves separately finding the mass and volume of the object, and then dividing the mass of the sample by its volume. In the following example, the mass is found directly by weighing, but the volume is found indirectly through length measurements.

density = \frac{mass}{volume}

EXAMPLE 1.1

Calculation of Density:

Gold—in bricks, bars, and coins—has been a form of currency for centuries. In order to swindle people into paying for a brick of gold without actually investing in a brick of gold, people have considered filling the centers of hollow gold bricks with lead to fool buyers into thinking that the entire brick is gold. It does not work: Lead is a dense substance, but its density is not as great as that of gold, 19.3 g/cm³. What is the density of lead if a cube of lead has an edge length of 2.00 cm and a mass of 90.7 g?

Solution:

The density of a substance can be calculated by dividing its mass by its volume. The volume of a cube is calculated by cubing the edge length.

v o l u m e o f l e a d c u b e = 2.00 c m \times 2.00 c m \times 2.00 c m = 8.00 {c m}^{3}

d e n s i t y = \frac{m a s s}{v o l u m e} = \frac{90.7 g}{8.00 {c m}^{3}} = \frac{11.3 g}{1.00 {c m}^{3}} = 11.3 g / {c m}^{3}

(We will discuss the reason for rounding to the first decimal place in the next section.)

Check Your Learning:

(a) To three decimal places, what is the volume of a cube (cm³) with an edge length of 0.843 cm?

(b) If the cube in part (a) is copper and has a mass of 5.34 g, what is the density of copper to two decimal places?

Answer:

(a) 0.599 cm³; (b) 8.91 g/cm³

LINK TO LEARNING

To learn more about the relationship between mass, volume, and density, use this interactive simulator to explore the density of different materials, like wood, ice, brick, and aluminum.

EXAMPLE 1.2

Using Displacement of Water to Determine Density:

This PhET simulation illustrates another way to determine density, using displacement of water. Determine the density of the red and yellow blocks.

Solution:

When you open the density simulation and select Same Mass, you can choose from several 5.00-kg colored blocks that you can drop into a tank containing 100.00 L water. The yellow block floats (it is less dense than water), and the water level rises to 105.00 L. While floating, the yellow block displaces 5.00 L water, an amount equal to the weight of the block. The red block sinks (it is more dense than water, which has density = 1.00 kg/L), and the water level rises to 101.25 L.

The red block therefore displaces 1.25 L water, an amount equal to the volume of the block. The density of the red block is:

d e n s i t y = \frac{m a s s}{v o l u m e} = \frac{5.00 k g}{1.25 L} = 4.00 k g / L

Note that since the yellow block is not completely submerged, you cannot determine its density from this information. But if you hold the yellow block on the bottom of the tank, the water level rises to 110.00 L, which means that it now displaces 10.00 L water, and its density can be found:

d e n s i t y = \frac{m a s s}{v o l u m e} = \frac{5.00 k g}{1 0.00 L} = 0.500 k g / L

Check Your Learning:

Remove all of the blocks from the water and add the green block to the tank of water, placing it approximately in the middle of the tank. Determine the density of the green block.

Answer:

2.00 kg/L

EXAMPLE 1.3

Experimental Determination of Density Using Water Displacement A piece of rebar is weighed and then submerged in a graduated cylinder partially filled with water, with results as shown.

This diagram shows the initial volume of water in a graduated cylinder as 13.5 milliliters. A 69.658 gram piece of metal rebar is added to the graduated cylinder, causing the water to reach a final volume of 22.4 milliliters

(a) Use these values to determine the density of this piece of rebar.

(b) Rebar is mostly iron. Does your result in (a) support this statement? How?

Solution:

The volume of the piece of rebar is equal to the volume of the water displaced:

v o l u m e = 22.4 m L - 13.5 m L = 8.9 m L = 8.9 {c m}^{3}

(rounded to the nearest 0.1 mL, per the rule for addition and subtraction)

The density is the mass-to-volume ratio:

d e n s i t y = \frac{m a s s}{v o l u m e} = \frac{69.658 g}{8.9 {c m}^{3}} = 7.8 g / {c m}^{3}

(rounded to two significant figures, per the rule for multiplication and division)

From Table 1.4 on the previous page, the density of iron is 7.9 g/cm³, very close to that of rebar, which lends some support to the fact that rebar is mostly iron.

