Transcript Uncertainty In Measurement

Uncertainty In
Measurement
Accuracy, Precision,
Significant Figures, and Scientific
Notation
ACCURACY
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A measure of how close a measurement comes
to the accepted or true value of whatever is
being measured
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Accepted value is a quantity used by general
agreement of the scientific community (usually found
in a reference manual)
PRECISION
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Measure of how close a series of
measurements are to one another
Measurements can:
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Be very precise without being accurate
Have poor precision and poor accuracy
Have good accuracy and good precision
ERROR
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Difference between experimental value
and accepted value
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Do you recall what accepted value is?
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Ea = | Observed – Accepted |
PERCENT ERROR
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Since it is close to impossible to measure
(through experimentation) anything and
reach the accepted value, there must be
some way to determine just how close you
actually got – that is called percent error.
Percent error is simply a mathematical
formula.
% Error = (Ea ÷ Accepted Value) ×100
SIGNIFICANT FIGURES
SIGNIFICANT FIGURES
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Measurement that includes all of the digits
that are known PLUS a last digit that is
estimated.
SIGNIFICANT FIGURE RULE #1
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Every nonzero digit is significant
Examples:
24.7 meters has 3 significant figures
0.473 meter has 3 significant figures
714 meters has 3 significant figures
245.4 meters has 4 significant figures
4793 meters has 4 significant figures
SIGNIFICANT FIGURES RULE #2
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Zeros between nonzero digits are
significant
Examples:
7003 meters has 4 significant figures
40.79 meters has 4 significant figures
0.40093 meters has 5 significant figures
SIGNIFICANT FIGURE RULE #3
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Zeros appearing in front of nonzero
digits are not significant
Examples:
0.032 meters has 2 significant figures
0.0003 meters has 1 significant figure
0.0000049 meters has 2 significant figures
SIGNIFICANT FIGURES RULE #4
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Zeros at the end of a number and to
the right of the decimal place are
always significant.
Examples:
43.00 meters has 4 significant figures
1.010 meters has 4 significant figures
9.000 meters has 4 significant figures
SIGNIFICANT FIGURES RULE #5
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Zeros at the end of a number but to
the left of the decimal are not
significant UNLESS they were
actually measured and not rounded.
To avoid ambiguity, use scientific notation
to show all significant figures if measured
amounts with no rounding.
THIS IS A DIFFICULT RULE TO
UNDERSTAND SO LET’S TALK FOR A BIT.
RULE #5 continued
300 meters (actually measured at 299) has
1 significant figure, but 300. meters
(actually measured at 300.) has 3
significant figures. The actual (not
rounded) amount should be shown as
3.00 x 102 meters.
The rounded 300 meters (299) can also be
shown in scientific notation but with only 1
significant figure: 3 x 102 meters.
CALCULATIONS USING
SIGNIFICANT FIGURES
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In all cases, round to the correct number
of significant figures as the LAST step.
Your final answer cannot be more precise
than the measured values used to obtain
it.
Scientific notation is often helpful in
rounding your final answer to the correct
number of significant figures.
ADDITION/SUBTRACTION RULE
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Answers will always be reported with the
same number of decimal places as the
measurement with the least number of
decimal places.
Example: 12.52 m + 349.0 m + 8.24 m
The “math” answer would be 369.76 m
However, the precise answer can only
have one decimal place:
369.8 m or 3.698 x 102 m
ADDITION/SUBTRACTION
EXAMPLES
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560.12 grams + 278.1 grams = 838.22 grams
Precise Answer would be 838.2 or 8.382 x 102 grams
454 cm + 2.15 cm + 200 cm = 656.15 cm
Precise Answer would be 656 or 6.56 x 102 cm
0.0010 meters – 0.123 m = - 0.122 m
Precise Answer would be -0.122 or -1.22 x 10-1 m
2.321 L – 1.1145 L = 1.2065 L
Precise Answer would be 1.207 or 1.207 x 100 m
MULTIPLICATION/DIVISION
RULE
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Round the final answer to the same
number of significant figures as the
measurement with the least number of
significant figures.
