Download Measurements and Their Uncertainty

Measurements and Their Uncertainty 3.1
3.1
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3.1
Measurements and Their
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Using and Expressing
Measurements
A measurement is a quantity that has both a
number and a unit.
Measurements are fundamental to the
experimental sciences. For that
reason, it is important to be able to
make measurements and to decide
whether a measurement is correct.
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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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Scientific Notation
Proper Scientific Notation Form:
M x 10N
1 < M <10
N is an integer
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Measurements and Their
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Calculator Use with
Scientific Notation
When using a Texas
Instrument Calculator,
look for the EE key. On a
graphing calculator it will
be a second function.
For 4.4 x 1025 you will see
4.4 E 25 on the display.
The calculator
understands E as
meaning x10
This method will give you
more reliable results than
using the carrot (^).
Second
Function
EE is
Second
Function on
comma key
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Measurements and Their
Uncertainty
Multiply and Divide
with Scientific Notation
To multiply –
>
3.6 x 1012 cm x 4.7 x 1015 cm =
16.920 x 1027 cm2 =
•Multiply the numbers
•Add the exponents
1.7 x 1028 cm2
•cm x cm = cm2
To divide –
•Divide the numbers
•Subtract the exponents
5.3 x 107 m3 / 3.5 x 102 m =
1.51428 x 105 m2 =
•m3 / m = m2
1.5 x 105 m2
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Measurements and Their
Uncertainty
Addition and
Subtraction with
Scientific Notation
Addition/Subtraction –
•Add/Subtract the numbers
•All numbers must be to the
same power of 10
•Keep the same power of 10
>
12.5 x 103 cm
13.7 x 103 cm
+3.4 x 103 cm
29.6 x 103 cm
•Keep the same units
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3.1
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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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Accuracy, Precision, and Error
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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3.1
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Accuracy, Precision, and Error
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Accuracy, Precision, and Error
Determining Error
• The accepted value is the correct value
based on reliable references.
• The experimental value is the value
measured in the lab.
• The absolute value of the difference between
the experimental value and the accepted
value is called the error.
Error = |Experimental – Accepted|*
*This is different than the formula in the textbook.
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Accuracy, Precision, and Error
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Measurements and Their
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Percent Error Calculation
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3.1
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Accuracy, Precision, and Error
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
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.
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Significant Figures
The significant figures in a
measurement include all of the
digits that are known, plus a last
digit that is estimated.
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Significant Figures in Measurements
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3.1
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Significant Figures in Measurements
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Significant Figures
1. All non-zero digits (1 to 9) are always
significant. (Ex. 245 has 3 SF)
2. Zeros sandwiched between non-zero
digits or other significant zeros are
always significant. (Ex. 101 has 3 SF –
also note example in #4 below)
3. Zeros to the left of all non-zero digits
are never significant. (Ex. 0.0025 has 2
SF and 0000001 has 1 SF)
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Significant Figures
4. Zeros to the right of all non-zero digits
must be to the right of the decimal to
be significant. (Ex. 250.0 has 4 SF
and 0.00370 has 3 SF)
5. Counting numbers (numbers with no
estimation) have infinite significant
figures. (Ex. There are ____ students
in this classroom)
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Significant Figures
6. Conversion factors also have infinite
significant figures. (Ex. 100 cm/1m or
60 sec/1 min.)
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Measurements and Their
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Please note in the next slide that the type
of measuring device has an impact on
the number of significant figures. (i.e.
the markings on the device determine the
number of known digits.
(There is always 1 estimated digit)
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3.1
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Significant Figures in Measurements
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Practice Problems for Conceptual Problem 3.1
Problem Solving 3.2 Solve Problem 2
with the help of an interactive guided
tutorial.
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Significant Figures in Calculations
In general, a calculated answer cannot
be more precise than the least precise
measurement from which it was
calculated.
The calculated value must be rounded to
make it consistent with the measurements
from which it was calculated.
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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.
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Rounding
1. If the number after the last significant
digit is less than 5, the number
remains unchanged.
2. If the number after the last significant
digit is 5 or greater, the last significant
digit is increased by 1.
Ex. 75.223 m to 3 SF = 75.2 m
75.277 m to 3 SF = 75.3 m
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Practice Problems for Sample Problem 3.1
Problem Solving 3.3 Solve Problem 3
with the help of an interactive guided
tutorial.
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3.1
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Significant Figures in Calculations
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.
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SAMPLE PROBLEM 3.2
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SAMPLE PROBLEM 3.2
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Practice Problems for Sample Problem 3.2
Problem Solving 3.6 Solve Problem 6
with the help of an interactive guided
tutorial.
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Significant Figures in Calculations
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.
• The position of the decimal point has nothing
to do with the rounding process when
multiplying and dividing measurements.
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SAMPLE PROBLEM 3.3
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Practice Problems
for Sample Problem 3.3
Problem Solving 3.8 Solve
Problem 8 with the help of an
interactive guided tutorial.
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Section Assessment
Assess students’ understanding
of the concepts in Section 3.1.
Continue to:
-or-
Launch:
Section Quiz
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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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3.1 Section Quiz
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
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