LSP 120: Quantitative Reasoning and Technological Literacy Section 903

LSP 120: Quantitative Reasoning and Technological Literacy Section 903 ÖzlemElgün

Making and Interpreting Graphs There are 3 types of graphs we use in Quantitative Reasoning class: • Pie Charts • Bar charts • 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 HIV_Adults_By_Country_2001.xlscontains 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.xlscontains 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:

IV. Deriving the formulas: Can you figure out (algebraically) why all three of these are just different versions of the same relationship? • Starting with the formula: Derive: • 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

LSP 120: Quantitative Reasoning and Technological Literacy Section 903