Section 2.2 ~ Dealing With Errors

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Section 2.2 Dealing With Errors

• Introduction to Probability and Statistics
• Ms. Young

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Objective
Sec. 2.2
To 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.
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Types of Error
Sec. 2.2

• Broadly speaking, measurement errors fall into
  two categories random errors and systematic
  errors
• Random errors occur because of random and
  inherently unpredictable events in the
  measurement process
• Examples
• weighing a baby that is shaking the scale
• Copying the measurement down wrong
• Reading a measuring device wrong
• Systematic errors occur when there is a problem
  in the measurement system that affects all
  measurements in the same way
• Examples
• An error in the calibration of any measuring
  device
• A scale that reads 1.2 pounds with nothing on it
• A clock that is 5 minutes slow

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How to deal with these errors
Sec. 2.2

• Random errors can be minimized by taking many
  measurements and averaging them
• Systematic errors are easy to fix when
  discovered, you can go back and adjust the
  measurements accordingly

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Example 1
Sec. 2.2

• 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.
• The first difficulty would most likely involve
  random errors because people undoubtedly made
  errors in reading the thermometer and recording
  the data
• The second difficultly would be an example of a
  systematic error since the excess heat in the
  urban would always cause the temperature to be
  higher than it would be otherwise.
• Refer to The Census case study on p.61 for
  another example

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Size of Errors Absolute versus Relative
Sec. 2.2

• Is the error big enough to be of concern or small
  enough to be unimportant?
• Scenario Suppose you go to the grocery store and
  buy what you think is 6 pounds of hamburger, but
  because the stores scale is poorly calibrated
  you actually get only 4 pounds. Youd probably
  be upset by this 2 pound error. Now suppose that
  you are buying hamburger for a huge town barbeque
  and you order 3000 pounds but only receive 2998
  pounds. You are short by the same 2 pounds as
  before, but in this case the error probably
  doesnt seem as important.
• The size of an error can differ depending on how
  you look at it
• Absolute error describes how far the claimed or
  measured value lies from the true value
• Example the 2-pound error on the scale at the
  grocery store
• Relative error compares the size of the
  absolute error to the true value and is often
  expressed as a percentage
• Example the case of buying only 4 pounds of
  meat because of the 2 pound error on the scale
  would result in a 50 relative error since the
  absolute error of 2 pounds is half the actual
  weight of 4 pounds

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Absolute Error
Sec. 2.2

• Example 2
• a. Your true weight is 100 pounds, but a scale
  says you weight 105 pounds.
• Find the absolute error.
• The measured weight is too high by 5 pounds
• b. The government claims that a program costs
  99.0 billion and the true cost
• is 100.0 billion. Find the absolute
  error.
• The claimed cost is too low by 1.0 billion
• A positive absolute error will occur when the
  measured value is higher than the true value
• A negative absolute error will occur when the
  measured value is lower than the true value

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Relative Error
Sec. 2.2

• Example 3
• a. Your true weight is 100 pounds, but a scale
  says you weigh 105 pounds.
• Find the relative error.
• Since the measured value was higher than the true
  value, the relative error is positive. The
  measured weight was too high by 5.

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Relative Error
Sec. 2.2

• Example 3
• b. The government claims that a program costs
  99.0 billion and the true cost
• is 100.0 billion. Find the relative
  error.
• Since the measured value was lower than the true
  value, the relative error is negative. The
  claimed cost was too low by 1.

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Describing Results Accuracy and Precision
Sec. 2.2

• Once a measurement is reported, we can evaluate
  it in terms of its accuracy and precision
• Accuracy describes how close a measurement lies
  to the true value
• Example A census count was 72,453 people, but
  the true population was 96,000 people. Not very
  accurate because it is nearly 25 smaller than
  the actual population
• Precision describes the amount of detail in a
  measurement
• Example census the value 72,453 is very
  precise as it seems to tell us the exact count as
  opposed to an estimate like 72,400

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Example 4
Sec. 2.2

• Suppose that your true weight is 102.4 pounds.
  The scale at the doctors 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 100.7 pounds. Which
  scale is more precise? Which is more accurate?
• The scale at the gym is more precise because it
  gives your weight to the nearest tenth of a pound
  as opposed to the nearest quarter of a pound.
• The scale at the doctors office is more accurate
  because its value is closer to your true weight.

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Summary
Sec. 2.2

• Two basic types of errors random and systematic
• The size of an error can be described as either
  absolute or relative
• Once a measurement is reported, it can be
  evaluated in terms of its accuracy and precision