Significant Figures

Part I: An Introduction

Objectives

• When you complete this presentation, you will be able to– distinguish between accuracy and precision– determine the number of significant figures there

are in a measured value

Introduction

• Chemistry is a quantitative science.– We make measurements.• mass = 47.28 g• length = 14.34 cm• width = 1.02 cm• height = 3.23 cm

Introduction

• Chemistry is a quantitative science.– We make measurements.– We get lots of numbers.– We use those numbers to calculate things.• volume = length × width × height

Introduction

• Chemistry is a quantitative science.– We make measurements.– We get lots of numbers.– We use those numbers to calculate things.• volume = 14.34 cm × 1.02 cm × 3.23 cm

Introduction

• Chemistry is a quantitative science.– We make measurements.– We get lots of numbers.– We use those numbers to calculate things.• volume = 47.244564 cm3

Introduction

• Chemistry is a quantitative science.– We make measurements.– We get lots of numbers.– We use those numbers to calculate things.• volume = 47.244564 cm3

• density = mass ÷ volume

Introduction

• Chemistry is a quantitative science.– We make measurements.– We get lots of numbers.– We use those numbers to calculate things.• volume = 47.244564 cm3

• density = 47.28 g ÷ 47.244564 cm3

Introduction

• Chemistry is a quantitative science.– We make measurements.– We get lots of numbers.– We use those numbers to calculate things.• volume = 47.244564 cm3

• density = 1.000750055 g/cm3

Introduction

• Chemistry is a quantitative science.– We make measurements.– We get lots of numbers.– We use those numbers to calculate things.• density = 1.000750055 g/cm3

Introduction

• Chemistry is a quantitative science.– We make measurements.– We get lots of numbers.– We use those numbers to calculate things.• density = 1.000750055 g/cm3

– What do these numbers mean?– Do we really know the density to the nearest

0.000000001 g/cm3 (= 1/1,000,000,000 g/cm3)?

Accuracy and Precision

• Accuracy and precision are often used to mean the same thing.– We expect that an accurate measurement is a

precise measurement.– Likewise, we expect that a precise measurement is

an accurate measurement.• They are related, but they are not the same

thing.

Accuracy and Precision

• The accuracy of a series of measurements is how close those measurements are to the “real” value.– The “real” value of a measurement is usually the

value accepted by scientists.– It is usually based on a large number of

measurements made by a large number of researchers over a long period of time.

Accuracy and Precision

• The accuracy of a series of measurements is how close those measurements are to the “real” value.

• An example of accuracy is how close you come to the bullseye when shooting at a target.

• Accurate shots come close to the bullseye.• Less accurate shots miss the bullseye.

Accuracy and Precision

• The precision of a series of measurements is how close the measurements are to each other.

• Precise shots come close to each other.• Non-precise shots are not close to each other.

– Which group has a greater accuracy?– The less precise group has a greater accuracy.

Accuracy and Precision

• When we are making a new measurement, we want to be as precise as possible.

• We also want to be accurate, but usually our measurement devise is already accurate.– In most cases, inaccuracy in chemistry labs is due

to misreading the instrument.

Accuracy and Precision

• When we are making a new measurement, we want to be as precise as possible.

• Normally, we increase precision by making many measurements.

• Then, we average the measurements.

Accuracy and Precision

• Example 1:– Ahab determines the density of a metal several

times.– His measurements are: 7.65 g/cm3, 7.62 g/cm3,

7.66 g/cm3, and 7.63 g/cm3.– He reports his average density as 7.64 g/cm3.

Accuracy and Precision

• Example 1:– Brunhilda determines the density of a metal

several times.– Her measurements are: 7.82 g/cm3, 8.02 g/cm3,

7.78 g/cm3, and 7.74 g/cm3.– She reports her average density as 7.84 g/cm3.

Accuracy and Precision

• Example 1:• Who is the

most accurate?• Accuracy is

related to how close you are to the accepted value.

