LSP 120: Quantitative Reasoning and
Technological Literacy
Section 202
Özlem Elgün
Making and Interpreting Graphs
There are 3 types of graphs we use in Quantitative Reasoning class:
1. Pie Charts
2. Bar charts
3. XY graphs (or line graphs)
How to describe the graphs
• Your graph should be able to stand alone without
any words to tell the reader what they are looking
at.
• If the reader doesn't understand the graph without
a caption (or story) then the graph isn't very good.
• In the paragraph describing the graph, you should
first tell your audience briefly what the graph is
about, and also point out what you want the
reader to know about the graph.
Misleading Graphs
• Difference between a bad graph and a
misleading graph
• Bad graph: a graph with incorrect information,
incorrectly designed. Etc…
• Misleading graph: data is "correctly" displayed
on the graph and yet gives different messages
depending on how it is displayed.
Graphing Checklist and Graphing
worksheet
• Graphing Worksheet –EXTRA CREDIT
Absolute and Relative Quantities
There are two ways to measure numerical data—
particularly if the goal is to measure the least and
greatest occurrence of some quantifiable variable.
• The absolute quantity is a measure of the absolute
occurrence of the variable. It is a “sheer” number.
It tells how many or how much. (number of adults
who have HIV)
• The relative quantity is the absolute quantity
divided by some other quantity. (percent of adults
who have HIV)
Absolute and Relative Quantities
• For example, the excel file IV_Adults_By_Country_2001.xls contains data
the number of adults with HIV in countries around the world.
• One could reorganize the data by sorting adults with HIV in descending
order.
• (To sort: First select the area on the spreadsheet you want to sort. Then
click on the Home Tab. Under the Editing Section choose custom Sort.
Choose the collumn you want to sorb by and then choose Sort Z to A
(largest to smallest).)
• One would find that South Africa, India, and Nigeria have the highest
number of adults with HIV.
• However, one cannot necessarily conclude that these countries have the
worst problem with HIV since there are vastly different populations.
• Another way to look at the data would be with relative quantities.
• One could calculate the percent of the adult population infected with HIV
by dividing the number of adults with HIV by the total adult population of
the country. If that calculation was done and that column was sorted,
Botswana, Lesotho, and Zimbabwe have the highest percents.
• This is a very different picture. Which is more informative? It depends on
what question you are attempting to answer. How and why did certain
countries “move” when sorting by absolute quantity (number of infected
adults) vs. relative quantity (percent of adults with HIV)?
Absolute and Relative Quantities
• India which had the second highest number of infected adults also has the
second highest adult population in the world so in some ways it is not
surprising that it has a high number of adults with HIV.
• Once the population is taken account by calculating the percent, India drops
down the list since only 1% of its population is infected. The high absolute
quantity yielded a low relative quantity because the number you divided by
(the total adult population) was so large.
• South Africa had the highest number of infected adults and also had a fairly
high percentage because its total population is not that large.
• Usually when dealing with state or country information, a relative quantity
is more informative because it allows you to compare states or countries to
each other. By calculating the relative quantity, you are taking the differing
populations of the country into account.
Different types of Relative Quantities
Fraction or Percent:
• Fractions or percents are used when comparing part to total of the same type of
variable. (example: percent of adults with AIDS/HIV) Percents can also be used
to show the relative change. Percent change is calculated by dividing the
absolute change by the original amount.
[Reminder: Percent change (new value –old value)/old value]
Rate:
• Rates are used compare different types of variables (example: tickets per
person, miles per hour, or crimes per 1000 people)
Ratio:
• Ratios are used to compare the same type of variable from two sources. For
example: California’s population is 33,872,000 and Oregon’s population is
3,421,000. Clearly CA’s population is larger but how many times larger?
33,872,000/3,421,000 = 9.90 Calculating the ratio of the populations tells us
that CA’s population is almost 10 times as large as OR’s population.
• The type of data you have will determine what type of relative quantity is
appropriate.
Absolute and Relative Quantities
• Another example: the excel file StateLotteries2000.xls contains data
on lottery ticket sales.
• One could organize the data by sorting ticket sales in descending
order.
•
One would find that Massachusetts, New York, Texas, and California
had the highest ticket sales and Montana, Nebraska, Vermont and
Idaho had the lowest.
• In this sense, ticket sales are absolute quantities, measuring the
absolute occurrence of the variable.
• Absolute quantities are “sheer” numbers.
Absolute and Relative Quantities
•
In the same example StateLotteries2000.xls , one could calculate ticket sales per person by
dividing absolute sales by the population of the state.
•
After making the calculation and sorting the data, one would find that Rhode Island, South
Dakota, Delaware, and Massachusetts had the highest ticket sales per person and Montana,
Nebraska, Arizona, and New Mexico had the lowest.
•
While both sorting by absolute and relative quantities gives us valid and accurate
information about ticket sales in each state, each gives a different view of the situation.
Which is more informative? It depends on what question you are attempting to answer.
•
Usually when dealing with state or country information, a relative quantity is more
informative because it allows you to compare states or countries.
•
If we only look at absolute sales, it is not surprising that states with larger populations (ie,
CA, NY, TX, FL) have larger sales and that states with smaller populations (ie VT, SD, MT, RI)
have lower ticket sales. By calculating the relative quantity of ticket sales per person, you
are taking the population of the state into account.
•
In this sense, ticket sales per person is a measure of the absolute occurrence of the variable
(ticket sales) in relation to some other quantifiable variable (state population).