Example: 7.55 m x 0.34 m
“Math” answer will be 2.567 m2
But, the precise answer will be 2.6 m2
because the measurement 0.34 m only
has 2 significant figures.
MULTIPLICATION/DIVISION
EXAMPLES
2.3 g/mL x 12.335 mL = 28.3705 g
Precise answer would be 28 or 2.8 x 101 grams
5.45 g/mL x 15.145 mL = 82.54025 g
Precise answer would be 82.5 or 8.25 x 101 grams
35.6 g / 2.3 mL = 15.47826087 g/mL
Precise answer would be 15 or 1.5 x 101 g/mL
15.565 g / 3.56 mL = 4.372191011 g/mL
Precise answer would be 4.37 or 4.37 x 100 g/mL
MEASUREMENTS
Writing them out!
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Scientific Notation: the product of two
numbers; a coefficient and 10 raised to a
power
“Product”: means multiplication
Coefficient always has one digit
followed by a decimal and then the
rest of the significant figures
Numbers to Scientific Notation
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To change any number to scientific
notation, move the decimal point
directly behind the very first digit,
counting how many places you move.
Look at these examples:
36,000 meters = 3.6 x 104 meters: I
moved the “understood” decimal 4
places to the left   
•245,000,000 buttons = 2.45 x 108 buttons: I moved
the understood decimal 8 places to the left.
•150. Grams = 1.50 x 102 grams: I moved the decimal 2
places to the left. Note: I also put a zero on the end
of my scientific notation.
These examples are all BIG numbers (or numbers
greater than one) so the exponents are positive.
Numbers to Scientific Notation
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0.036 meters = 3.6 x 10-2 meters: I
moved the decimal 2 places to the
right
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0.0000245 liters = 2.45 x 10-5 liters: I
moved the decimal 5 places to the
right
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Small to Scientific Notation
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0.150 Grams = 1.50 x 10-1 grams: I
moved the decimal 1 place to the
right. Note: I also put a zero on the
end of my scientific.
These examples are all small numbers
(or numbers less than one) so the
exponents are negative.
Work-out these problems in
your notes:
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Determine the number of significant figures:
1) 0.502
6) 1362205.2
2) 0.0000455
7) 450.0 x 103
3) 0.000984
8) 1000 x 10-3
4) 0.0114 x 104
9) 1.29
5) 2205.2
10) 0.982 x 10-3
Bell Ringer
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Please take out a sheet of paper and
number down to 10
You will have 8 minutes
Bell Ringer
To Scientific Notation:
To decimal:
1) 3427 3.427 x 103
6) 1.56 x 104 15600
2) 0.00456 4.56 x 10-3
7) 0.56 x 10-2 0.0056
3) 123,453 1.23453x 105
8) 0.000459 x 10-1
0.0000459
9) 0.0209 x 10-3
0.0000209
4) 3100.0 x 102
3.1000 x 105
5) 1362205.2
1.3622052 x 106
10) 0.00259 x 103
2.59
Bell Ringer #2
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Please take out a sheet of paper and
number down to 10
You will have 8 minutes
Bell Ringer #2
To Scientific Notation:
To decimal:
1) 4005
6) 4.58 x 104
2) 0.000698
7) 0.321 x 10-4
3) 25,514
8) 0.000895 x 10-3
4) 814,524
9) 0.0114 x 103
5) 23,564.12
10) 5.124 x 103
Work-out these problems in
your notes:
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Addition and subtraction rule
1) 6.18 x 10-4 + 4.72 x 10-4
2) 9.10 x 103 + 2.2 x 106
3) 1913.0 - 4.6 x 103
4) 4.25 x 10-3 - 1.6 x 10-2
5) 2.34 x 106 + 9.2 x 106
Work-out these problems in
your notes:
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Multiplication and Division rule
1) 8.95 x 107/ 1.25 x 105
2) (4.5 x 102)(2.45 x 1010)
3) 3.9 x 6.05 x 420
4) 14.1 / 5
5) (1.54 x 105)(3.5 x 106)