In grades 5-8, the mathematics curriculum should include exploration of statistics in real-world situations so that students can--

• systematically collect, organize, and describe data;
• construct, read, and interpret tables, charts, and graphs;
• make inferences and convincing arguments that are based on data analysis;
• evaluate arguments that are based on data analysis;
• develop an appreciation for statistical methods as powerful means for decision making.

Focus

In this age of information and technology, an ever-increasing need exists to understand how information is processed and translated into usable knowledge. Because of society's expanding use of data for prediction and decision making, it is important that students develop an understanding of the concepts and processes used in analyzing data. A knowledge of statistics is necessary if students are to become intelligent consumers who can make critical and informed decisions.

In grades K-4, students begin to explore basic ideas of statistics by gathering data appropriate to their grade level, organizing them in charts or graphs, and reading information from displays of data. These concepts should be expanded in the middle grades. Students in grades 5-8 have a keen interest in trends in music, movies, fashion, and sports. An investigation of how such trends are developed and communicated is an excellent motivator for the study of statistics. Students need to be actively involved in each of the steps that comprise statistics, from gathering information to communicating results.

Identifying the range or average of a set of data, constructing simple graphs, and reading data points as answers to specific questions are important activities, but they reflect only a very narrow aspect of statistics. Instead, instruction in statistics should focus on the active involvement of students in the entire process: formulating key questions; collecting and organizing data; representing the data using graphs, tables, frequency distributions, and summary statistics; analyzing the data; making conjectures; and communicating information in a convincing way. Students' understanding of statistics is also enhanced by evaluating others' arguments. This exercise is of particular importance to all students, since advertising, forecasting, and public policy are frequently based on data analysis.

Discussion

Middle school students' curiosity about themselves, their peers, and their surroundings can motivate them to study statistics. The data to be gathered, organized, and studied should be interesting and relevant; students' interest in themselves and their peers, for example, can motivate them to investigate the "average" student in the class or school. First, students can formulate questions to determine the characteristics of an "average" student--age, height, eye color, favorite music or TV show, number of people in family, pets at home, and so on. Although numerous categories are possible, some discussion will help students to develop a survey instrument to obtain appropriate data. Sampling procedures are a critical issue in data collection. Which students should be surveyed to determine Mr. or Ms. Average? Must every student be questioned? If not, how can randomness in the sampling be assured and how many samples are needed to accumulate enough data to describe the average student?

Random samples, bias in sampling procedures, and limited samples all are important considerations. For instance, would collecting data from the men's and women's basketball teams provide needed information to determine the average height of a college student? Will a larger sample reveal a more accurate picture of the percentage of students with brown hair? The graph in figure 10.1 illustrates the results of increasing the sample size.

Fig. 10.1. Graph of brown-haired students

Data can be presented in many forms: charts, tables (fig. 10.2), plots (e. g., stem-and-leaf, box-and-whiskers, and scatter), and graphs [e.g., bar, circle, or line). Each form has a different impact on the picture of the information being presented, and each conveys a different perspective. The choice of form depends on the questions that are to be answered. Using the same data, students can develop graphs with different scales to show how the change of scale can dramatically alter the visual message that is communicated.

Fig. 10.2. Data table

Computer software can greatly enhance the organization and representation of data. Data-base programs offer a means for students to structure, record, and investigate information; to sort it quickly by various categories; and to organize it in a variety of ways. Other programs can be used to construct plots and graphs to display data. Scale changes can be made to compare different views of the same information. These technological tools free students to spend more time exploring the essence of statistics: analyzing data from many viewpoints, drawing inferences, and constructing and evaluating arguments.

A particular point that should be raised with students is how average relates to numerical and nonnumerical data. Although there are several measures of central tendency, students are generally exposed only to the mean or median, yet the mode might be the best "average" for a set of nonnumerical data.

Students should also explore the concepts of center and dispersion of data. The following activity includes all the important elements of this standard and illustrates the use of box-and-whisker plots as an effective means of describing data and showing variation.

A class is divided into two large groups and then subdivided into pairs. One student in each pair estimates when one minute has passed, and the other watches the clock and records the actual time. All the students in one group concentrate on the timing task, while half the students in the second group exert constant efforts to distract their partners. The box plots show that the median times for the two groups are about the same but the times for the distracted group have greater variation. Note that in the distracted group, one data point is far enough removed from the others to be an outlier. See figure 10.3.

Fig. 10.3. Time estimates

Sports statistics and other real data are settings in which students can generate new data and investigate a variety of conjectures. The table in figure 10.4 contains some information from an NBA championship game between Los Angeles and Boston.

Fig. 10.4. NBA championship game statistics

Using the table, students can generate such new information as points/minute, rebounds/minute, points/field goals attempted. Who is the best percentage shooter? From another source, they can find the height of each player and determine rebounds/inch of height or points/inch of height. A problem like this is ideally suited to the curious nature of middle school students and opens up a world of questions and investigations to them.

Formulating key questions, interpreting graphs and charts, and solving problems are important goals in the study of statistics. Statistics can help answer questions that do not lend themselves to direct measurement. Once data are collected and organized, such questions as the following can guide students in interpreting the data:

• What appears most often in the data?
• What trends appear in the data?
• What is the significance of outliers?
• What interpretations can we draw from these data, and can we use our interpretations to make predictions?
• What difficulties might we encounter when extending the interpretations or predictions to other related problems?
• What additional data can we collect to verify or disprove the ideas developed from these data?