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Measurement: Accuracy, Precision, and Significant Digits

Chapter 2, Section 3

Using Scientific Measurements

5-2020 L. Blanchard Byrne

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Target 02-01 Analyze and interpret data and mathematical relationships.

Target 02-03 Collect quantitative and qualitative data with accuracy and precision.

Target 02-04 Use dimensional analysis to solve problems.

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Measuring

observation: fact, not changing, known truth, accepted – you have experienced it

Compare Observation vs. Data

data: are recorded observations or items of information

qualitative data: descriptions rather than measurements

quantitative data: recorded measurements, which are sometimes organized into tables and graphs

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Measuring: Length

Length is a linear measurement.

measured using metric rulers

Example

Find the length of the gray rod in cm.

Answer: 9.6 cm

Answer: 8.49 cm

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Measuring: Mass

Mass is the amount of “stuff’ that makes up matter or an object.

measured using beam balances

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Measuring: Mass

3. Move the masses largest to smallest.

4. Round to 1 uncertain digit.

What’s the mass of the liquid and beaker?

1. Make sure the balance is level at zero.

2. Place the object of interest on the balance.

Answer:

114.496 g

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Measuring: Volume

Volume is the space occupied by any sample of matter.

measured using graduated cylinders, pipettes, burets, syringes, etc.

always read from the BOTTOM of the meniscus – the curved surface of a liquid in a tube

must read meniscus at EYE LEVEL

use all certain digits and one uncertain digit

common units: liter (L), milliliter (mL), cubic centimeter (cm³)

length x width x height

Beakers are not used generally because they only measure APPROXIMATE – they should not be used to find specific volumes.

The glass cylinder (graduated cylinder) has etched marks to indicate volumes, a pouring lip, and quite often, a plastic bumper to prevent breakage.

Parallax errors arise when a meniscus or needle is viewed from an angle rather than from straight-on at eye level. AVOID

There are two unlabeled graduations below the meniscus, and each graduation represents 1 mL, so the certain digits of the reading are . . . ?

52 mL

Estimate the uncertain digit and take a reading.

The meniscus is about eight tenths of the way to the next graduation, so the final digit in the reading is . . . ?

0.8 mL

The volume in the graduated cylinder is . . .

52.8 mL

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Measuring: Temperature

Temperature is a measure of the average amount of kinetic energy of the particles in the material

measured using thermometers

K= ^oC+ 273

^oC = K - 273

Temperature Scales	Freezing Point	Boiling Point
Fahrenheit °F	32°F	212°F
Celsius °C	0°C	100°C
Kelvin (K) SI Units	273 K	373 K

hold in solution – do not touch bottom or sides

read at EYE LEVEL

use all certain digits and one uncertain digit

87.5 C
Certain digits are determined from the calibration marks on the thermometer.

an object’s temperature determines the direction of heat transfer – heat always moves from the object at a higher temperature to the object at a lower temperature – EXAMPLE: holding ice in hand
almost all substances expand with an increase in temperature and contract when temperature decreases – exception is water
liquid-in-glass thermometers contain a liquid (alcohol or mineral spirits) that expands and contracts more than the volume of the glass, producing changes in the column height of the liquid

Celsius scale: named after the Swedish astronomer Anders Celsius is the scale of the metric system

two readily determined temperatures used as references are the freezing (0°C) and boiling point (100°C) at 100 equal intervals or degrees Celsius

Kelvin scale: used in the physical sciences and is aka the absolute scale named for Lora Kelvin, a Scottish physicist and mathematician

the freezing point of water is 273.15 kelvins (K) and the boiling point is 373.15 kelvins (K)

does not use the degree sign

the zero point on the Kelvin scale, 0 K, or absolute zero, is equal to -273.15 °C (can round to -273 for problems)

one degree on the Celsius scale is equivalent to 1 K, so converting from one temperature to another is easy 🡪 K = °C + 273 OR °C = K - 273

Fahrenheit – room temp = 68C; body temp 98.6

Celsius- room temp = 20 ; body temp 37 C

Kelvin – room temperature = 293 K; body temp 310 K

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Measuring: Temperature

Equations

^oF = (1.8 x ^oC ) +32

^oF = 1.8 (K – 273) + 32

^oC = K – 273

^oC = .556 (^oF- 32)

K = .556 (^o F – 32) + 273

K= ^oC+ 273

Examples

Convert 32°F to °C
^oC = 0.556 (^oF- 32) 0^oC

2. Convert 102°C to K

K= ^oC+ 273 375 K

Convert 39°C to °F

🡪 ^oF = (1.8 x ^oC ) +32 102.2 °F

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Measuring: Volume

What’s the volume of the liquid in the graduated cylinder?

Answer: 6.62 mL

Answer: 11.5 mL

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Measuring: Temperature

What’s the temperature in degrees Celsius?

