In grades 9-12, the mathematics curriculum should include the continued study of data analysis and statistics so that all students can--

• construct and draw inferences from charts, tables, and graphs that summarize data from real-world situations;
• use curve fitting to predict from data;
• understand and apply measures of central tendency, variability, and correlation;
• understand sampling and recognize its role in statistical claims;
• design a statistical experiment to study a problem, conduct the experiment, and interpret and communicate the outcomes;
• analyze the effects of data transformations on measures of central tendency and variability;

and so that, in addition, college-intending students can--

• transform data to aid in data interpretation and prediction;
• test hypotheses using appropriate statistics.

Focus

Collecting, representing, and processing data are activities of major importance to contemporary society. In the natural and social sciences, data are also summarized, analyzed, and transformed. These activities involve simulations and/or sampling, fitting curves, testing hypotheses, and drawing inferences. To enhance their social awareness and career opportunities, students should learn to apply these techniques in solving problems and in evaluating the myriad statistical claims they encounter in their daily lives.

The study of statistics in grades 9-12 should consolidate, deepen, and build on student understandings of methods of exploratory data analysis as developed in the elementary and middle grades. Students should be encouraged to apply statistical tools to other academic subjects through the exploration of such data as student-opinion polls for social studies, word or letter counts for English, and plant-growth records for biology. Out-of-school activities such as athletics provide further opportunities for data analysis, the results of which can be seen to be immediately useful.

It is essential that students come to understand the difference between the right-or-wrong quality characteristic of most mathematical thinking and the qualified nature of outcomes in statistical analysis. It is equally important, however, that students do not extrapolate beyond this fact to reject statistical thinking because it allows counterexamples. Instead, they should recognize that statistics plays an important intermediate role between the exactness of other mathematical studies and the equivocal nature of a world dependent largely on individual opinion.

Computing technology allows students to represent data in graphs quickly (with curve fitting done for them) and to calculate statistical measures with remarkable precision using single computer keystrokes. What is missing--and what their study of statistics should provide--is an understanding of which measures are appropriate for a given problem and what such measures as mean, variance, and correlation can tell them about a problem. Furthermore, it is essential that students learn to interpret results intelligently.

Discussion

This standard should not be viewed as advocating, or even prescribing, a statistics course; rather, it describes topics that should be integrated with other mathematics topics and disciplines. For example, curve fitting is a statistical topic that integrates easily into the study of linear and higher-order equations. Students could investigate the possible relationship between car age and mileage by collecting data from the school parking lot and constructing a scatter plot (fig. 10.1).

Fig. 10.1. Car mileage by model year

Since the points seem to lie in a reasonably narrow band, students can identify a "best" line that fits their data. Techniques for constructing such a line can range from the most basic, such as placing a piece of uncooked spaghetti or a string on the graph so that approximately the same number of data points fall above as below, to approaches involving medians of grouped data or the technique of least squares. Many calculators provide the capability to generate an equation for the regression line (one of these "best" lines) as well as the associated correlation coefficient. Students then can use either their graph or their equation to predict, for example, the expected mileage of a 1980 car. They should be encouraged to write a summary paragraph about the information displayed in the table or graph and include inferences they believe are supported by their analysis of the data.

Students should also come to realize that curve fitting is not appropriate for all data sets. In paperback books, for example, there is so little relationship between the number of pages and the price that further analysis would not yield useful information.

Communication plays a central role in statistical problems. Quantitative results require careful exposition and interpretation if they are to have meaning. In particular, it is often true that different modes of data representation convey quite different messages. A regression line (calculated directly from data without reference to a scatter plot) might be strongly influenced by a few aberrant points, for example, whereas the scatter plot for the same data might suggest that these outliers represent anomalies that may be due to mistakes in data handling. A further investigation of these specific points might lead to their rejection, a new curve fit, and an improved correlation.

Students must acquire intuitive notions of randomness, representativeness, and bias in sampling to enhance their ability to evaluate statistical claims. These understandings would give students the appropriate tools for rejecting such television advertising claims as one that portrays a series of people choosing the same commercial toothpaste. (Here the implication of representativeness clearly is not fulfilled.) If they are to grasp the concepts of sampling, the central limit theorem, and confidence intervals, students should have experience constructing and analyzing sampling distributions through simulations. These experiences provide students with the tools and the perspective they need to interpret such claims in the media as, "The polls indicate that 55% of the voters, with an error of 3%, prefer candidate X (with 90% confidence)." Statistics and probability threads are interwoven here. College-intending students also should apply this understanding of sampling in designing their own experiments to test hypotheses.

Students should be aware that bias can arise in the interpretation of results as well as in sampling: the interpreter's predisposition or expectation may strongly affect the message derived from the statistical results. This often occurs in the presentation and interpretation of data gathered for political purposes.

College-intending students should become familiar with such distributions as the normal, Student's t, Poisson, and chi square. Students should be able to determine when it is appropriate to use these distributions in statistical analysis (e.g., to obtain confidence intervals or to test hypotheses). Instructional activities should focus on the logic behind the process in addition to the "test" itself.

In recent years, nonparametric methods or distribution-free methods like the chi-square test in the cola example in the standard on probability (Standard 11) have increasingly been used as alternatives to statistical tests that assume a particular (often normal) distribution. Nonparametric techniques (which also include such measures as the sign test, the Mann-Whitney U test, and Spearman's rank correlation test) are extremely versatile, easy to use, often derive their power directly from combinatorics and the binomial distribution (of the statistics, not the sample), and are particularly well suited to small samples. As these methods continue to gain in popularity, it is expected that they will become an integral part of the evolving statistics curriculum.

All students should be encouraged to discover generalizations that relate the effect of modifying a set of data by addition or scalar multiplication on the mean, median, mode, and variance. For example, class test scores could be transformed by increasing each score by 10 points (or by multiplying each score by 1.1). Technology provides an easy means by which students can compute the statistics for the transformed data, which, on analysis, lead to generalizations that the mean, median, and mode are increased by 10 (multiplied by 1.1) and the variance is unchanged (multiplied by (1.1)² ). College-intending students should be able to derive these results algebraically.

Statistical data, summaries, and inferences appear more frequently in the work and everyday lives of people than any other form of mathematical analysis. It is therefore essential that all high school graduates acquire, at the appropriate level, the capabilities identified in this standard. This expectation will require that statistics be given a more prominent position in the high school curriculum.