measurement and its uncertainties

Measurement and Its

Uncertainties

© Copyright Pearson Prentice Hall

Measurements and Their Uncertainty

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Using and Expressing Measurements

Using and Expressing Measurements

How do measurements relate to science?

3.1



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3.1 Using and Expressing Measurements

A measurement is a quantity that has both a number and a unit.

It is important to make measurements and to decide whether a measurement is correct.



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3.1 Using and Expressing Measurements

In scientific notation, a given number is written as the product of two numbers: a coefficient and 10 raised to a power.

The number of stars in a galaxy is an example of an estimate that should be expressed in scientific notation.



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Accuracy, Precision, and Error

Accuracy, Precision, and Error

How do you evaluate accuracy and precision?

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3.1 Accuracy, Precision, and Error

Accuracy and Precision

• Accuracy is a measure of how close a measurement comes to the actual or true value of whatever is measured.

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



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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.



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Determining Error

• The accepted value is the correct value based on reliable references.

• The experimental value is the value measured in the lab.

• The difference between the experimental value and the accepted value is called the error.



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The percent error is the absolute value of the error divided by the accepted value, multiplied by 100%.



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Accuracy, Precision, and Error3.1



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Just because a measuring device works, you cannot assume it is accurate. The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate.



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Significant Figures in Measurements


Why must measurements be reported to the correct number of significant figures?

3.1



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Suppose you estimate a weight that is between 2.4 lb and 2.5 lb to be 2.46 lb. The first two digits (2 and 4) are known. The last digit (6) is an estimate and involves some uncertainty. All three digits convey useful information, however, and are called significant figures.

The significant figures in a measurement include all of the digits that are known, plus a last digit that is estimated.

3.1



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The calculated answers often depend on the number of significant figures in the values used in the calculation.

3.1



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Significant Figures in Measurements3.1

Rules for determining significant figures:

1.) Every nonzero digit in a measurement is

significant.

examples: all these measurements have 3

significant figures in them:

24.7 meters

0.743 meter

714 meters



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2.) Zeros appearing between nonzero digits are significant.

examples: These have 4 in them:

7003 meters

40.79 meters

1.503 meters



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3.) Leftmost zeros appearing in front of nonzero digits are

not significant.

examples: these have only 2:

0.0071 meter

0.42 meter

0. 0000099 meter



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4.) Zeros at the end of a number and to the right of a

decimal point are always significant.

examples: these have 4 in them:

43.00 meters

1.010 meters

9.000 meters



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5.) Zeros at the rightmost end of a measurement that lie

to the left of an understood decimal point are not

significant if they show as placeholders.

examples:

300 meters = 1

7000 meters = 1

27210 meters = 4



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6.) All digits in a scientific notation is significant.

ex. 1.28 x 10 3 = 3

5. 1 x 10 4 = 2

6.000245 x 10 9 = 7


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> Significant Figures in Measurements

Animation 2

See how the precision of a calculated result depends on the sensitivity of the measuring instruments.


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Practice Problems

Problem Solving 3.2 Solve Problem 2 with the help of an interactive guided tutorial.

for Conceptual Problem 3.1



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3.1 Significant Figures in Calculations

Rounding

To round a number, you must first decide how many significant figures your answer should have. The answer depends on the given measurements and on the mathematical process used to arrive at the answer.


SAMPLE PROBLEM

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3.1


SAMPLE PROBLEM

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3.1


SAMPLE PROBLEM

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3.1


SAMPLE PROBLEM

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3.1


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Practice Problems


for Sample Problem 3.1



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Addition and Subtraction

The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places.


SAMPLE PROBLEM

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3.2


SAMPLE PROBLEM

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3.2


SAMPLE PROBLEM

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3.2


SAMPLE PROBLEM

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3.2


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Practice Problems for Sample Problem 3.2




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Multiplication and Division

• In calculations involving multiplication and division, you need to round the answer to the same number of significant figures as the measurement with the least number of significant figures.


SAMPLE PROBLEM

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3.3


SAMPLE PROBLEM

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3.3


SAMPLE PROBLEM

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3.3


SAMPLE PROBLEM

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3.3


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Practice Problems for Sample Problem 3.3



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Section Quiz

-or-Continue to: Launch:

Assess students’ understanding of the concepts in Section 3.1.

Section Assessment


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3.1 Section Quiz

1. In which of the following expressions is the number on the left NOT equal to the number on the right?

a. 0.00456 10–8 = 4.56 10–11

b. 454 10–8 = 4.54 10–6

c. 842.6 104 = 8.426 106

d. 0.00452 106 = 4.52 109


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3.1 Section Quiz

2. Which set of measurements of a 2.00-g standard is the most precise?

a. 2.00 g, 2.01 g, 1.98 g

b. 2.10 g, 2.00 g, 2.20 g

c. 2.02 g, 2.03 g, 2.04 g

d. 1.50 g, 2.00 g, 2.50 g


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3. A student reports the volume of a liquid as 0.0130 L. How many significant figures are in this measurement?

a. 2

b. 3

c. 4

d. 5

3.1 Section Quiz

END OF SHOW

measurement and its uncertainties

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