Transcript Section 2.2 ~ Dealing With Errors Introduction to Probability and Statistics Ms. Young

Section 2.2 ~
Dealing With Errors
Introduction to Probability and Statistics
Ms. Young
Sec. 2.2
Objective
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.
Sec. 2.2
Types of Error

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
Sec. 2.2
How to deal with these errors


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
Sec. 2.2
Example 1

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
Sec. 2.2
Size of Errors: Absolute versus Relative

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 store’s scale is poorly
calibrated you actually get only 4 pounds. You’d 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 doesn’t 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
Sec. 2.2
Absolute Error
Absolute error = claimed or measured value - actual value

Example 2:

a. Your true weight is 100 pounds, but a scale says you weight 105 pounds.
Find the absolute error.
Absolute error = 105 lb - 100 lb
Absolute error = 5 lb


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.
Absolute error = $99.0 billion - $100.0 billion
Absolute error = - $1.0 billion



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
Sec. 2.2
Relative Error
Relative error =

Absolute error
claimed or measured value - actual value
100%
actual value
Example 3:

a. Your true weight is 100 pounds, but a scale says you weigh 105 pounds.
Find the relative error.
claimed or measured value - actual value
Relative error =
100%
actual value
Relative error =
105 lb- 100 lb
100%
100 lb
Relative error =
5 lb
100%
100 lb
Relative error = 5%

Since the measured value was higher than the true value, the relative error is
positive. The measured weight was too high by 5%.
Sec. 2.2
Relative Error
Relative error =

Absolute error
measured value - actual value
100%
actual value
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.
claimed or measured value - actual value
Relative error =
100%
actual value
Relative error =
$99.0 billion - $100.0 billion
100%
$100.0 billion
Relative error =
-$1.0 billion
100%
$100.0 billion
Relative error = -1%

Since the measured value was lower than the true value, the relative error is
negative. The claimed cost was too low by 1%.
Sec. 2.2
Describing Results: Accuracy and Precision

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
Sec. 2.2
Example 4

Suppose that your true weight is 102.4 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 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 doctor’s office is more accurate because its value is
closer to your true weight.

Sec. 2.2
Summary
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