Measurement Ahab Brunhilda

1 7.65 g/cm3 7.82 g/cm3

2 7.62 g/cm3 8.02 g/cm3

3 7.66 g/cm3 7.78 g/cm3

4 7.63 g/cm3 7.74 g/cm3

Average 7.64 g/cm3 7.84 g/cm3

The accepted value is 7.84 g/cm3.

Accuracy and Precision

• Example 1:• Who is the

most accurate?• Ahab’s data

gives a value 0.20 g/cm3 from the accepted value.

Measurement Ahab Brunhilda

1 7.65 g/cm3 7.82 g/cm3

2 7.62 g/cm3 8.02 g/cm3

3 7.66 g/cm3 7.78 g/cm3

4 7.63 g/cm3 7.74 g/cm3

Average 7.64 g/cm3 7.84 g/cm3

The accepted value is 7.84 g/cm3.

Accuracy and Precision

• Example 1:• Who is the

most accurate?• Brunhilda’s

data gives a value the same as the accepted value.

Measurement Ahab Brunhilda

1 7.65 g/cm3 7.82 g/cm3

2 7.62 g/cm3 8.02 g/cm3

3 7.66 g/cm3 7.78 g/cm3

4 7.63 g/cm3 7.74 g/cm3

Average 7.64 g/cm3 7.84 g/cm3

The accepted value is 7.84 g/cm3.

Accuracy and Precision

• Example 1:• Who is the

most accurate?• Therefore,

Brunhilda is the most accurate.

Measurement Ahab Brunhilda

1 7.65 g/cm3 7.82 g/cm3

2 7.62 g/cm3 8.02 g/cm3

3 7.66 g/cm3 7.78 g/cm3

4 7.63 g/cm3 7.74 g/cm3

Average 7.64 g/cm3 7.84 g/cm3

The accepted value is 7.84 g/cm3.

Accuracy and Precision

• Example 1:• Who is the

most precise?• Ahab’s data

varies from 7.62 to 7.66 g/cm3 - a spread of 0.04 g/cm3.

Measurement Ahab Brunhilda

1 7.65 g/cm3 7.82 g/cm3

2 7.62 g/cm3 8.02 g/cm3

3 7.66 g/cm3 7.78 g/cm3

4 7.63 g/cm3 7.74 g/cm3

Average 7.64 g/cm3 7.84 g/cm3

The accepted value is 7.84 g/cm3.

Accuracy and Precision

• Example 1:• Who is the

most precise?• Brunhilda’s

data varies from 7.74 to 8.02 g/cm3 - a spread of 0.28 g/cm3.

Measurement Ahab Brunhilda

1 7.65 g/cm3 7.82 g/cm3

2 7.62 g/cm3 8.02 g/cm3

3 7.66 g/cm3 7.78 g/cm3

4 7.63 g/cm3 7.74 g/cm3

Average 7.64 g/cm3 7.84 g/cm3

The accepted value is 7.84 g/cm3.

Accuracy and Precision

• Example 1:• Who is the

most precise?• Ahab’s data

was the most precise.

Measurement Ahab Brunhilda

1 7.65 g/cm3 7.82 g/cm3

2 7.62 g/cm3 8.02 g/cm3

3 7.66 g/cm3 7.78 g/cm3

4 7.63 g/cm3 7.74 g/cm3

Average 7.64 g/cm3 7.84 g/cm3

The accepted value is 7.84 g/cm3.

Significant Figures

• But, what does this have to do with significant figures?

• EVERYTHING!

Significant Figures

• The measurements we use in our calculations have a built-in precision.– When we find that the mass of an object is 47.28

g, we are saying that we know the mass of the object to a precision of 0.01 g (1/100 g).

– So, we know the mass to be• (4×10) g + (7×1) g + (2×0.1) g + (8×0.01) g

– We know the mass to 4 significant figures.

1 2 3 4

Significant Figures

• The measurements we use in our calculations have a built-in precision.– When we find that the mass of an object is 47.28

g, we are saying that we know the mass of the object to a precision of 0.01 g (1/100 g).

