1. Section 3C Dealing with Uncertainty
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2. Significant Digits – How we state measurements
42 years old
42.0 years old
42.3 years old
Zeros are ‘significant’ when they represent actual measurements, but not when they serve only to locate the decimal point.
One key consideration in dealing with uncertainty is how we state measurements.

3. When are digits significant?
Type of Digit
Significance
Nonzero digit ( )
Always significant
Zeros that follow a nonzero digit and lie to the right of the decimal point (4.20 or 3.00)
Zeros between nonzero digits (4002 or 3.06) or other significant zeros (first zero in 30.0)
Zeros to the left of the first nonzero digit (0.006 or )
Never significant
Zeros to the right of the last nonzero digit but before the decimal point (40,000 or 210)
Not significant unless stated otherwise

4. Counting Significant Digits
Examples:
13.80 seconds
= 1.380×10 seconds
4 significant digits
(implies a measurement to the nearest .01 seconds)
meters
= 3.45×10-4 meters
3 significant digits
(implies a measurement to the nearest meter)
= 1 millionth

5. Counting Significant Digits
Examples:
26,000 students
=2.6×104
2 significant digits
(implies a measurement to the nearest 1000 people)
1.170  106 people
=(1,170,000 people)
4 significant digits
(implies a measurement to the nearest 1,000 people)

6. 3-C
Dealing with Errors
Errors can occur in many ways, but generally can be classified as one of two basic types: random or systematic errors.
Whatever the source of an error, its size can be described in two different ways: as an absolute error, or as a relative error.
Once a measurement is reported, we can evaluate it in terms of its accuracy and its precision.

7. Two Types of Measurement Error
3-C
Two Types of Measurement Error
Random errors occur because of random and inherently unpredictable events in the measurement process.
Systematic errors occur when there is a problem in the measurement system that affects all measurements in the same way, such as making them all too low or too high by the same amount.
Understanding Errors:
If you discover a systematic error, you might be able to go back and adjust the affected measurements. In contrast, the unpredictable nature of random errors makes it impossible to correct for them. However, you can minimize the effects of random errors b y making many measurements and averaging them. For example, if you measure Tukey’s weight 10 times, your measurements will probably be too high in some cases and too low in others. Thus when you average the 10 measurements, you are likely to get a value that better represents his true weight.

8. 3-C
Which type of error?
Researchers studying the progression of the AIDS epidemic need to know how many people are suffering from AIDS, which they can try to determine by studying medical records. Two of the many problems they face in this research are that (1) some people who are suffering from AIDS are misdiagnosed as having other diseases, and vice versa, and (2) some people with AIDS never seek medical help and therefore do not have medical records.

9. 3-C
Which type of error?
Before taking off, a pilot is supposed to set the aircraft altimeter to the elevation of the airport. A pilot leaves Denver (the mile high city) with her altimeter set to 2500 feet. Explain how this affects the altimeter readings throughout the flight.
What kind of error is this?
1 mile = 5280 ft

10. 3-C
Size of an Error
Suppose a scale says Tukey weighed 4.0 lbs but he actually weighed 3.5 lbs.
The same scale said my husband weighs lbs, but he actually weighs only 185 lbs.
That same ½ lb is very important in considering Tukey – but not so important with regard to Steve’s weight.
Relative Error Tukey: 14.3%
Relative Error Steve: .27%
Size of an error

11. Absolute Error vs. Relative Error
3-C
Absolute Error vs. Relative Error
absolute error
relative error
Size of an error

12. Absolute Error vs. Relative Error
3-C
Absolute Error vs. Relative Error
Example: A projected budget surplus of $17 billion turns
out to be $25 billion at the end of the fiscal year.
absolute error
= measured value – true value
= $17 billion – $25 billion = $-8 billion
relative error

13. Absolute Error vs. Relative Error
3-C
Absolute Error vs. Relative Error
Example: The label on a bag of dog food says “20 pounds,” but the true weight is only 18 pounds.
absolute error
= measured value – true value
= 20 lbs – 18 lbs = 2 lbs
relative error

14. 3-C
Accuracy vs. Precision
Accuracy describes how closely a measurement approximates a true value. An accurate measurement is very close to the true value.
Precision describes the amount of detail in a measurement.

15. Which measurement is more precise? More accurate?
Your true height is inches.
A tape measure that can be read to the nearest
⅛ inch gives your height as 62⅜ inches.
A new laser device at the doctor’s office that gives reading to the nearest 0.05 inches gives your height as inches.
1/8 inch = .125 inch /8 = inches
62.5 – = – 92.9 = -.4inches so the tape measure is more accurate.
But the laser device is more precise.

16. Precision Tape measure: read to nearest ⅛ inch = .125 inches
Laser device: read to nearest .05 inches
So the Laser is more precise.

17. Accuracy Actual height = 62.50 inches
Tape measure: 62⅜ inches = inches
(absolute difference = inches)
Laser device: inches
(absolute difference = .4 inches)
So the tape measure is more accurate.

18. Combining Measured Numbers
Rounding rule for addition or subtraction: Round your answer to the same precision as the least precise number in the problem.
Rounding rule for multiplication or division: Round your answer to the same number of significant digits as the measurement with the fewest significant digits.
Note: You should do the rounding only after completing all the operations – NOT during the intermediate steps!!!

19. Combining Measured Numbers
A book written in 1975 states that the oldest Mayan ruins are 2000 years old. How old are they now (in 2005)?
The book is about 30 years old.
2000 is the least precise (of 2000 and 30).
So we round our answer to the nearest 1000 years.
2030 rounds to
2000 years old.

20. 41.5 millions has 3 significant digits 82,000 has 2 significant digits
The government in a town of 82,000 people plans to spend $41.5 million this year. Assuming all this money must come from taxes, what average amount must the city collect from each resident?
$41,500,000 ÷ 82,000 people
= $ per person
41.5 millions has 3 significant digits
82,000 has 2 significant digits
So we round our answer to 2 significant digits.
$ rounds to $510 per person.

21. Homework for Wednesday:
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# 12, 20, 26, 30, 43, 46, 50, 56, 66