Check Your Learning:

An irregularly shaped piece of a shiny yellowish material is weighed and then submerged in a graduated cylinder, with results as shown.

This diagram shows the initial volume of water in a graduated cylinder as 17.1 milliliters. A 51.842 gram gold colored rock is added to the graduated cylinder, causing the water to reach a final volume of 19.8 milliliters

(a) Use these values to determine the density of this material.

(b) Do you have any reasonable guesses as to the identity of this material? Explain your reasoning.

Answer:

(a) 19 g/cm³; (b) It is likely gold; the right appearance for gold and very close to the density given for gold in Table 1.4.

Key Concepts and Summary

Measurements provide quantitative information that is critical in studying and practicing chemistry. Each measurement has an amount, a unit for comparison, and an uncertainty. Measurements can be represented in either decimal or scientific notation. Scientists primarily use SI (International System) units such as meters, seconds, and kilograms, as well as derived units, such as liters (for volume) and g/cm³ (for density). In many cases, it is convenient to use prefixes that yield fractional and multiple units, such as microseconds (10⁻⁶ seconds) and megahertz (10⁶ hertz), respectively.

Key Equations

T_{°C} = \frac{5}{9} \times (T_{°F} - 32)

T_{°F} = (\frac{9}{5} \times T_{°C}) + 32

T_{K} = °C + 273.15

T_{°C} = K - 273.15

density = \frac{mass}{volume}

Measurement Uncertainty, Accuracy, and Precision

By the end of this section, you will be able to:

Define accuracy and precision
Distinguish exact and uncertain numbers
Correctly represent uncertainty in quantities using significant figures
Apply proper rounding rules to computed quantities

Counting is the only type of measurement that is free from uncertainty, provided the number of objects being counted does not change while the counting process is underway. The result of such a counting measurement is an example of an exact number. By counting the eggs in a carton, one can determine exactly how many eggs the carton contains. The numbers of defined quantities are also exact. By definition, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilogram. Quantities derived from measurements other than counting, however, are uncertain to varying extents due to practical limitations of the measurement process used.

Significant Figures in Measurement

The numbers of measured quantities, unlike defined or directly counted quantities, are not exact. To measure the volume of liquid in a graduated cylinder, you should make a reading at the bottom of the meniscus, the lowest point on the curved surface of the liquid.

This diagram shows a 25 milliliter graduated cylinder filled with about 20.8 milliliters of fluid. The diagram zooms in on the meniscus, which is the curved surface of the water that is visible when the graduated cylinder is viewed from the side. You make the reading at the lowest point of the curve of the meniscus. — Figure 1.17 To measure the volume of liquid in this graduated cylinder, you must mentally subdivide the distance between the 21 and 22 mL marks into tenths of a milliliter, and then make a reading (estimate) at the bottom of the meniscus.

Refer to the illustration in Figure 1.17. The bottom of the meniscus in this case clearly lies between the 21 and 22 markings, meaning the liquid volume is certainly greater than 21 mL but less than 22 mL. The meniscus appears to be a bit closer to the 22-mL mark than to the 21-mL mark, and so a reasonable estimate of the liquid’s volume would be 21.6 mL. In the number 21.6, then, the digits 2 and 1 are certain, but the 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 22-mL mark and estimate this digit to be 7. Note that it would be pointless to attempt to estimate a digit for the hundredths place, given that the tenths-place digit is uncertain. In general, numerical scales such as the one on this graduated cylinder will permit measurements to one-tenth of the smallest scale division. The scale in this case has 1-mL divisions, and so volumes may be measured to the nearest 0.1 mL.

This concept holds true for all measurements, even if you do not actively make an estimate. If you place a quarter on a standard electronic balance, you may obtain a reading of 6.72 g. The digits 6 and 7 are certain, and the 2 indicates that the mass of the quarter is likely between 6.71 and 6.73 grams. The quarter weighs about 6.72 grams, with a nominal uncertainty in the measurement of ± 0.01 gram. If the coin is weighed on a more sensitive balance, the mass might be 6.723 g. This means its mass lies between 6.722 and 6.724 grams, an uncertainty of 0.001 gram. Every measurement has some uncertainty, which depends on the device used (and the user’s ability). All of the digits in a measurement, including the uncertain last digit, are called significant figures or significant digits. Note that zero may be a measured value; for example, if you stand on a scale that shows weight to the nearest pound and it shows “120,” then the 1 (hundreds), 2 (tens) and 0 (ones) are all significant (measured) values.

