1.04 Accuracy, Precision and Significant Figures
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1.04 Accuracy, Precision and Significant Figures
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Uncertainty in numbers Story: Taxi driver points to a pyramid "...these pyramids are 4506 years old". After a quick calculation, the tourist asked, “how can you be so certain that these were completed in 2506 BC ?”
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1.04 Accuracy, Precision and Significant Figures
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Uncertainty in Measurements It is important to realize that any measurement will always contain some degree of uncertainty. The uncertainty of the measurement is determine by the scale of the measuring device.
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1.04 Accuracy, Precision and Significant Figures
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Precision vs. Accuracy Precision: Indication of how close individual measurements agree. Accuracy: How close individual measurements agree with true value. In general, experimental measurements are taken numerous times to improve precision; more precise ε more accurate.
….. but this is not always true.
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1.04 Accuracy, Precision and Significant Figures
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High Precision Low Accuracy High precision grouping is tight. Low Accuracy - but the mark misses the target.
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1.04 Accuracy, Precision and Significant Figures
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Low Precision High Accuracy High Accuracy - mark is averaged around the target. Low Precision grouping is scattered.
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1.04 Accuracy, Precision and Significant Figures
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Low Precision Low Accuracy Low Precision grouping is scattered. Low Accuracy mark misses the target. 7
1.04 Accuracy, Precision and Significant Figures
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High Precision High Accuracy High Accuracy mark is averaged around the target. High precision grouping is tight.
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1.04 Accuracy, Precision and Significant Figures
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Accuracy, Precision & Significant Figures Measurements in the lab
a) Precise Accurate (laudable) Promote the analyst
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b) Precise Inaccurate (avoidable) Repair the instrument
c) Imprecise Accurate (by accident) Train the analyst
1.04 Accuracy, Precision and Significant Figures
d) Imprecise Inaccurate (lamentable) Hire a new analyst
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Making a measurement
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1.04 Accuracy, Precision and Significant Figures
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Reading a temperature measurement
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1.04 Accuracy, Precision and Significant Figures
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Reading a length measurement Accurate number small errors Precise number small uncertainty
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0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
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1.04 Accuracy, Precision and Significant Figures
Δ = 1cm error is tenth 1/10 8
4.4 cm + .1
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Reading a length measurement Accurate number small errors Precise number small uncertainty 0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
13
Δ = 1cm error is tenth 1/10 8
4.4 cm + .1
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1.04 Accuracy, Precision and Significant Figures
8.17.09
Reading a length measurement Accurate number small errors Precise number small uncertainty
14
0
1
2
3
4
5
6
7
0
1
2
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5
6
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1.04 Accuracy, Precision and Significant Figures
Δ = 1cm error is tenth 1/10 8
4.4 cm + .1
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Reading a length measurement Accurate number small errors Precise number small uncertainty 0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
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1.04 Accuracy, Precision and Significant Figures
Δ = 1cm error is tenth 1/10 8
4.4 cm + .1
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Examples Read the measurement of each.
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1.04 Accuracy, Precision and Significant Figures
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Significant Figures Alternative method of regarding uncertainty. Scientist have found it useful to tell the degree of certainty of a measured number; merely writing down all the digits that are certain and not writing down any that are not certain.
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1.04 Accuracy, Precision and Significant Figures
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How many significant figures do these numbers contain ? 12.000 Significant Figures
? 5
0.100
.0005320
4500
?4
? 2
?3
Nonzero integers - always count as significant figures. Zeros Leading zeros - are zeros that precede all of the nonzero digits. They never count as significant figures. Captive zeros - are zeros that fall between nonzero digits. They all count as significant figures. Trailing zeros - are zeros at the right end of the number. They are significant only if the number contains a decimal point.
Exact numbers - such as tallies or conversion factors have unlimited number of significant figures. 18
1.04 Accuracy, Precision and Significant Figures
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A simpler way An easy method to remember significant figures rule is as follows:
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1.04 Accuracy, Precision and Significant Figures
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Now determine the number of significant figures 12.000 Significant Figures
?
? 5
0.100
? ? 3
.0005320
4500
? ? 4
? ? 2
12.000 five signifciant figures three signifciant figures 0.100 .0005320 four signifciant figures 4500 two signifciant figures 20
1.04 Accuracy, Precision and Significant Figures
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Significant Digits, the different type of Uncertainty
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1.04 Accuracy, Precision and Significant Figures
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Manipulation of Significant Figures: (add and subtract)
Least precise
2
0
.
4
.
3
8
1 3 0
.
6
1 0
5
.
3
1 0
5
.
Addition and Subtraction: 2
2
2Uncertainty of answer
(Significant figures of answer)
Answ: = 105.
is limited to the value with the 2
2 2least precise value (number
with fewest digit after decimal place - the number 83 in our example).
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1.04 Accuracy, Precision and Significant Figures
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Difference between Significant figures Precision What is the difference between significant figures and precision. For example in the example shown, which has the fewest significant figures and which is the least precise ?
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1.04 Accuracy, Precision and Significant Figures
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Manipulation of Significant Figures: (mult and divide)
Fewest number of significant figures
44.4 0.17
3 S. F. 2 S. F.
7 . 548
2 S. F.
7 .5 24
Answer with 2 s.f.
Multiplication and division: Uncertainty of answer (Significant. Figure) is limited to the value with the fewest significant figures. In our example, 0.17 limits the value.
1.04 Accuracy, Precision and Significant Figures
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Exception to significant figure rules: These type of numbers contain unlimited significant figures
(do not influence the number of signif.. figures in the final answer).
• Number of Tallies, i.e., 5 fingers, 176 students, • Definition of numbers i.e., Exactly 1 m = 100 cm, or 1 in = 2.54 cm
• Power of 10 in exponential notation i.e., 106 but no such thing as 10 6.4 25
1.04 Accuracy, Precision and Significant Figures
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Written digits of results must have proper number of significant figures. Can determine number of significant figures for any number using U.S. map analogy. • Addition/Subtraction Least precise number in the data determines number of signif. figures in the result • Multiplication/Division data with fewest signif. figures determines number of signif. figures in the result. Exception to the rules of significant figures 26
1.04 Accuracy, Precision and Significant Figures
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