Significant Figures

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



– We get lots of numbers .

– We use those numbers to calculate things.

• volume = length × width × height

Introduction





• volume = 14.34 cm × 1.02 cm × 3.23 cm

Introduction





• volume = 47.244564 cm 3

Introduction





• volume = 47.244564 cm 3

• density = mass ÷ volume

Introduction





• volume = 47.244564 cm 3

• density = 47.28 g ÷ 47.244564 cm 3

Introduction





• volume = 47.244564 cm 3

• density = 1.000750055 g/cm 3

Introduction





• density = 1.000750055 g/cm 3

Introduction





• density = 1.000750055 g/cm 3

– What do these numbers mean?

– Do we really know the density to the nearest

0.000000001 g/cm 3 (= 1 /

1,000,000,000 g/cm 3 )?

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.


• 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.


• 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.


• 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.


• 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.


• 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.


• Example 1:

– Ahab determines the density of a metal several times.

– His measurements are: 7.65 g/cm 3 , 7.62 g/cm 3 ,

7.66 g/cm 3 , and 7.63 g/cm 3 .

– He reports his average density as 7.64 g/cm 3 .


• Example 1:

– Brunhilda determines the density of a metal several times.

– Her measurements are: 7.82 g/cm 3 , 8.02 g/cm 3 ,

7.78 g/cm 3 , and 7.74 g/cm 3 .

– She reports her average density as 7.84 g/cm 3 .


• Example 1:

• Who is the most accurate?

Measurement

1

Ahab Brunhilda

7.65 g/cm 3 7.82 g/cm 3

• Accuracy is related to how close you are to the accepted value.

2

3

4

Average

7.62 g/cm 3

7.66 g/cm 3

7.63 g/cm 3

7.64 g/cm 3

8.02 g/cm 3

7.78 g/cm 3

7.74 g/cm 3

7.84 g/cm 3

The accepted value is 7.84 g/cm 3 .


• Example 1:


• Ahab’s data gives a value

0.20 g/cm 3 from the accepted value.

Measurement

1

2

3

4

Average

Ahab Brunhilda

7.65 g/cm 3 7.82 g/cm 3

7.62 g/cm 3

7.66 g/cm 3

7.63 g/cm 3

7.64 g/cm 3

8.02 g/cm 3

7.78 g/cm 3

7.74 g/cm 3

7.84 g/cm 3



• Example 1:


Measurement

1

Ahab Brunhilda

7.65 g/cm 3 7.82 g/cm 3

• Brunhilda’s data gives a value the same as the accepted value.

2

3

4

Average

7.62 g/cm 3

7.66 g/cm 3

7.63 g/cm 3

7.64 g/cm 3

8.02 g/cm 3

7.78 g/cm 3

7.74 g/cm 3

7.84 g/cm 3



• Example 1:


• Therefore,

Brunhilda is the most accurate.

Measurement

1

2

3

4

Average

Ahab Brunhilda

7.65 g/cm 3 7.82 g/cm 3

7.62 g/cm 3

7.66 g/cm 3

7.63 g/cm 3

7.64 g/cm 3

8.02 g/cm

7.78 g/cm

7.74 g/cm

7.84 g/cm

3

3

3

3



• Example 1:

• Who is the most precise?

• Ahab’s data varies from

7.62 to 7.66 g/cm 3 - a spread of 0.04 g/cm 3 .

Measurement

1

2

3

4

Average

Ahab Brunhilda

7.65 g/cm 3 7.82 g/cm 3

7.62 g/cm 3

7.66 g/cm 3

7.63 g/cm 3

7.64 g/cm 3

8.02 g/cm

7.78 g/cm

7.74 g/cm

7.84 g/cm

3

3

3

3



• Example 1:


• Brunhilda’s data varies from 7.74 to

8.02 g/cm 3 - a spread of 0.28 g/cm 3 .

Measurement

1

2

3

4

Average

Ahab Brunhilda

7.65 g/cm 3 7.82 g/cm 3

7.62 g/cm 3

7.66 g/cm 3

7.63 g/cm 3

7.64 g/cm 3

8.02 g/cm

7.78 g/cm

7.74 g/cm

7.84 g/cm

3

3

3

3



• Example 1:


• Ahab’s data was the most precise.

Measurement

1

2

3

4

Average

Ahab Brunhilda

7.65 g/cm 3 7.82 g/cm 3

7.62 g/cm 3

7.66 g/cm 3

7.63 g/cm 3

7.64 g/cm 3

8.02 g/cm

7.78 g/cm

7.74 g/cm

7.84 g/cm

3

3

3

3


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

1 2 3 4

• (4×10) g + (7×1) g + (2×0.1) g + (8×0.01) g

– We know the mass to 4 significant figures.

Significant Figures



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



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



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 × 10 5

➠

2 significant figures

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0.03552 = 3.552 × 10 −2

➠


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14 = 1.4 × 10 1

➠


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1,400 = 1.4 × 10 3

➠


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0.014 = 1.4 × 10 -2

➠


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13.0 = 1.30 × 10 1

➠


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0.004200 = 4.200 × 10 -3

➠


Examples

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

1. 4,210 m

2. 0.0002543 s

4.21×10 3 m

2.543×10 -4 s

➠


➠


3. 5,100,000 kg

4. 0.745 mL

5. 4.324 cm

6. 0.00700 L

5.1×10 6 kg

7.45×10 -1 mL

4.324×10 0 cm

7.00×10 -3 L

➠


➠


➠


➠


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.

Significant Figures

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