Counting Significant Figures

A measurement result is properly reported when its significant digits accurately represent the certainty of the measurement process. But what if you were analyzing a reported value and trying to determine what is significant and what is not? Well, for starters, all nonzero digits are significant, and it is only zeros that require some thought. We will use the terms “leading,” “trailing,” and “captive” and the general rules for counting significant figures can be summarized as follows:

Non-zero digits are always significant.
Captive zeroes are always significant.
Leading zeroes are never significant.
Trailing zeroes are significant only if there is a decimal point in the number

Let's understand the terms, “leading,” “trailing,” and “captive” zeroes and where the rules come from. Leading zeroes are all the zeroes that are at the very beginning before the first non-zero digit starts in number. Captive zeroes are zeroes between two non-zero digits. Trailing zeroes are all the zeroes at the come at the end, after the last non-zero digit. See the picture below where the zeroes are labeled.

The left diagram uses the example of 3090. The zero in the hundreds place is labeled “captive” and the zero in the ones place is labeled trailing. The right diagram uses the example 0.008020. The three zeros in the ones, tenths, and hundredths places are labeled “leading.” The zero in the ten-thousandths place is labeled “captive” and the zero in the millionths place is labeled “trailing.”

Starting with the first nonzero digit on the left, count this digit and all remaining digits to the right. This is the number of significant figures in the measurement unless the last digit is a trailing zero lying to the left of the decimal point. In the diagram below on the left, we start at 1 and count all the way to 7. In the diagram below on the right, we start at 5 and count all the way to the last zero. The last zero in the figure below on the right, is included because it is a decimal zero which indicates the instrument's precision of making a measurement to the tenth's place. Because decimal zeroes are a result of the measurement, they are significant.

The left diagram uses the example of 1267 meters. The number 1 is the first nonzero figure on the left. 1267 has 4 significant figures in total. The right diagram uses the example of 55.0 grams. The number 5 in the tens place is the first nonzero figure on the left. 55.0 has 3 significant figures. Note that the 0 is to the right of the decimal point and therefore is a significant figure.

Captive zeros also result from measurement and are therefore always significant (see figure below on the left). Leading zeros, however, are never significant—they merely tell us where the decimal point is located. The leading zeros in the figure below on the right are not significant. We could use exponential notation (as described in Appendix B) and express the number as 8.32407 $\times$ 10⁻³; then the number 8.32407 contains all of the significant figures, and 10⁻³ locates the decimal point.

The left diagram uses the example of 70.607 milliliters. The number 7 is the first nonzero figure on the left. 70.607 has 5 significant figures in total, as all figures are measured including the 2 zeros. The right diagram uses the example of 0.00832407 M L. The number 8 is the first nonzero figure on the left. 0.00832407 has 6 significant figures.

The number of significant figures is uncertain in a number that ends with a zero to the left of the decimal point location. The zeros in the measurement 1,300 grams could be significant or they could simply indicate where the decimal point is located (ie. the zeroes may act as place holders for ones and tens place in this example). The ambiguity can be resolved with the use of exponential notation: 1.3 $\times$ 10³ (two significant figures), 1.30 $\times$ 10³ (three significant figures, if the tens place was measured), or 1.300 $\times$ 10³ (four significant figures, if the ones place was also measured). In cases where only the decimal-formatted number is available, it is prudent to assume that all trailing zeros are not significant.

This figure uses the example of 1300 grams. The one and the 3 are significant figures as they are clearly the result of measurement. The 2 zeros could be significant if they were measured or they could be placeholders.

When determining significant figures, be sure to pay attention to reported values and think about the measurement and significant figures in terms of what is reasonable or likely when evaluating whether the value makes sense. For example, the official January 2014 census reported the resident population of the US as 317,297,725. Do you think the US population was correctly determined to the reported nine significant figures, that is, to the exact number of people? People are constantly being born, dying, or moving into or out of the country, and assumptions are made to account for the large number of people who are not actually counted. Because of these uncertainties, it might be more reasonable to expect that we know the population to within perhaps a million or so, in which case the population should be reported as 3.17 $\times$ 10⁸ people.

