1. Copyright © 2009 Pearson Education, Inc.
2.2 Dealing with Errors
LEARNING GOAL
Understand the difference between random and systematic errors, be able to describe errors by their absolute and relative sizes, and know the difference between accuracy and precision in measurements.
Page 59
Copyright © 2009 Pearson Education, Inc.

2. Copyright © 2009 Pearson Education, Inc.
Types of Error: Random and Systematic
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.
Page 59
Copyright © 2009 Pearson Education, Inc.

3. Copyright © 2009 Pearson Education, Inc.
By the Way ...
A systematic error in which a scale’s measurements differ consistently
from the true values is called a calibration error. You can test the
calibration of a scale by putting known weights on it, such as 0-, 5-, 10-, and 20-pound weights, and making sure that the scale gives the expected readings.
Page 59
Copyright © 2009 Pearson Education, Inc.
Slide

4. Copyright © 2009 Pearson Education, Inc.
Page 60
Copyright © 2009 Pearson Education, Inc.
Slide

5. Copyright © 2009 Pearson Education, Inc.
TIME OUT TO THINK
Go to a Web site (such as that gives the current time. How far off is your clock or watch? Describe the possible sources of random and systematic errors in your timekeeping.
Page 60
Copyright © 2009 Pearson Education, Inc.
Slide

6. Copyright © 2009 Pearson Education, Inc.
EXAMPLE 1 Errors in Global Warming Data
Scientists studying global warming need to know how the average temperature of the entire Earth, or the global average temperature, has changed with time. Consider two difficulties in
trying to interpret historical temperature data from the early 20th century: (1) Temperatures were measured with simple thermometers and the data were recorded by hand, and (2) most
temperature measurements were recorded in or near urban areas, which tend to be warmer than surrounding rural areas because of heat released by human activity. Discuss whether each of these two difficulties produces random or systematic errors, and consider the implications of these errors.
Page 60
Copyright © 2009 Pearson Education, Inc.
Slide

7. Copyright © 2009 Pearson Education, Inc.
EXAMPLE 1 Errors in Global Warming Data
The first difficulty involves random errors because people undoubtedly made occasional errors in reading the thermometers, in calibrating the thermometers, and in recording temperature readings. There is no way to predict whether any individual reading is correct, too high, or too low. However, if there are several readings for the same region on the same day, averaging these readings can minimize the effects of the random errors.
The second difficulty involves a systematic error because the excess heat in urban areas always causes the temperature reading to be higher than it would be otherwise. If the
researchers can estimate how much this systematic error affected the temperature readings, they can correct the data for this problem.
Solution:
Page 60
Copyright © 2009 Pearson Education, Inc.
Slide

8. Copyright © 2009 Pearson Education, Inc.
By the Way ...
The fact that urban areas tend to be warmer than they would be in the absence of human activity is often called the urban heat island effect. Major causes of this effect include heat released by burning fuel in automobiles, homes, and industry and the fact that pavement and large masonry buildings tend to retain heat from sunlight.
Page 60
Copyright © 2009 Pearson Education, Inc.
Slide

9. Copyright © 2009 Pearson Education, Inc.
TIME OUT TO THINK
The question of whether the Census Bureau should be allowed to adjust its “official” count on the basis of statistical surveys is very controversial. The Constitution calls for an “actual enumeration” of the population (Article 1, Section 2, Subsection 2). Do you believe that this wording precludes or allows the use of statistical surveys in the official count? Defend your opinion. Also discuss reasons why Democrats tend to favor the use of sampling methods while Republicans tend to oppose it.
Page 61. This follows the Case Study on The Census.
Copyright © 2009 Pearson Education, Inc.
Slide

10. Copyright © 2009 Pearson Education, Inc.
Size of Error: Absolute versus Relative
Absolute and Relative Errors
The absolute error describes how far a claimed or measured value lies from the true value:
absolute error = claimed or measured value – true value
The relative error compares the size of the absolute error to the true value. It is often expressed as a percentage:
relative error = x 100%
Page 61-62
absolute error
true value
Copyright © 2009 Pearson Education, Inc.
Slide

11. Copyright © 2009 Pearson Education, Inc.
EXAMPLE 2 Absolute and Relative Error
Find the absolute and relative error. Your true weight is 100 pounds, but a scale says you weigh 105 pounds.
Solution: The measured value is the scale reading of 105 pounds and the true value is 100 pounds.
absolute error = measured value – true value
= 105 lb – 100 lb
= 5 lb
Page 62
relative error = x 100% =
absolute error
true value
5 lb
100 lb
x 100% = 5%
Copyright © 2009 Pearson Education, Inc.
Slide

12. Copyright © 2009 Pearson Education, Inc.
Describing Results: Accuracy and Precision
Definitions
Accuracy describes how closely a measurement approximates a true value. An accurate measurement is close to the true value. (Close is generally defined as a small relative error, rather than a small absolute error.)
Precision describes the amount of detail in a measurement.
Page 63
Copyright © 2009 Pearson Education, Inc.
Slide

13. Copyright © 2009 Pearson Education, Inc.
By the Way ...
In 1999, NASA lost the $160 million Mars Climate Orbiter when engineers sent it very precise computer instructions in English units (pounds) but the spacecraft software interpreted them in metric units (kilograms). In other words, the loss occurred because the very precise instructions were actually quite inaccurate! NASA learned its lesson and has since sent four spacecraft to Mars successfully, with another (called Phoenix) scheduled to land on Mars in 2008.
Page 63
Copyright © 2009 Pearson Education, Inc.
Slide

14. Copyright © 2009 Pearson Education, Inc.
EXAMPLE 3 Accuracy and Precision in Your Weight
Suppose that your true weight is pounds. The scale at the doctor’s office, which can be read only to the nearest quarter pound, says that you weigh 102¼ pounds. The scale at the gym, which gives a digital readout to the nearest 0.1 pound, says that you weigh pounds.
Which scale is more precise? Which is more accurate?
Solution: The scale at the gym is more precise because it gives your weight to the nearest tenth of a pound, whereas the doctor’s scale gives your weight only to the nearest quarter pound. However, the scale at the doctor’s office is more accurate because its value is closer to your true weight.
Page 63
Copyright © 2009 Pearson Education, Inc.
Slide

15. Copyright © 2009 Pearson Education, Inc.
TIME OUT TO THINK
In Example 3, we need to know your true weight to determine which scale is more accurate. But how would you know your true weight? Can you ever be sure that you know it? Explain.
Page 63
Copyright © 2009 Pearson Education, Inc.
Slide

16. Copyright © 2009 Pearson Education, Inc.
By the Way ...
The digits in a number that were actually measured are called significant digits. All the digits in a number are significant except for zeros that are included so that the decimal point can be properly located. For example, has four significant digits; the zeros are required for proper placement of the decimal point. The number 1,234,000,000 has four significant digits for the same reason. The number has four significant digits; the zero is significant because it is not required for the proper placement of the decimal point and therefore implies an actual measurement of zero tenths.
Page 64
Copyright © 2009 Pearson Education, Inc.
Slide

17. Copyright © 2009 Pearson Education, Inc.
Summary: Dealing with Errors
The ideas we’ve covered in this section are a bit technical, but very important to understanding measurements and errors. Let’s briefly summarize how the ideas relate to one another.
• Errors can occur in many ways, but generally can be classified into one of two basic types: random errors 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.
Page 64
Copyright © 2009 Pearson Education, Inc.
Slide

18. Copyright © 2009 Pearson Education, Inc.
The End
Page 13
Copyright © 2009 Pearson Education, Inc.
Slide