uncertainty in measurement professor bob kaplan university department of science 1
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Uncertainty in Measurement Professor Bob Kaplan University Department of Science
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Limitations of the Instrument
Individual Skill
Random Conditions
( not under control )
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Numbers reported should have:
1) All Digits Known
2) One Estimated Digit
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All known digits
andEstimated digit
( uncertain digit )
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Smallest unit of measurement
on glassware, ruler, scale, etc.
Place value of increment or unit
in the reported number.
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Degree of uncertainty generally depends on the smallest division or increment of the measuring instrument used (e.g. ruler, graduated cylinder, etc.).
But it also depends on:
Skill of the individual
One person may feel comfortable splitting the
Division in half ( +/- 0.5 unit )
Another person may feel confident splitting the
Division in tenths ( +/- 0.1 unit )
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Reported number: 34.746 meters
Uncertainty level: +/- 0.001 meters
= +/-- 1 mm
Reported number: 34.73579 meters
Uncertainty level: +/- 0.00001 meters
= +/-- 0.01 mm
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In measured numbers, the “sig figs” include:
All the reported numbers including
the estimated digit.
When we do calculations, we will need to count the significant figures in each of the numbers used .
In each individual number,
all non-zero numbers are counted as “sig figs”.
Zeros may or may not be significant,
depending on their position in the number.
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Zeros between integers
Always significant [ e.
g. 1004 ] 13
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Zeros that precede
integers in decimals
Never
significant [ e.g. 0.0001234
]
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Zeros that follow integers -
End of a number
Never
significant [ e.g.
1,004,000 ]
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The purpose of significant figures is to tell
you where to round off the number.
If the first digit to be dropped is 4 or less,
it and all the following digits
should be dropped.
If the first digit to be dropped is 5 or greater,
the last retained digit of the number is
increased by 1.
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Answer can be no more precise
than the least precise quantity.
Example: Solution of nitric acid that is precise to
+/-0.0001 molarity (moles / liter).
Mix that with another solution that was
not measured precisely at all.
Is the precision of my original solution retained ?
Of course not !!!!
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Result should contain the same number of sig figs
as the measurement that has the least number of
sig figs.18
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3 * 1, 465, 876 = 4, 000,000
3 * 1.465876 = 4.0
32 * 550 = 17,600 = 18,000
32 * 560 = 18,920 = 19,000
32 * 568 = 18,176 = 18,000
32 * 575 = 18,400 = 18,000
32 * 580 = 18,560 = 19,000
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“Limiting term” :
Term with fewest
decimal places.
The result is rounded off the
same as number with
fewest decimal places.
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57898.32
+ 33.34567
_____________
57931.66
57931.67
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Consider the number: 564.32
What is the place value of the 5 ?
Hundreds is correct
In a number like 564.32, what single thing determines the uncertainty ???
Last number (or digit) is correct !!
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What about the last digit is important ???
Place Value !
What is the place value of the last digit
in the number 546.32 ?
Hundredths
So what is the level of uncertainty ????
+/- .01
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Precision :
Measures repeatability
Accuracy :
Distance from true value 24
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True value:
32.146
Accurate measurements:
32.132 , 32.150 , 32.161
Precise measurements:
36.456 , 36.468 , 36.345
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Systematic Error
Instrumentation
Calibration
Standards of
Measurement 26
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