Answer: 87.5°C

Answer: 35.0°C

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Density

Density is the ratio of the mass of an object to its volume.

common units: g/cm³

is an intensive property – only depends on the composition of the substance, not on the size of the sample

exception is water

in general as temperature increases, density decreases (becomes less dense) and vice versa

Use mini candy bars (musketeers vs. snickers) and unwrap and drop in clear water/jar. One sinks 🡪 Why? Then get larger musketeers . . . Will it sink or float?

Nuts are dense. The Snickers, packed with peanuts, sinks to the bottom, while the nut-less 3 Musketeers floats.

You could test your students’ understanding of density by asking them to predict what would happen with other size 3 Musketeers bars. For example, would a king size candy bar float or sink? Some might think it will sink because it’s bigger, but remember, density is an intrinsic property, it doesn’t depend upon size.

An intensive property is a bulk property, meaning that it is a physical property of a system that does not depend on the system size or the amount of material in the system. Examples of intensive properties include temperature, refractive index, density, and hardness of an object. With a mixture, density can vary because the composition of a mixture can vary – EX. Corn syrup in table 3.6

EXCEPTION = water – the volume of water increases as its temperature decreases so ice or solid water floats because it is less dense than liquid water.

Greek symbol for density is called “rho”

WHAT IF – ice were MORE DENSE THAN WATER??? It would be easier to pour water in glass from ice water but what reprocussions would this have??? Lakes would feeze solid from the bottom up; aquatic life would be destroyed; possible climate changes.

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In the case of a solid floating or sinking in a liquid- Sinking and floating is a density relationship between the solid and the liquid. If the solid is less dense than the liquid, it floats. If the solid is more dense than the liquid (and you fully submerge it under the liquid....no cheating like they do with boats!) then it sinks.

It's important to note that when a solid floats, a certain percent of it is above the liquid (floating) and a certain percent of it is in the liquid (sinking). Take ice for example. The density of ice is 0.92 grams per cubic centimeter and the density of liquid water is 1.0 grams per cubic centimeter. Thus, 92% of the ice will be sinking, and 8% of the ice will be floating. Look to your left if you don't believe me.

This relationship can be summarized mathematically as follows: Density of the Solid x 100 = % of the object that sinks

Density of the liquid

That means that if the ice were put into a liquid with a density of 2.0 grams per cubic centimeter only 46% of it would sink in the liquid and 54% of it would be floating above. At the same time, if you put ice into a liquid with a density of 0.68 grams per cubic centimeter 100% of it would sink.

The following 2 simulations will help you better understand these concepts.

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Density

Examples

What’s the density of a standard gold bar if it has a mass of 12,400 g and a volume of 12,979.822 cm³ ? In kg/L?

There’s 237 cm³ in a cup of water. The density of liquid water at 20°C is 1.000 g/ cm³. What is its mass?

What is the mass of 1.55 L of a liquid with a density of 0.93 g/mL?

Answer: 237 g

Answer: 0.955 g/cm³

Answer: 1400 g

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Density

Examples

What’s the density of a standard gold bar if it has a mass of 12,400 g and a volume of 12,979.822 cm³ ? In hg/mL?

Answer: 0.955 g/cm³

There’s 237 cm³ in a cup of water. The density of liquid water at 20°C is 1.000 g/ cm³. What is its mass?

Answer: 237 g

What is the mass of 1.55 L of a liquid with a density of 0.93 g/mL?

Answer: 1400 g

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Review: Density

Examples

Equal amounts of mercury (13.6 g/cm³), water (1.000 g/cm³), and corn oil (0.922 g/cm³) are added to a beaker.

Describe the arrangement of the layers of liquids in the beaker.

Answer: corn oil on top of water on top of mercury

b. A small sugar cube is added to the beaker. Describe its location.

Answer: it floats between the layers of mercury and water – it’s density is greater than the density of water and less than the density of mercury

c. What change will occur to the sugar cube over time?

Answer: it will dissolve in the water over time

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Uncertainty in Measurement

Measurement is a quantity that has both a number and a unit.

is performed using instruments
is always some degree of uncertainty
a digit that is estimated is called uncertain

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Uncertainty in Measurement

The meter sticks are used to measure the red wire. What’s the length of the wire?

Measurement: 1.2 m

Measurement: 1.25 m

RED = CERTAIN

BLUE = UNCERTAIN

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Uncertainty in Measurement

Accuracy is a measure of how close a measurement comes to the actual, true or accepted value.

Precision is a measure of how close a series of measurements are to one another.

example: darks on a dartboard – “bull’s eye” represents the true or correct value of what you are measuring – the closeness of a dart to the bull’s-eye corresponds to the degree of accuracy; the closer it comes to the bull’s-eye, the more accurately the dart was thrown and the closeness of several darts to one another corresponds to the degree of precision

to evaluate the accuracy of a measurement, the measured value must be compared to the correct value

to evaluate the precision of a measurement, you must compare the values of two or more repeated measurements

Values for precision are not necessary and do not need to be close to the accepted value.

If measurements turn out inaccurate but precise . . . . Most likely the instrument is calibrated incorrectly or the operator/measurer is consistent just inaccurate . . . Not using the tool appropriately.

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Uncertainty in Measurement

Example

Actual Height: 1.50 m (4’9”)

Measured Height: 1.55 m (5’1”)

Calculate % error.