– In the same way, we measure the length (14.34 cm), width (1.02 cm), and height (3.23 cm) to a precision of 0.01 cm.

– We know the length to 4 significant figures.

Significant Figures

• The measurements we use in our calculations have a built-in precision.– When we find that the mass of an object is 47.28

g, we are saying that we know the mass of the object to a precision of 0.01 g (1/100 g).

– In the same way, we measure the length (14.34 cm), width (1.02 cm), and height (3.23 cm) to a precision of 0.01 cm.

– We know the width to 3 significant figures.

Significant Figures

• The measurements we use in our calculations have a built-in precision.– When we find that the mass of an object is 47.28

g, we are saying that we know the mass of the object to a precision of 0.01 g (1/100 g).

– In the same way, we measure the length (14.34 cm), width (1.02 cm), and height (3.23 cm) to a precision of 0.01 cm.

– We know the height to 3 significant figures.

Significant Figures

• It should be simple to tell how many significant figures there are in a measurement.

• For example: if we measure the length of the room to be 14 meters, we have 2 significant figures.

• But 14 m = 1400 cm– 1400 cm has 4 digits– It still has only 2 significant figures.

Significant Figures

• It should be simple to tell how many significant figures there are in a measurement.

• For example: if we measure the length of the room to be 14 meters, we have 2 significant figures.

• And, 14 m = 0.014 km– 0.014 has 4 digits– It still has only 2 significant figures.

Significant Figures

• We have rules for determining the number of significant figures in a measurement.

• There is an easy way to determine the number of significant figures in a measurement.– We convert the number to scientific notation, and

count the number of significant figures.

450,000 = 4.5 × 105 2 significant figures➠

Significant Figures

• We have rules for determining the number of significant figures in a measurement.

• There is an easy way to determine the number of significant figures in a measurement.– We convert the number to scientific notation, and

count the number of significant figures.

0.03552 = 3.552 × 10−2 4 significant figures➠

Significant Figures

• We have rules for determining the number of significant figures in a measurement.

• There is an easy way to determine the number of significant figures in a measurement.– We convert the number to scientific notation, and

count the number of significant figures.

14 = 1.4 × 101 2 significant figures➠

Significant Figures

• We have rules for determining the number of significant figures in a measurement.

• There is an easy way to determine the number of significant figures in a measurement.– We convert the number to scientific notation, and

count the number of significant figures.

1,400 = 1.4 × 103 2 significant figures➠

Significant Figures

• We have rules for determining the number of significant figures in a measurement.

• There is an easy way to determine the number of significant figures in a measurement.– We convert the number to scientific notation, and

count the number of significant figures.

0.014 = 1.4 × 10-2 2 significant figures➠

Significant Figures

• We have rules for determining the number of significant figures in a measurement.

• There is an easy way to determine the number of significant figures in a measurement.– We convert the number to scientific notation, and

count the number of significant figures.

13.0 = 1.30 × 101 3 significant figures➠

Significant Figures

• We have rules for determining the number of significant figures in a measurement.

• There is an easy way to determine the number of significant figures in a measurement.– We convert the number to scientific notation, and

count the number of significant figures.

0.004200 = 4.200 × 10-3 4 significant figures➠

Examples

• How many significant figures are in each of the following numbers?

1. 4,210 m2. 0.0002543 s3. 5,100,000 kg4. 0.745 mL5. 4.324 cm6. 0.00700 L

4.21×103 m ➠ 3 significant figures

2.543×10-4 s ➠ 4 significant figures

5.1×106 kg ➠ 2 significant figures

7.45×10-1 mL ➠ 3 significant figures

4.324×100 cm ➠ 4 significant figures

7.00×10-3 L ➠ 3 significant figures

Summary

• Accuracy relates to how close a value is to an accepted value.

• Precision relates to how close individual measurements are to each other.

• Significant figures are a measure of the precision of our measurements.