Section 3C Dealing with Uncertainty Pages 168-178

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.

When are digits significant? Type of Digit Significance Nonzero digit (123.457) 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 0.00052) 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

Counting Significant Digits Examples: 13.80 seconds = 1.380×10 seconds 4 significant digits (implies a measurement to the nearest .01 seconds) .000345 meters = 3.45×10-4 meters 3 significant digits (implies a measurement to the nearest .000001 meter) .000001 = 1 millionth

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)

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.

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.

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.

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

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 185.5 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

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

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

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

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.

Which measurement is more precise? More accurate? Your true height is 62.50 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 62.90 inches. 1/8 inch = .125 inch 62 3/8 = 62.375inches 62.5 – 2.375 = -.125 62.5 – 92.9 = -.4inches so the tape measure is more accurate. But the laser device is more precise.

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

Accuracy Actual height = 62.50 inches Tape measure: 62⅜ inches = 62.375 inches (absolute difference = -.125 inches) Laser device: 62.90 inches (absolute difference = .4 inches) So the tape measure is more accurate.

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!!!

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.

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 = $506.10 per person 41.5 millions has 3 significant digits 82,000 has 2 significant digits So we round our answer to 2 significant digits. $506.10 rounds to $510 per person.

Homework for Wednesday: Pages 178-180 # 12, 20, 26, 30, 43, 46, 50, 56, 66