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Section 2.3Using Scientific Measurements
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Accuracy & Precision
Accuracy
Refers to the closeness of measurements to the correct
or accepted value of the quantity measured
Closeness to the correct value
Example:
Data Collected: Actual length= 5.1 cm
5.2 cm
5.1 cm
5.0 cm
5.1 cm
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Accuracy & Precision
Precision
Refers to the closeness of a set of measurements of the
same quantity made in the same way
Closeness of values to each other
Example:
Data Collected: Actual length= 5.1 cm
4.2 cm4.1 cm
4.0 cm
4.1 cm
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Accuracy & Precision
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Percentage Error
Percentage Error Formula:
Used to determine the accuracy of a value by comparing
it quantitatively to the correct or accepted value
% Error= ValueexperimentalValue accepted x100
Value accepted
Percentage Error Formula: Used to determine the accuracy of a value by comparing
it quantitatively to the correct or accepted value
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Percentage Error
% Error= ValueexperimentalValue accepted x100
Value accepted
Example: What is the percentage error for a mass measurement of
17.7 g, given that the correct value is 21.2 g?
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Percentage Error
% Error= ValueexperimentalValue accepted x100
Value accepted
Example: A volume is measured experimentally as 4.26 mL. What
is the percentage error, given that the correct value is
4.15 mL?
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Uncertainty in Measurements
Uncertainty always exists in any measurement
What affects the precision and/or accuracy of a
measurement? Skill of measurer (human errorcorrect it!)
Conditions of measurement
Instruments
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Uncertainty in Measurements What is the length of the nail?
Definitely between 2.8 and 2.9 cm
Value is about halfway between 2.8 and 2.9 cm
Hundredths place is somewhat uncertain
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Uncertainty in Measurements
When measuring record all certainnumbers and one uncertain (estimated)
number.
Length of Nail = 2.85 Estimated Number
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Measurement with a
Graduated Cylinder
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Measurement with a
Graduated Cylinder
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Measurement with an
Electronic Balance
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Significant Figures
Significant Figures are all of the numbers recorded in ameasurement, including all the certain numbers plus the first
estimated number
Why are significant figures important when taking data in thelaboratory?
Significant figures indicate the precision of the measured value to
someone looking at the data.
Example:Mass= 1100 grams (mass has been rounded to the nearest hundred
grams)
Mass= 1100.0 grams (mass has been rounded to the nearest tenth of a
gram)
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Rules for Determining
Significant Figures1. Zeros appearing between nonzero digits are significant.
Examples: 40.7 L 3 sig figs
87 009 km 5 sig figs
2. Zeros appearing in front of all nonzero digits are notsignificant.
Examples: 0.095 897 m 5 sig figs
0.0009 kg 1 sig fig
3. Zeros at the end of a number and to the right of a decimalpoint are significant.
Examples: 85.00 g 4 sig figs
9.000 000 000 mm 10 sig figs
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Rules for Determining
Significant Figures4. Zeros at the end of a number but to the left of a decimal point
may or may not be significant.
If a zero has not been measured or estimated but is just a
placeholder it is not significant. (no decimal point)
Examples: 2000 m 1 sig fig
29 310 cg 4 sig figs
A decimal point placed after zeros indicates that they are
significant.Examples: 2000. m 4 sig figs
29310. cg 5 sig figs
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Determining the Number of
Significant Figures5. All numbers in the coefficient of a number expressed in scientific notationare significant, including zeros.
Example: 7.3021 x 10-4 5 sig figs
6. Counting numbers and defined conversion factors within the same systemof measurement are exact and have an infinite number of significant figures.
Example: 2 students infinite sig figs
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Determining the Number of
Significant Figures
Pretend that the number you are evaluating is sitting on a map of the
United States
If a decimal point is present in the number, you are going towards thePacific Ocean. Start counting from the right and stop when you reach
the last non-zero digit.
Example: 1.20 ________ significant figures
190.113 ________ significant figures
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Determining the Number of
Significant Figures
If a decimal point is absent in the number, you are
going towards the Atlantic Ocean. Start counting from the left and
stop when you reach the last non-zero digit.
Examples: 250 ________ significant figures
601,820 ________ significant figures
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Practice 1: How many significant figuresare in each of the following numbers?
1. 92
2. 78.04
3. 405.34
4. 0.23
5. 23.40
6. 15.40
7. 1.2 x 103
8. 210
9. 0.00120
10. 801.5
11. 0.0478
12. 230
13. 230.
14. 54.00
15. 0.00610
16. 0.0102
17. 1,000
18. 9.010 x 10-6
19. 101.0100
20. 2,370.0
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Practice 1: How many significant figuresare in each of the following numbers?
21. Why are significant figures important when taking data in
the laboratory?
22. Why are significant figures NOT important when solving
problems in math class?
23. Using two different instruments, a student measured thelength of their foot to be 27 cm and 27.00 cm. Explain the
difference between these two measurements.