Absolute and Relative Change
• We use absolute change to describe the actual
increase or decrease from a reference (or
old/earlier) value to a new (or later) value:
• Absolute Change = new value – reference value
• We use relative change to compare the absolute
change to the reference value:
• Relative Change =
=
• For communication purposes, we convert relative change,
which is a fraction (converted to a decimal number) to a
percentage (percentage change). The following are three
ways to convert a fraction (decimal number) to a
percentage:
• Move the decimal place to the right two places
• Multiply by 100%
• Use the button
in Excel
• For this course, we will generally show percentages
formatted to two (2) decimal places. (Right click on the cell,
format cell)
Examples:
• During a 6-month period Nokia’s stock doubled in price from $10 to $20.
– Calculate the absolute, relative and percentage changes in the stock price.
– When a quantity doubles, what is its relative change? percentage change?
– Will you get the same answer if the stock started at $5 and doubled to $10? If a service
fee doubled from $25 to $50?
• The number of DVD players in homes in the United States increased dramatically
from 1999 to 2001, from 5.4 million to 25.1 million. By how many percent did it
rise from 1999 to 2001?
• In 2001, there were 654 drive in theaters in the US, 77% fewer than in 1985.
Approximately how many drive in theaters were there in 1985?
• Enrollment at DePaul University has grown by 36% from 1990 to 2001. If the Fall
2001 enrollment was 21,363 students then what was the total enrollment in 1990?
• The January 14, 2002 issue of Sports Illustrated reported that the average home
attendance for the Washington Wizards in 2000-01 season was 16,075. Since
Michael Jordan’s return as a NBA player, the average attendance at Wizards home
games has grown to 20,874 for the 2001-02 season. What percent increase does
this represent?
NOW…
• When we were calculating the slope in linear
models, does the rate of change (slope) involve
absolute change or relative change?
– Rate of change(slope): (newY-oldY/newX-oldX)
: Absolute change in Y/ Abs. change in X
• When we were calculating the r (percent change) in
exponential models.. is that an absolute change or
relative change?
– Percent change: (newY-oldY)/oldY
: Relative change in Y
Percentage of…
• Understanding “Percentage of” in 3 ways:
I. The Formula:
part
%
whole
(where the % is written as a decimal)
II. Visually: Whole means the entire pie. Part means one of the shaded regions or
pieces of the pie.
III. There are three ways to think about this relationship:
part
%
whole
part
 whole
%
% * whole  part
IV. Deriving the formulas: Can you figure out (algebraically) why all
three of these are just different versions of the same relationship?
a.
Starting with the formula:
Derive:
b.
Starting with the formula:
Derive:
V. Solving Problems
• There are two approaches to solving the following problems. The first
approach is to identify the two given numbers. Then decide which version of
the part/whole relationship will help you answer the question.
– If you are given part and whole, then use the first version.
– If you are given part and % then use the second version.
– And, finally, if you are given whole and % then use the third version.
• The second approach is to remember the first formula, fill in the information
you are given and then solve for the missing variable.
• For all problems, remember to use the decimal version of the %.
VI. Number Drills
• 2 is what percentage of 10?
2/10= 1/5= 0.2 = 20%
• 20% of what number is 2?
20%= 0.2
2/0.2= 10
• What is 20% of 10?
20% = 0.2
0.2*10= 2
• 17 is 32% of what number?
17/0.32 = 53.125
• 67.2 is what percentage of 150?
67.2/150= 0.448= 44.8%
• What is 233% of 71?
233%=2.33
2.33*71= 165.43
• What is .7% of 50?
0.7% = 0.007
0.007*50= 0.35
• 35 is 9% of what number?
35/0.09= 388.8889
• 10,003 is what percentage of
1,762,325?
10003/1762325= 0.005676 =
0.5676%
• one million three hundred
thousand is what percentage of
one billion?
one million three hundred
thousand =1.3 million
one billion= 1000 million
1.3/1000= 0.0013 = 0.13%
• one thousand is what percentage
of two thousand three hundred
and six?
• 1000/2306= 0.433651= 43.37%
VII. Applications:
• In Chicago in the year 2000, there were
approximately 1.053 million African Americans, 907
thousand whites (non-Hispanic), and 754 thousand
Hispanics, and 181 thousand others (other races or
two or more races). What percentage of
Chicagoans in 2000 were of Hispanic origin?
Total: 1.053 + 0.907 +0.754 + 0.181= 2.895 million
0.754/2.895 = 0.260449 = 26%
• DePaul’s undergraduate student body is
approximately 21,000 students. 54% of the student
body is female. Approximate how many females
attend DePaul?
54% = 0.54
0.54* 21,000= 11,340
• In 1993, 248.7 million people in the United States
were born in the United States, and the rest, 19.8
million were foreign born. What percentage of the
total population of the US was foreign born?
Total population = 248.7 + 19.8 = 268.5
19.8/268.5 = 0.073743 = 7.37%
• The sales tax is 8.75% in most counties of Illinois. If
you purchase a new car for $15,000, what is the
sales tax you will pay?
8.75%= 0.0875
0.0875*15000= $1312.5
• You are in another state (not Illinois). You are
buying a computer at Best Buy. The price before
taxes is $949. When the cashier wrings up your
purchase you owe $1005.94. What is the sales tax
in this state? (You might be in Connecticut or
Pennsylvania)
1005.94-949=56.94
56.94/949=0.06= 6%
• At one point, the Tribune article refers to a subtotal
of murders “with only 10% of the year yet to go.”
10% of the year is how many months?
10% = 0.1
0.1*12= 1.2 months