Two-Way Frequency Tables Summarize categorical data

Focus 6 Learning Goal – (HS. S-ID. A. 1, HS. S-ID. A. 2, HS. S-ID. A Focus 6 Learning Goal – (HS.S-ID.A.1, HS.S-ID.A.2, HS.S-ID.A.3, HS.S-ID.B.5) = Students will summarize, represent and interpret data on a single count or measurement variable. 4 3 2 1 In addition to level 3.0 and above and beyond what was taught in class,  the student may: · Make connection with other concepts in math · Make connection with other content areas. The student will summarize, represent, and interpret data on a single count or measurement variable. - Comparing data includes analyzing center of data (mean/median), interquartile range, shape distribution of a graph, standard deviation and the effect of outliers on the data set. - Read, interpret and write summaries of two-way frequency tables which includes calculating joint, marginal and relative frequencies. The student will be able to: - Make dot plots, histograms, box plots and two-way frequency tables. - Calculate standard deviation. - Identify normal distribution of data (bell curve) and convey what it means.   With help from the teacher, the student has partial success with summarizing and interpreting data displayed in a dot plot, histogram, box plot or frequency table. Even with help, the student has no success understanding statistical data.

Numerical vs. Categorical Data Numerical data includes data that are numbers: height, weight, age, price, temperature, test scores, etc. Categorical data that isn’t numeric would be like: eye color, hair color, favorite ice cream, favorite superhero, city of birth, etc. Two-Way Table a table listing two categorical variables whose values have been paired.

Data was collected from men and women about their favorite DC comic book superheroes: Wonder Woman, Batman, or Superman. 100 men and 100 women were surveyed. The data is displayed in a two-way frequency table. Talk with the person next to you to determine how to read the table. What do the labels mean?

Data was collected from men and women about their favorite DC comic book superheroes: Wonder Woman, Batman, or Superman. We learn some basic information from the table when it is displayed in this format. 87 people prefer Wonder Woman. Of the 87 people who prefer Wonder Woman, 62 of them were women.

Types of Frequencies: Joint Frequency: The data in the middle cells. (The yellow section.) These are the ratios of the frequency in a particular category and the grand total number of data values. It is called joint frequency because you are joining one variable from the row and one variable from the column. Marginal Frequency: The data on the far right and the bottom. (The blue section.) These are the ratios of the sum of the joint relative frequency in a row or column and the grand total number of data values. Conditional relative frequency: The ratio of a joint relative frequency and related marginal relative frequency. This is what you did in Linear Algebra.

Conditional Frequency: (A refresher from last year .) To calculate Conditional Frequency, you divide the cell total by the row total. Calculate the conditional frequency of males who like Wonder Woman. 25/100 = 25%. Calculate the conditional frequency of females who like Wonder Woman. 62/100 = 62%. Calculate the conditional frequency of males who like Batman. 40/100 = 40% Fill in the rest of the middle section of the table. 25% 40% 35% 62% 23% 15%

Conditional Frequency: (A refresher from last year .) To fill in the row total for Male, divide Male Total by the grand Total. (100/200) To fill in the row total for Female, divide Female Total by the grand Total. (100/200) To fill in the column total for WW, divide WW Total by the grand Total. (87/200) To fill in the column total for BW, divide BM Total by the grand Total. (63/200) To fill in the column total for SM, divide SM Total by the grand Total. (50/200) To fill in the column total for TOTAL, add up the percents in that column or in that row. 25% 40% 35% 50% 62% 23% 15% 50% 43.5% 31.5% 25% 100%

Conditional Frequency: These are the types of questions that can be answered from conditional frequency tables. What percent of males prefer Batman? What percent of females prefer Superman? What percent of the males surveyed prefer Wonder Woman? A question you cannot answer from this table is, “What percent of the total people surveyed were women who liked Wonder Woman?” The answer of 62% is not correct. That is the percent of JUST the women who like Wonder Woman.

Joint “Relative” Frequency: Relative or Joint Relative Frequency will allow you to answer the question, “What percent of the people surveyed were women who like Wonder Woman?” Calculate Relative Frequency by dividing the cell total by the grand total. Divide male votes for Wonder Woman by the grand total. 25/200 = 0.125 or 12.5% Divide female votes for Wonder Woman by the grand total. 62/200 = 0.310 or 31% Finish filling in the table. 12.5% 20% 17.5% 31% 11.5% 7.5%

Calculate Relative Frequencies: 12.5% 20% 17.5% 50% 31% 11.5% 7.5% 50% 43.5% 31.5% 25% 100% To fill in the row total, add up all of the percents in that row. To fill in the column total, add up all of the percents in that column. Now that the table has been changed to percents, you can make predictions. What is the probability that a person picked at random from this group will be a female that likes Wonder Woman? What is the probability that a person picked at random from this group will be a male who likes Superman? 31% 17.5%

Marginal Frequencies These are the ratios of the sum of the joint relative frequency in a row or column and the grand total number of data values. These frequencies were calculated when you made the Relative Frequency table. The marginal frequencies answer these types of questions: What percent of the people surveyed were female? What percent of the people surveyed liked Superman? What percent of the people surveyed like Batman? 12.5% 20% 17.5% 50% 31% 11.5% 7.5% 50% 43.5% 31.5% 25% 100%