ACTIVITY 5.1 Constructing and Interpreting Two-Way Frequency Tables Ms. Carter is an athletic coordinator at Liberty Middle School. She is developing an after-school sports program. Ms. Carter has a budget to follow and needs to determine which sports she will include in the program. She surveys students in her eighth-grade class in order to determine which sports the students prefer. The results are shown in the table. Name Gender Favorite Sport Name Gender Favorite Sport Sue Female Soccer Jorge Male Basketball Alex Male Baseball Maria Female Volleyball Tamika Female Volleyball Sarah Female Basketball Beth Female Soccer Sam Male Soccer Eric Male Volleyball Marcus Male Basketball Carla Female Baseball Ben Male Soccer Will Male Basketball Yasmin Female Basketball Paulos Male Volleyball Jon Male Soccer Rose Female Volleyball Donna Female Volleyball Suzi Female Soccer Kayla Female Basketball Ashley Female Soccer Devon Male Basketball Carson Male Baseball Dawn Female Volleyball Eryn Female Soccer Harley Male Soccer Abigail Female Basketball Jordan Male Basketball Nicole Female Volleyball Bert Male Baseball 1. Identify the variables in the table. M2-332 TOPIC 4: Patterns in Bivariate Data
Previously, you explored relationships between two variables whose data were quantitative or numerical. Not all data are numerical. Data that can be grouped into categories are called categorical (or qualitative) data. One method of organizing categorical data is in a two-way table. A two-way table displays categorical data that shows the number of data points that fall into each group for two variables. One variable is divided into rows, and the other is divided into columns. WORKED EXAMPLE The two-way table displays the favorite sports of students in Ms. Carter s eighth-grade class. For example, you can use the twoway table to record the females who preferred soccer. There are two groups for the variable gender. The two groups are male and female. There are four groups for the variable sport. The four groups are baseball, basketball, soccer, and volleyball. Favorite Sports of Students in Ms. Carter s Eighth-Grade Class Baseball Basketball Soccer Volleyball Gender Male Female To record information in the two-way table, you can use tally marks to ensure each variable of a data point is recorded. Therefore, since five females prefer soccer, you would use 5 tally marks in the category for female and soccer. Analyzing the data is an important step in the statistical process. Analyzing includes organizing the data in a way to make interpreting the data most effective. 2. Complete the two-way table in the worked example for all students in Ms. Carter s class. LESSON 5: Would You Rather...? M2-333
Once you have recorded all of the data with tally marks, you can count the tally marks in each cell of the two-way table, and then you can write the frequency for each variable. The frequency of a variable is the number of times it appears in a data set. 3. Record the frequency in the two-way table. Then, calculate the totals. Favorite Sports of Students in Ms. Carter s Eighth-Grade Class Sport Baseball Basketball Soccer Volleyball Total Gender Male Female Total This two-way table has three rows and five columns now. Why is it important to know the totals for the data set? 4. Add the total number of males and females. Then, add the total number of students who preferred baseball, basketball, soccer, and volleyball. What does this tell you? 5. Which sport is the least favorite and most favorite a. of the students? b. of the male students? c. of the female students? M2-334 TOPIC 4: Patterns in Bivariate Data
6. According to the survey, which sports should not be included in Ms. Carter s after-school program? Explain your reasoning. 7. What other conclusions can you draw about the favorite sports of students in Ms. Carter s eighth-grade class? 8. Construct a two-way frequency table to compare the fears of heights reported by students in your class who chose the ability to fly versus those students who chose the ability to read minds. What conclusions can you draw? LESSON 5: Would You Rather...? M2-335
ACTIVITY 5.2 Constructing and Interpreting Two-Way Relative Frequency Tables A recent study has estimated that between 70% and 90% of the world s population is right-handed. Another study suggests that almost 90% of athletes are right-handed. And yet another study shows that left-handed people have a higher percentage of participants in individual sports, such as wrestling or golf. Mr. Harris s math class thinks that these figures may be incorrect. They decide to conduct a random survey to determine which hand is favored, and whether the favored hand affects if a person participates in certain types of sports, or no sports at all. The results are shown in the two-way table. Sports Participation Individual Team Does Not Play Total Left 3 13 8 Hand Favored Right 6 23 4 Mixed 1 3 2 Total 1. Name the two variables displayed in the table. M2-336 TOPIC 4: Patterns in Bivariate Data
2. Which hand was favored most in the survey? 3. Which hand was favored least in the survey? 4. Calculate the total for each row and each column in the table. a. How many total people participated in the survey? b. Out of all the people surveyed, how many were left-handed? right-handed? mixed-handed? c. Out of all of the people surveyed, how many participated in individual sports? participated in team sports? did not participate in any sports? LESSON 5: Would You Rather...? M2-337
You cannot verify that the studies figures are correct by simply looking at the frequencies. Instead you must determine the relative frequencies. A relative frequency is the ratio or percent of occurrences within a category to the total of the category. To determine the ratio of each category, determine the part to the whole of each category. To determine the percent of each category, set up a fraction with the denominator being the total number of each row. 5. Complete the relative frequencies for each favored hand category. Round decimals to the nearest thousandth. What does the total for a row mean? Hand Favored Sports Participation Individual Team Does Not Play Total Left 3 24 5 0.125 Right 23 33 5 0.70 Mixed 6. Interpret each of the relative frequencies for each category. a. left-handed people b. right-handed people c. mixed-handed people M2-338 TOPIC 4: Patterns in Bivariate Data
7. Determine the percent of people who participated in individual and team sports. a. left-handed people b. right-handed people If you are determining the percent of people who participated in two sports categories, would you add or multiply the percents? c. mixed-handed people 8. Which group of people had the greatest number participate in sports? LESSON 5: Would You Rather...? M2-339
9. Complete the relative frequencies for each sports participation. Round decimals to the nearest thousandth. Left 3 10 Sports Participation Individual Team Does Not Play 5 0.3 13 39 5 0.33 Hand Favored Right 4 14 5 0.29 Mixed Total 10 10 5 1 10. Interpret each of the relative frequencies for the type of sports participation. 11. In completing the relative frequency tables, did the studies figures seem accurate? M2-340 TOPIC 4: Patterns in Bivariate Data
TALK the TALK NOTES Going Back in Time 1. What is the difference between numerical data and categorical data? 2. What is a relative frequency of categorical data? LESSON 5: Would You Rather...? M2-341