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A K–8 resource to support deep comprehension of math skills and concepts
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#### June 9, 2016 6-8-statistics-and-probability

Making two-way tables for bivariate categorical data is a standard in the Grade 8 Common Core State Standards math curriculum. I have had some great experiences with my Grade 8 social studies teacher in developing cross-curricular projects for bivariate categorical data. This post highlights some of the two-way frequency table data sets we have been developing for a global studies project this spring.

Bivariate data — the analysis of data for two characteristics per observation — is the theme for the Statistics and Probability strand for Grade 8. In earlier grades, only univariate data (one characteristic per subject) was used in data analysis. The Grade 8 study of linear relationships and their graphs permits students to think about the association between two variables. Standards 8.SP.A.1 through 8.SP.A.3 of the Grade 8 data standards have to do with these associations, but standard 8.SP.A.4 is specifically about bivariate categorical data and two-way tables. With this standard students need to construct and interpret two-way tables.

Using the two-way frequency tables data sets developed with my social studies colleague, students construct the data tables using the raw bivariate categorical data that we provided. We give students a choice of data on various topics from the World Bank*, some of which are available as a download. This post uses an example with data about birth rates from 2011.

The table below shows the two bivariate categorical data variables—income (high, upper-middle, lower-middle, or low) of the country and whether the country has a high or low birth rate (birth rate lower than 20 per 1000 women or greater than 20 per 1000 women). After providing a table of data for 204 countries, students construct the table using the raw data. This can be done by hand or with the downloadable Excel spreadsheet provided with this post. The tallied numbers are called the joint frequencies.

 Birth Rate < 20 per 1000 Birth Rate > 20 per 1000 High Income Countries 66 6 Upper-Middle Income Countries 28 22 Lower-Middle Income Countries 11 40 Low Income Countries 1 30

Having constructed the table, there are three ways students need to be able to analyze the two-way tables.

1) Students should find the row totals (marginal frequencies) and calculate the percentage by row in order to answer questions such as:

• What percentage of high-income countries have a birthrate greater that 20 per 1000?

• What percentage of low-income countries have a birthrate lower than 20 per 1000?

Each row adds up to 100%.

 Birth Rate < 20 per 1000 Birth Rate > 20 per 1000 Row Totals High Income Countries 66/72 = 92% 6/72 = 8% 72 Upper-Middle Income Countries 28/50 = 56% 22/50 = 54% 50 Lower-Middle Income Countries 11/51 = 22% 40/51 = 78% 51 Low Income Countries 1/31 = 3% 30/31 = 97% 31

2) Students should find the column totals (marginal frequencies) and calculate the percentages by column in order to answer question such as:

• What percentage of countries with birth rates less than 20 per 1000 are upper middle income countries?

• What percentage of countries with birth rates greater than 20 per 1000 are lower middle-income countries.

Each column adds up to 100%.

 Birth Rate < 20 per 1000 Birth Rate > 20 per 1000 High Income Countries 66/106 = 62% 6/98 = 6% Upper-Middle Income Countries 28/106 = 26% 22/98 = 22% Lower-Middle Income Countries 11/106 = 10% 40/98 = 41% Low Income Countries 1/106 = 1% 30/98 = 31% Column Totals 106 98

3) Students should calculate the table total and calculate percentages for the table using this total in order to answer questions such as:

• What percentage of the countries are lower middle income with a birth rate greater than 20 per 1000?

All of the joint frequency percentages should add up to 100%.

 Birth Rate < 20 per 1000 Birth Rate > 20 per 1000 Row Totals High Income Countries 66/204 = 32% 6/204 = 3% 72 Upper-Middle Income Countries 28/204 = 14% 22/204 = 11% 50 Lower-Middle Income Countries 11/204 = 5% 40/204 = 20% 51 Low Income Countries 1/204 = 0.5% 30/204 = 15% 31 Column Totals 106 98 204

Notice the extra bold outline around the box for the table total — this is one area where Grade 8 students can make mistakes. The sum of the row totals should equal the sum of the column totals, which is also a great way for students to check their work! The table total is the matching total (204) and not the sum of all of the rows and columns (which would be 408).

The final step of the process is to interpret the two-way tables. By looking at any of the three tables generated above, students should be able to conclude that higher income countries have lower birth rates than lower income countries. This interpretation meets the higher-order thinking goal of analyzing the two-way frequency table data sets.

One way that my social studies colleague and I can help students meet math, ELA, and world history standards is by writing a paragraph about the two-way table, justifying their conclusions by citing evidence from the table. Students meet Mathematical Practice 3 by using results of the analysis of the two-way tables in constructing arguments and communicating them to others.

Students also meet Grade 8 History/Social Studies writing standards by introducing a claim about a topic; organizing the reasons and evidence logically; supporting claims with logical reasoning and relevant, accurate data; and providing a concluding statement or section that follows from and supports the argument presented.

I am lucky that I can collaborate with my social studies colleague so that students can work on a single project that will receive a math grade from me as well as a social studies writing grade. If you would like to have students work on such a project, click on the button below for a download that includes four two-way frequency table data sets that students can use to construct, analyze and interpret bivariate categorical data. The economic data sets from 2011 are provided by the World Bank and include information about worldwide Internet use, mobile telephone subscriptions, the cost of starting up a business, and ATM availability.

*Data set from The World Bank: World Development Indicators Database: Birth rate, crude (per 1,000 people)

Original Sources: (1) United Nations Population Division. World Population Prospects, (2) United Nations Statistical Division. Population and Vital Statistics Report (various years), (3) Census reports and other statistical publications from national statistical offices, (4) Eurostat: Demographic Statistics, (5) Secretariat of the Pacific Community: Statistics and Demography Programme, and (6) U.S. Census Bureau: International Database.

Used in accordance with the license posted at: http://web.worldbank.org/WBSITE/EXTERNAL/0,,contentMDK:22547097~pagePK:50016803~piPK:50016805~theSitePK:13,00.html