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Practice 2: Rounding Using
Significant Figures
Value # of significantfigures
Rounded
Value
24 4.31589 3
25 83,692.1 2
26 0.00574800 3
27 2,591.7742 5
28 0.0219983 4
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Practice 2: Rounding Using
Significant Figures
Value # of significantfigures
Rounded
Value
29 0.000123 2
30 23.842 4
31 7,563,874.5748 9
32 32.847 3
33 0.291 1
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Practice 2: Rounding Using
Significant Figures
Value # of significantfigures
Rounded
Value
34 0.00473 2
35 382,739.47362 6
36 83.75 3
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Rounding using Significant
Figures
Decide how many significant figures are needed
Round to that many digits, counting from the left (start counting
from the first nonzero digit)
Examine the number to the right of the last significant figure-If the digit is less than 5Leave the last significant figure
alone
-If the digit is 5 or greaterRound the last significant figure
up by 1
Change the remaining digits to zeros if the number is greater than 1
or- eliminate the digits if the number is less than 1
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Significant Figures in
Calculations
A calculated answer cannot be more
precise than the measuring tool so a
calculated answer must match the leastprecise measurement.
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Significant Figures in Addition
and Subtraction In addition and subtraction problems,
the answer must have the same
number of figures to the right of the
decimal point as there are in the
measurement having the fewest
figures to the right of the decimalpoint.
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Significant Figures in Addition
and SubtractionUse the following problem to answer the questions below.
12.52 g + 349.0 g + 8.24 g
What is the rule for rounding the answer to a calculation involvingaddition and subtraction of measurements?
Identify the place value (tens, ones, tenths, hundredths) of the lastsignificant digit in each measurement.
12.52 g ________ 349.0 g ________ 8.24 g ________
To what place value should the answer to this calculation be rounded?
Calculated answer without rounding:
Final answer with the appropriate number of significant figures:
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Practice 3: Complete the followingcalculations and round to the appropriate number of
significant figures.37. 12.52 g + 349.0 g + 8.24 g
38. 1.327 mg + 9.45 mg + 103.38 mg
39. 56.1 cm - 2.001 cm3.11 cm
40. 101.004 g + 45.0 g75.34 g
41. 8 g + 2.981 g + 8.217 g
42. 114.21 g + 3041.2 g + 0.042 g + 349.5 g
43. 78.43 g + 21.019 g + 83 g
44. 90.023 cm5.90 cm
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Significant Figures in
Multiplication and Division
In multiplication and division problems,
the answer cannot have more significant
figures than the measurement in theproblem with the fewest significant
figures.
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Significant Figures in
Multiplication and DivisionUse the following problem to answer the questions below.
8.913 m x 20.005 m
What is the rule for rounding the answer to a calculation involvingmultiplication and division of measurements?
8.913 m ____ sig figs 20.005 ____ sig figs
How many significant figures should the answer to this calculation
have?
Calculated answer without rounding
Final answer with the appropriate number of significant figures:
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45. 3.11 m x 56.1 m x 2.001 m
46. 904 L 83.41 L
47. 9.345 dg 7.2 dg 320 dg
48. 410 mm x 178.8 mm x 321 mm
49. 56.3 g x 1.7346 g 100.2 g
50. 8.3 hL x 2.27 hL
51. 2.56 cm x 4.652 cm x 8.70 cm
52. 0.81 mg 450 mg
Practice 3:Complete the followingcalculations and round to the appropriate number
of significant figures.
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Scientific Notation
Used to express very large or very small numbers
Written as a coefficient between 1 and 10 (not including10) multiplied by 10 raised to an exponent.
General Format: M x 10n
M= coefficient 1< M
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Scientific Notation
When converting between real numbers and scientific notation:
Positive Exponent = Large Real #
(greater than 10)
Negative Exponent = Small Real #
(smaller than 1)
*If the exponent is 0 dont move the decimal
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Real Number to Scientific
NotationWhen converting from a real number to scientific notation
a) move the decimal so that one number is to the left
b) count the number of times the decimal place was
moved to get the value of the exponent
c) if the real number is large than exponent is +
if the real number is small than exponent is -
Example 1:
93,000,000
Example 2:
0.000167
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Practice: Convert the following real
numbers to scientific notation.
1) length of a football field, 91.4 meters
2) diameter of a carbon atom, 0.000 000 000 154 meter
3) radius of Earth, 6 378 888 meters
4) the diameter of a human hair, 0.000 008 meter
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Scientific Notation to Real
NumbersWhen converting from scientific notation to a real number
a) if the exponent is + make the number large
if the exponent ismake the number smaller
b) number of times the decimal is moved = exponentc) add zeros in the empty spaces
Example 1:
2.11 x 10-9
Example 2:
7.3418 x 102
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Convert the following from scientific
notation to real numbers.
1) average distance between the center of the sun and the center of
Earth, 1.496 x 1011meters
2) mass of a fly, 3.27 x 10-1gram
3) the number of atoms of hydrogen in a gram, 6.02 x 1023atoms
4) the number of stars in a galaxy, 1.1 x 1010stars
dd b
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Addition & Subtraction Using
Scientific Notation These operations can be performed only if the values have the
same exponent (n factor)
Make sure the answer is in correct scientific notation format
Example:
4.2 X 104 kg + 7.9 x 103kg
l l S f
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Multiplication Using Scientific
Notation M factors are multiplied and the exponents are added
algebraically
Example: 5.23 x 106m x 7.1 x 10-2m
Di i i U i S i ifi
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Division Using Scientific
Notation M factors are divided and the exponent of the denominators is
subtracted from that of the numerator
Example:
5.44 x 107 g
8.1 x 104 mol