Significant Figures in Calculations

A second important principle of uncertainty is that results calculated from a measurement are at least as uncertain as the measurement itself. Take the uncertainty in measurements into account to avoid misrepresenting the uncertainty in calculated results. One way to do this is to report the result of a calculation with the correct number of significant figures, which is determined by the following three rules for rounding numbers:

When adding or subtracting numbers, round the result to the same number of decimal places as the number with the least number of decimal places (the least certain value in terms of addition and subtraction).
When multiplying or dividing numbers, round the result to the same number of digits as the number with the least number of significant figures (the least certain value in terms of multiplication and division).
Treat exact quantities as having infinite number of significant digits (i.e., exact numbers have inifnite precision). Example 1 in = 2.54 cm is an exact conversion factor. The numbers 1 and 2.54 are exact and should be treated as having infinite significant digits. All metric conversion factors are also exact.
If the digit to be dropped (the one immediately to the right of the digit to be retained) is less than 5, “round down” and leave the retained digit unchanged; if it is more than 5, “round up” and increase the retained digit by 1; if the dropped digit is 5, round up or down, whichever yields an even value for the retained digit. (The last part of this rule may strike you as a bit odd, but it’s based on reliable statistics and is aimed at avoiding any bias when dropping the digit “5,” since it is equally close to both possible values of the retained digit.)

The following examples illustrate the application of this rule in rounding a few different numbers to three significant figures:

0.028675 rounds “up” to 0.0287 (the dropped digit, 7, is greater than 5)
18.3384 rounds “down” to 18.3 (the dropped digit, 3, is less than 5)
6.8752 rounds “up” to 6.88 (the dropped digit is 5, and the retained digit is even)
92.85 rounds “down” to 92.8 (the dropped digit is 5, and the retained digit is even)

Let’s work through these rules with a few examples.

EXAMPLE 1.3

Rounding Numbers

Round the following to the indicated number of significant figures:

(a) 31.57 (to two significant figures)

(b) 8.1649 (to three significant figures)

(d) 0.90275 (to four significant figures)

Solution:

(a) 31.57 rounds “up” to 32 (the dropped digit is 5, and the retained digit is even)

(b) 8.1649 rounds “down” to 8.16 (the dropped digit, 4, is less than 5)

(d) 0.90275 rounds “up” to 0.9028 (the dropped digit is 5, and the retained digit is even)

Check Your Learning:

Round the following to the indicated number of significant figures:

(a) 0.424 (to two significant figures)

(b) 0.0038661 (to three significant figures)

(d) 28,683.5 (to five significant figures)

Answer:

(a) 0.42; (b) 0.00387; (c) 421.2; (d) 28,684

EXAMPLE 1.4

Addition and Subtraction with Significant Figures

Rule: When adding or subtracting numbers, round the result to the same number of decimal places as the number with the fewest decimal places (i.e., the least certain value in terms of addition and subtraction).

(a) Add 1.0023 g and 4.383 g.

(b) Subtract 421.23 g from 486 g.

Solution:

(a) $\begin{array}{l} \begin{array}{l} \frac{\begin{matrix} 1.0023 g \\ + 4.383 g \end{matrix}}{5.3853 g} \end{array} \end{array}$

Answer is 5.385 g (round to the thousandths place; three decimal places)

(b) $\begin{array}{l} \begin{array}{l} \end{array} \\ \frac{\begin{array}{l} 486 g \\ - 421.23 g \end{array}}{64.77 g} \end{array}$

Answer is 65 g (round to the ones place; no decimal places)

Figure A shows 1.0023 being added to 4.383 to yield the answer 5.385. 1.0023 goes to the ten thousandths place, but 4.383 goes to the thousandths place, making it the less precise of the two numbers. Therefore the answer, 5.3853, should be rounded to the thousandths, to yield 5.385. Figure B shows 486 grams minus 421.23 grams, which yields the answer 64.77 grams. This answer should be round to the ones place, making the answer 65 grams.

Check Your Learning:

(a) Add 2.334 mL and 0.31 mL.

(b) Subtract 55.8752 m from 56.533 m.

Answer:

(a) 2.64 mL; (b) 0.658 m

EXAMPLE 1.5

Multiplication and Division with Significant Figures Rule: When multiplying or dividing numbers, round the result to the same number of digits as the number with the fewest significant figures (the least certain value in terms of multiplication and division).

