Unit 2. Section D
Page 1 of 3
Unit 2 – Numbers, Number Sense and Computation
Section D – Estimation
Base Objectives: Proportion, percent, estimation, roundoff error
Relative:
• It doesn’t say much to someone that you lost $1000 on the stock market, because it is all about how
much you had invested. A $1000 loss if you have $1 million invested isn’t bad compared to an
investment of, say, $2000.
• The same could be said for weight loss. Losing 5 pounds will seem much different to someone who
is 100 lbs as compared to someone who is 300 lbs.
• What is really important is the percentage change, which is found by
new amount − original amount
change
, or similarly
original amount
original amount
1000
1000
• Using this formula, the stock market loss is
= 10−3 = 0.01 = 1% vs.
= 0.5 = 50% .
6
10
2000
5
5
And the weight loss is
= 5% vs.
= 1.67%
100
300
• This is also interesting to note in advertising, where they give misleading information
• For example, did you know that when bacteria exist (on your counter, in your laundry, etc) their cells
exist in the millions to billions? E-coli has a doubling time of 20 minutes, which means every 20
minutes there are twice the number of cells as before. And it only takes about 10 cells to make you
sick. That means if you started with one cell, seven hours later there would be about 2 million cells.
Even if you happen to kill 99.9% of them with a “great” disinfectant, there would still be over 2000
cells left! 200 times the amount that could make you sick!
The Advantages of “About”:
• In the above example, notice that I used the word “about” frequently
• What is the advantage of giving someone an estimate instead of the actual amount?
• There are many types of estimations. For example, front end estimation is rounding with the left
most or largest digit given. Can you think of others?
C. Bellomo, revised 30-Mar-06
Unit 2. Section D
Page 2 of 3
So What If I’m a Bit Off? The Disadvantages:
• The Patriot missile defense system used during the Gulf War was rendered ineffective due to round
off error. The system used an integer timing register which was incremented at intervals of 0.1 s (10
ticks per second). However, the integers were converted to decimal numbers by multiplying by the
binary approximation of 0.1,
209715
0.000110011001100110011002 =
≈ 0.099999904
2097152
min
sec
ticks
As a result, after 100 hours there were 100hr × 60
× 60
× 10
= 3.6 × 106 ticks
hr
min
sec
This error accumulated, and after 100 hours the difference of
209715 ⎞
⎛1
6
⎜ −
⎟ ⋅ 3.6 ×10 ≈ 0.34332264sec
⎝ 10 2097152 ⎠
had accumulated. As a result, an Iraqi Scud missile could not be targeted and was allowed to
detonate on a barracks, killing 28 people.
Source: http://mathworld.wolfram.com/RoundoffError.html
•
An example close to home… calculating your grades for this class.
Your grades are in for the semester, and are represented by the middle column in the chart below
Worth of
Total Grade
Participation
0.1
9 Projects
0.04444444
0.04444444
0.04444444
0.04444444
0.04444444
0.04444444
0.04444444
0.04444444
0.04444444
4 Quizzes
0.1
0.1
0.1
0.1
Final Project
0.1
% Grade
Earned
96
83
87
95
91
86
81
94
95
91
89
81
87
94
92
Pts
Earned
9.6
3.688889
3.866667
4.222222
4.044444
3.822222
3.6
4.177778
4.222222
4.044444
8.9
8.1
8.7
9.4
9.2
Final Percentage: 89.58889
•
•
Recall that participation is worth 10%, Projects are worth a total of 40% (there are 9), Quizzes are
worth 40% (there are 4) and the final project is worth 10%.
40
Without rounding, each project is worth
% ≈ 4.44% . Using this value your grade is calculated to
9
be an 89.58889%, which is “rounded” to a 90% (A-).
C. Bellomo, revised 30-Mar-06
Unit 2. Section D
Page 3 of 3
•
With rounding, each project is worth about 4%. Using this value your grade is calculated to be an
86%
Worth of % Grade
Pts
Total Grade Earned Earned
Participation
0.1
96
9.6
9 Projects
0.04
83
3.32
0.04
87
3.48
0.04
95
3.8
0.04
91
3.64
0.04
86
3.44
0.04
81
3.24
0.04
94
3.76
0.04
95
3.8
0.04
91
3.64
4 Quizzes
0.1
89
8.9
0.1
81
8.1
0.1
87
8.7
0.1
94
9.4
Final Project
0.1
92
9.2
Final Percentage: 86.02
Which is not even a B+!
When Does This Most Often Occur?:
• The formal definition of round off error is: the error produced in a computation by rounding
results at one or more intermediate steps, resulting in a result different from that which would be
obtained using exact numbers.
• The most common problems resulting from roundoff error occur either when many steps are
involved with rounding occurring at each step, when two quantities very close to each other are
subtracted, or when a number is divided by a number which is close to zero.
C. Bellomo, revised 30-Mar-06