Prior to the middle grades, students should have had experiences collecting, organizing, and representing sets of data. They should be facile both with representational tools (such as tables, line plots, bar graphs, and line graphs) and with measures of center and spread (such as median, mode, and range). They should have had experience in using some methods of analyzing information and answering questions, typically about a single population.

In grades 6–8, teachers should build on this base of experience to help students answer more-complex questions, such as those concerning relationships among populations or samples and those about relationships between two variables within one population or sample. Toward this end, new representations should be added to the students' repertoire. Box plots, for example, allow students to compare two or more samples, such as the heights of students in two different classes. Scatterplots allow students to study related pairs of characteristics in one sample, such as height versus arm span among students in one class. In addition, students can use and further develop their emerging understanding of proportionality in various aspects of their study of data and statistics.

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them

Middle-grades students should formulate questions and design experiments or surveys to collect relevant data so that they can compare characteristics within a population or between populations. For example, a teacher might ask students to examine how various design characteristics of a paper airplane—such as its length or the number of paper clips attached to its nose—affect the distance it travels and its consistency of flight. Students would then plan experiments in which they collect data that would allow them to compare the effects of particular design features. In addition to helping students design their experiments logically, the teacher should help them consider other factors that might affect the data, such as wind or inconsistencies in launching the planes.

Because laboratory experiments involving data collection are part of the middle-grades science curriculum, mathematics teachers may find it useful to collaborate with science teachers so that they are consistent in their design of experiments. Such collaboration could be extended so that students might collect the data for an experiment in science class and analyze it in mathematics class.

In addition to collecting their own data, students should learn to find relevant data in other resources, such as Web sites or print publications. Consumer Reports, for example, regularly compares the characteristics of various products, such as the quality of peanut butter; the longevity of rechargeable batteries; or the cost, size, and fuel efficiency of automobiles. When using data from other sources, students need to determine which data are appropriate for their needs, understand how the data were gathered, and consider limitations that could affect interpretation.

Select and use appropriate statistical methods to analyze data

Students also need to think about measures of center in relation to the spread of a distribution. In general, the crucial question is, How do changes in data values affect the mean and median of a set of data? To examine this question, teachers could have students use a calculator to create a table of values and compute the mean and median. Then they could change one of the data values in the table and see whether the values of the mean and the median are also changed. These relationships can be effectively demonstrated using software through which students can control a data value and observe how the mean and median are affected. For example, using software that produces line plots for data sets, students could plot a set of data and mark the mean and median on the line. The students could then change one data value and observe how the mean and median change. By repeating this process for various data points, they can notice that changing one data value usually does not affect the median at all, unless the moved value is at the middle of the data set or moves across the middle, but that every change in a value affects the mean. Thus, the mean is more likely to be influenced by extreme values, since it is affected by the actual data values, but the median involves only the relative positions of the values. Other similar problems can be useful in helping students understand the different sensitivities of the mean and median; for example, the mean is very sensitive to the addition or deletion of one or two extreme data points, whereas the median is far less sensitive to such changes.

Students should consider how well different graphs represent important characteristics of data sets. For example, they might notice that it is easier to see symmetry or skewness in a graph than in a table of values. Graphs, however, can lose some of the features of the data, as can be demonstrated by generating a family of histograms for a single set of data, using different bin sizes: the different histograms may convey different pictures of the symmetry, skewness, or variability of the data set. Another example is seen when comparing a histogram and a box plot for the same data, such as those for the one-clip plane in figures 6.27 and 6.28. Box plots do not convey as much specific information about the data set, such as where clusters occur, as histograms do. But box plots can provide effective comparisons between two data sets because they make descriptive characteristics such as median and interquartile range readily apparent.

Develop and evaluate inferences and predictions that are based on data

In collecting and representing data, students should be driven by a desire to answer questions on the basis of the data. In the process, they should make observations, inferences, and conjectures and develop new » questions. They can use their developing facility with rational numbers and proportionality to refine their observations and conjectures. For example, when considering the relative-frequency histogram in figure 6.27, most students would observe that "the paper plane goes between 15 and 21 feet about as often as it goes between 24 and 33 feet," but such an observation is not very precise about the frequency. A teacher could press students to make more-precise statements: "The plane goes between 15 and 21 feet about 45 percent of the time."

Box plots are useful when making comparisons between populations. A teacher might pose the following question about the box plots in figure 6.28:

From the box plots (in fig. 6.28), which type of plane appears to fly farther? Which type of plane is more consistent in the distance it flies?

Understand and apply basic concepts of probability

Teachers should give middle-grades students numerous opportunities to engage in probabilistic thinking about simple situations from which students can develop notions of chance. They should use appropriate terminology in their discussions of chance and use probability to » make predictions and test conjectures. For example, a teacher might give students the following problem:

Suppose you have a box containing 100 slips of paper numbered from 1 through 100. If you select one slip of paper at random, what is the probability that the number is a multiple of 5? A multiple of 8? Is not a multiple of 5? Is a multiple of both 5 and 8?

Although simulations can be useful, students also need to develop their probabilistic thinking by frequent experience with actual experiments. Many can be quite simple. For example, students could be asked to predict the probability of various outcomes of flipping two coins sixty times. Some students will incorrectly expect that there are three equally likely outcomes of flipping two coins once: two heads, two tails, and one of each. If so, they may predict that each of these will occur about twenty times. If groups of students conducted this experiment, they could construct a relative-frequency bar graph from the pooled data for the entire class. Then they could discuss whether the results of the experiment are consistent with their predictions. If students are accustomed to reasoning from and about data, they will understand that » discrepancies between predictions and outcomes from a large and representative sample must be taken seriously. The detection of discrepancies can lead to learning when students turn to classmates and their teacher for alternative ways to think about the possible results of flipping two coins (or other similar compound events). Teachers can then introduce students to various methods—organized lists, tree diagrams, and area models—to help them understand and compute the probabilities of compound events.

Using a problem like the following, a teacher might assess students' understanding of probability in a manner that includes data analysis and reveals possible misconceptions:

For the one-clip paper airplane, which was flight-tested with the results shown in the relative-frequency histogram (in fig. 6.27), what is the probability that exactly one of the next two throws will be a dud (i.e., it will travel less that 21 feet) and the other will be a success (i.e., it will travel 21 feet or more)?

To solve this problem, students would need to understand the data representation in figure 6.27 and use ratios to estimate that there is about a 45 percent chance that a throw will be a dud and about a 55 percent chance that it will be a success. Then they would need to use some method for handling the compound event and deal with the fact that there are two ways it might occur. Students who understand all that is required might produce a tree diagram like the one in figure 6.30 to show that the total probability is 198/400, or .495, since each of the two possibilities—"dud first, then success" and "success first, then dud"—has a probability of 99/400.

Fig. 6.30. A tree diagram for determining the probability of a compound event, given simple data.