LSP 120: Quantitative Reasoning and Technological Literacy

Topic 4: Absolute and Relative Quantities

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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 (i.e. number of adults who have HIV)

• The relative quantity is the absolute quantity divided by some other quantity (i.e. percentage of adults who have HIV)

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• 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

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• 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.

Absolute and Relative Quantities

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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.

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• 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

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• 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 Quantities

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• 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 = =

Absolute and Relative Change

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Absolute and 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)

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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?

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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)

• When we were calculating the r (percent change) in exponential models, is that an absolute change or relative change?– Percent change: (newY-oldY)/oldY

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