Statistical Studies: Statistical Investigations III.A Student Activity Sheet 5: Histograms

1. What does the histogram in the opening of the lesson represent 1. What does the histogram in the opening of the lesson represent? Describe the distribution as completely and accurately as possible with regard to center, shape, and spread. This histogram represents the average SAT math score at various colleges. For example, it appears that approximately 27 schools have an average SAT math score of 450. (Answers may vary.) A few schools have a very high or very low average SAT math score (even a few schools with very high scores could be considered outliers). Most schools have an average SAT math score that is between about 420 and 600. The average score for all the schools seems to be about 500. The distribution appears very roughly symmetric or possibly just a little bit skewed right.

2. Refer to the research cycle 2. Refer to the research cycle. Where in the cycle is examining histograms located? The Collect and Analyze portions of the cycle is where data will be gathered, organized, and presented in some way so that information about the data set is conveyed graphically, analytically, or both.

3. Without discussing with your classmates, answer the following questions—write one answer on each of your 12 slips of paper. Place each slip in the appropriate paper bag. a. What year were you born? b. What is your gender? c. How many text messages did you send yesterday? d. What does your cell plan charge for texting? e. How many people under the age of 18 live in your house? f. Which is your favorite food from the following choices: pizza, hamburgers, sushi, salad, chicken, other? g. Do you have a job that pays by the hour? h. If yes, how many hours do you work in an average week? (If no, put 0.) i. List how many hours you work a week (again) and then the average number of hours that you study a week. j. List your gender (again) and your shoe size. k. List your gender (again) and number of text messages (again). l. What is your favorite kind of music?

4. While your classmates finish Question 3, jot down your thoughts about the following questions: • What do you think categorical data means? Categorical is related to the word categories , so categorical data is information about how study subjects fit into different categories (for example, Republican, Democrat, or Independent; male or female; blonde, brunette, bald, and so on). •What do you think quantitative data means? Quantitative is related to the word quantity, so quantitative data is information that is recorded numerically (for example, height, age, income). • What do you think univariate means? Univariate means one variable , so information about one variable is recorded for each participant (for example, height). • What do you think bivariate means? Bivariate means two variables , so information about two variables is recorded for each participant (for example, height and weight).

5. Does the histogram illustrate univariate data or bivariate data 5. Does the histogram illustrate univariate data or bivariate data? Categorical or quantitative? The histogram represents univariate data—one piece of information (each person’s semester average) is recorded for each participant. Since the semester average is represented numerically, the data are quantitative. 6. Would you describe the histogram as symmetric? Why or why not? this graph is, in fact, considered fairly symmetric, as the two sides roughly reflect each other. 7. Describe the distribution of grades for the class. Justify any estimates you make. Some students may eyeball the average at about 82 or 83. Others may compute an approximate class average as: [(2 • 65) + (7 • 75) + (9 • 85) + (3 • 95)]/21 = 81.2

8. Consider the new histogram showing bus ridership at a local high school. This histogram is not symmetric; it is skewed to the right. You know this because the distribution has a tail out toward the right side. What is happening with this population that causes the distribution to be skewed to the right? As students get older, they get their driver’s licenses and drive cars to school, or they ride with friends. In addition, there may not be as many 18-year-old students at the school if many 18-year-olds have graduated. 9. Estimate the average age of students who ride the bus. This average is the center of the distribution. Justify your estimate. Some students may eyeball the average at about 15. Others may compute from estimates: [(14 • 880) + (15 • 700) + (16 • 330) + (17 • 180) + (18 • 90)]/2,180 = 15.04 years

10. Suppose the histogram of bus ridership looked like this instead 10. Suppose the histogram of bus ridership looked like this instead. Data values that are distant from most of the other values are generally thought of as outliers. What may have happened? Does this data distribution affect your answers to Questions 8 and 9? Either some very bright 10-year-olds are going to the high school or there was a datarecording error. You need to check the data to see if correction is needed. If there are really 10-year-olds at the school, you need to refigure our average age and make a note of the outlier.