Friday, September 17, 2021

Algebra 1 PARCC: home price stats


The following two-part multiple-choice question, explained here in hopes of helping algebra 1 students and their parents in Maryland and Illinois prepare for the PARCC test near the end of this school year, appears on the released version of PARCC’s Spring 2016 test for algebra 1 (#25):

A real estate agent recorded the home prices, in thousands of dollars, for 50 randomly selected homes in two communities, A and B. The dot plots display the recorded data.

Part A

Which statement best describes the relationship between the home prices in community A and community B?

A. The homes in community A are typically more expensive and more consistent in price than those in community B.

B. The homes in community A are typically more expensive and less consistent in price than those in community B.

C. The homes in community A are typically less expensive and more consistent in price than those in community B.

D. The homes in community A are typically less expensive and less consistent in price than those in community B.

Part B

Which of the listed home prices most likely represents the third quartile for the 50 home prices in community B?

A. $160,000
B. $165,000
C. $167,000
D. $170,000

Answer and references

For part A, the correct answer is D, and for part B, the correct answer is C.

Common Core Math, High School Statistics & Probability, Interpreting Categorical & Quantitative Data

The high school statistics standards in the Common Core, especially, for this problem, HSS.ID.A.1, “Represent data with plots on the real number line (dot plots, histograms, and box plots),” HSS.ID.A.2, “Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets,” and HSS.ID.A.3, “Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers),” are well tested by these two questions.

Students are required to evaluate the shape and center of the distribution of home prices in two communities and compare those features of the data. In addition, part B requires that students understand how to find the interquartile range in asking them what the third quartile of one of the distributions might most likely be.

Solution strategy (there are others)

For part A, examine and describe the median and range or interquartile range of the data.

The dot plot for community A is very spread out, with home prices ranging from a low of a little more than $90,000 to a high of about $183,000. The range for the 50 randomly selected homes in community B is much tighter, ranging from a little more than $155,000 to about $175,000. From this we can conclude that the “consistency” of price is more in community B, because homes sell for about $165,000 and don’t seem to stray very far from that.

In community A, on the other hand, it’s difficult to predict the mean home price from the graph, but since the data is fairly evenly distributed across the range—at least compared to home prices in community B, which seems to reach a peak at about $160,000 and tail off to the right but not very far—we can see that most of the homes in community A sold for between $140,000 and $160,000.

That would put the mean of this distribution, most likely, in the $150,000 range, excluding the low and high outliers. That’s less than the mean in community B, which is probably in the neighborhood of $160,000, or possibly a little higher, depending on how big the tail is to the right of the peak.

For part B, know what the third quartile means and count the dots.

Part B asks a very straightforward question about determining the third quartile of the data in community B. We are told there are 50 dots on the graph, representing the sale price of 50 homes in community B. One-fourth of that is about 13 homes, so the third quartile will be the home price represented by the 13th home price from the top of the distribution.

Counting dots, we find only eight at $170,000 or above, eliminating D. We find one at $169,000, two at $168,000, and two at $167,000. One of the dots at $167,000 is the 13th dot from the top, meaning the third quartile of data occurs at $167,000.

Analysis of this question and online accessibility

The question is valid in that it tests students’ ability to evaluate the central tendency and spread of data plotted on a dot plot.

The question can be delivered online or on paper. However, as shown, the question is better tested on paper than online, since reading the same long sentence for each option choice in part A, with just two little words changed, may lead to errors in responding. With an online test delivery platform, which is how most students take the PARCC test, the question in part A is better delivered using drop-down menus for the words less and more. As a result, validity and reliability measures for the item may differ under different test-taking environments.

No special accommodation challenges can be identified with this question, so the question is considered fair.

Resources for further study

Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has developed a nice page dealing with the interquartile range. Her explanation is a little different from mine, because she uses box-and-whiskers plots. Her lesson begins here.

In addition, Sal Khan, an engineer who developed the Khan Academy, a set of thousands of tutorial videos about math and a few other subjects, has created a nice written essay about choosing the most appropriate measure of central tendency, between mean and median, given characteristics of the data, beginning here.

Ron Larson, Laurie Boswell, Timothy D Kanold, Lee Stiff. Algebra 2, Illinois edition. Evanston, Ill.: McDougal Littell, a division of Houghton Mifflin Company, 2008. The book is used in several algebra classes taught in Illinois high schools.

Foerster, Paul A. Algebra and Trigonometry: Functions and Applications, revised edition. Addison-Wesley, 1980, 1984. The book is used in several algebra classes taught in middle and high schools in both Illinois and Maryland.

Purpose of this series

To help algebra 1 students and their parents prepare for the PARCC test in math, as administered in at least six states, or to just master content on that test, we provide an analysis of every algebra 1 math problem PARCC released in 2016. The series can be found here.

Paul Katula
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

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