% error =

|1.50 m - 1.55 m| | 100% = 3.3%

1.50 m |

How to Determine Error

actual, accepted or theoretical value: correct value based on reliable references

measured or experimental value: what’s recorded in the lab

percent (relative) error =

|expected - achieved| | 100%

expected value |

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Review: Accuracy and Precision

If these sets of measurements were made of the boiling point of some liquid under similar conditions, explain which set is the most precise.

73°C, 76°C, 75°C
77°C, 78°C, 78°C
80°C, 81°C, 82°C

What would have to be known to determine which set is the most accurate?

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Uncertainty in Measurement is Communicated Using Significant Figures

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Significant Figures

a measurement includes both a # and a unit

the number includes all digits that are known certain digits, plus ONE last digit that is estimated – uncertain digit

calculated answers depend on the # of significant figures in the values used in the equation

in general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated

Examples

If

7.7 m x 5.4 m = ?

was typed into the calculator, the answer would be 41.58 - four sig figs.

The measurements used in the calculation is expressed to only two significant figures so the answer must also have two significant figures : 42m²

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Significant Figures

Rules to Determine the Number of SIG FIGS

all nonzero numbers are significant

zeros between nonzero digits are significant

trailing zeroes to the right of a decimal are significant
zeros at the rightmost end that lie to the left of a decimal point are NOT significant

leading zeroes are NOT significant

exact quantities have an infinite amount of sig figs

Examples

3,422 = 4 sig figs

1.02 = 3 sig figs

3./4. 100.30 or 0.34500 = 5 sig figs

3,000 = 1 sig fig

5) 0.000032 = 2 sig figs

6. 1 inch = 2.54 cm = infinite sig figs

By writing the measurements in scientific notation, you can eliminate such place holding zeros: in this case 7.1 x 10^-3 meter, 4.2 x 10^-1 meter, and 9.9 x 10^-5 meter.

in in addition and subtraction, the answer should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places
in multiplication and division, the answer should be rounded to the same number of significant figures as the measurement with the least number of significant figures

DEFINE certain vs. UNCERTAIN digits.

measurements must always be reported to the correct number of significant figures 🡪 the calculated value MUST BE ROUNDED so only sig figs are reported

Why must a given measurement always be reported to the correct number of signifcant figures????

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Rounding

Rules for Rounding Numbers

If the last digit is:

- 5, followed by nonzero digit(s)

- greater than 5

- less than 5

- 5, not followed by nonzero digit (s) and preceded by an odd digit

- 5, not followed by nonzero digit(s), and the preceding significant digit is even

Examples

42.68 g → 42.7 g

17.32 m → 17.3 m

2.7851 cm → 2.79 cm

4.635 kg → 4.64 kg

(because 3 is odd)

78.65 mL → 78.6 mL

(because 6 is even)

Last Digit Should:

increased by 1

stay the same

increased by 1

(odd rounds up)

stay the same

(down is same, both are four letters)

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Review

How many sig figs in each of the following?

1.0070 m

18.10 kg

100,720 L

3.29 x 10³ s

0.0054 cm

3,200,000

4 donuts

ANSWER

5 sig figs

4 sig figs

5 sig figs

3 sig figs

2 sig figs

2 sig figs

infinite number

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Significant Figures in Calculations

Rules to Determine the Number of SIG FIGS

– in CALCULATIONS

Multiplication and Division

round the answer to the same # of significant figures as the measurement with the least # of significant figures

Examples

6.38 x 2.0 = 12.76

ANSWER: 13 (2 sig figs)

100.0 g / 23.7 cm³

CALC: 4.219409283 g/cm³ ANSWER: 4.22 g/cm³

0.02 cm x 2.371 cm

CALC: 0.04742 cm² ANSWER: 0.05 cm²

710 m / 3.0 s

CALC: 236.6666667 m/s ANSWER: 240 m/s

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Significant Figures in Calculations

Rules to Determine the Number of SIG FIGS

– in CALCULATIONS

Addition and Subtraction

Round the answer to the same number of decimal places (not digits) as the measurement with the least # of decimal places

Examples

6.8 + 11.934 = 18.734

ANSWER: 18.7

3.24 m + 7.0 m

CALC: 10.24 m ANSWER: 10.2 m

100.0 g – 23.73 g

CALC: 76.27 g ANSWER: 76.3 g

0.02 cm + 2.371 cm

CALC: 2.391 ANSWER: 2.39 cm

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Review: Significant Figures

CALC ANSWER

5872.786 lb*ft 5870 lb*ft

709.228 L 709.2 L

0.35888 g/mL 0.359 g/mL

1821.57 lb 1821.6 lb

0.16 mL 0.160 mL

0.000192 mm² 0.0002 mm²

PROBLEM

1818.2 lb x 3.23 ft

713.1 L – 3.872 L

1.030 g / 2.87 mL

1818.2 lb + 3.37 lb

2.030 mL – 1.870 mL

0.0006 mm x 0.32 mm

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Fin