(a) Multiply 0.6238 cm by 6.6 cm.

(b) Divide 421.23 g by 486 mL.

Solution:

(a) $\begin{array}{l} \begin{array}{l} 0.6238 cm \times 6.6 cm = 4.11708 {cm}^{2} ⟶ result is 4.1 {cm}^{2} (round to two significant figures) \\ four significant figures \times two significant figures ⟶ two significant figures answer \end{array} \end{array}$

(b) $\begin{array}{l} \frac{421.23}{g} = 0.86728... g/ mL ⟶ result is 0.867 g/mL (round to three significant figures) \\ \frac{five significant}{figures} figures ⟶ three significant figures answer \end{array}$

Check Your Learning:

(a) Multiply 2.334 cm and 0.320 cm.

(b) Divide 55.8752 m by 56.53 s.

Answer:

(a) 0.747 cm² (b) 0.9884 m/s

In the midst of all these technicalities, it is important to keep in mind the reason for these rules about significant figures and rounding—to correctly represent the certainty of the values reported and to ensure that a calculated result is not represented as being more certain than the least certain value used in the calculation.

EXAMPLE 1.6

Calculation with Significant Figures One common bathtub is 13.44 dm long, 5.920 dm wide, and 2.54 dm deep. Assume that the tub is rectangular and calculate its approximate volume in liters.

Solution:

\begin{array}{l} V & = & l \times w \times d \\ = & 13.44 dm \times 5.920 dm \times 2.54 dm \\ = & 202.09459 ... {dm}^{3} (value from calculator) \\ = & {202 dm}^{3}, or 202 L (answer rounded to three significant figures) \end{array}

Check Your Learning:

What is the density of a liquid with a mass of 31.1415 g and a volume of 30.13 cm³?

Answer:

1.034 g/mL

Accuracy and Precision

Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to evaluate both the precision and the accuracy of their results. Measurements are said to be precise if they yield very similar results when repeated in the same manner. A measurement is considered accurate if it yields a result that is very close to the true or accepted value. Precise values agree with each other; accurate values agree with a true value. These characterizations can be extended to other contexts, such as the results of an archery competition (Figure 1.18).

Figures A through C each show targets with holes where the arrows hit. The archer in figure A was both accurate and precise as all 3 arrows are clustered in the center of the target. In figure B, the archer is precise but not accurate, as all 3 arrows are clustered together but to the upper right of the center of the target. In Figure C, the archer is neither accurate nor precise as the 3 holes are not close together and are located both to the upper right and right of the target. — Figure 1.18 (a) These arrows are close to both the bull’s eye and one another, so they are both accurate and precise. (b) These arrows are close to one another but not on target, so they are precise but not accurate. (c) These arrows are neither on target nor close to one another, so they are neither accurate nor precise.

Suppose a quality control chemist at a pharmaceutical company is tasked with checking the accuracy and precision of three different machines that are meant to dispense 10 ounces (296 mL) of cough syrup into storage bottles. She proceeds to use each machine to fill five bottles and then carefully determines the actual volume dispensed, obtaining the results tabulated in Table 1.5.

Volume (mL) of Cough Medicine Delivered by 10-oz (296 mL) Dispensers
Dispenser #1	Dispenser #2	Dispenser #3
283.3	298.3	296.1
284.1	294.2	295.9
283.9	296.0	296.1
284.0	297.8	296.0
284.1	293.9	296.1

Table 1.5

Considering these results, she will report that dispenser #1 is precise (values all close to one another, within a few tenths of a milliliter) but not accurate (none of the values are close to the target value of 296 mL, each being more than 10 mL too low). Results for dispenser #2 represent improved accuracy (each volume is less than 3 mL away from 296 mL) but worse precision (volumes vary by more than 4 mL). Finally, she can report that dispenser #3 is working well, dispensing cough syrup both accurately (all volumes within 0.1 mL of the target volume) and precisely (volumes differing from each other by no more than 0.2 mL).

Key Concepts and Summary

Quantities can be defined or measured. Measured quantities have an associated uncertainty that is represented by the number of significant figures in the quantity’s number. The uncertainty of a calculated quantity depends on the uncertainties in the quantities used in the calculation and is reflected in how the value is rounded. Quantities are characterized with regard to accuracy (closeness to a true or accepted value) and precision (variation among replicate measurement results).