Meaning and Importance

Understanding probability and statistics is essential in the modern world with the print and electronic media full of statistical information and its interpretation. The goal of mathematical instruction in this area should be to make students sensible, critical, users of probability and statistics, able to apply their processes and principles to real-world problems. We do not want students to think that those people who did not win the lottery yesterday have a greater chance of winning today! We also do not want them to believe an argument merely because various statistics are offered. We would like them to understand the issues underlying the reliability of election polls. In short, they should be able to judge whether the statistics are meaningful and are used appropriately.

K-12 Development and Emphases

In the area of probability, young children start out simply learning to use probabilistic terms correctly. Words like possibly, probably, and certainly have definite meanings, implying increasing likelihood of an event happening, and it takes children some time to begin to use them correctly. Beyond that, though, elementary age children are certainly able to understand the probability of an event. Starting with phrases like once in six tosses, children progress to more sophisticated probabilistic language like chances are one out of six, and finally to standard fractional notation for the expression of a probability. To motivate and encourage that maturation, students should be regularly engaged in predicting and determining probabilities.

Experiments leading to discussions about the difference between experimental and theoretical probability can also be done by older elementary and middle grades students. Theoretical probability is the probability based on an analysis of the physical properties and behavior of the objects involved in the event. For instance, we know enough about the properties of a fair die to know that each face is equally likely to wind up on top. Experimental probabilities are those determined by data gathered experimentally. For example, students may be able to determine the experimental probabilities of rolling a sum of seven or a sum of four with two dice long before they can explain why the two probabilities are different from each other from a theoretical point of view.

Older students should understand the difference between simple and compound events (like rolling one die versus rolling two dice), and the difference between independent and dependent events (like flipping one coin repeatedly five times versus picking five marbles out of a bag of ten). Again, the best way to approach this content is with open-ended investigations that allow the students to arrive at their own conclusions through experimentation and discussion. Eventually, students should feel comfortable representing real-life events using probability models.

In statistics, young children can start out as early as kindergarten with data collection, organization, and graphing and a focus on those skills, with obviously increasing sophistication, should last throughout their schooling. Students must be able to understand the tables, charts, and graphs used to present data, and they must be able to organize their own data into formats which make it easier to understand. While young students can do exhaustive surveys about some interesting question for all of the members of the class, older students should focus some time and energy on the questions involved with sampling. Identifying and obtaining data from a well-defined sample of the population is the most important job of a professional pollster.

As students progress through the elementary grades, an increased focus on central tendency and later on variance and correlation are appropriate. Students should be able to use mean, median, and mode and should know the differences in their uses. Measures of the variance from the center of a set of data, or dispersion, also provide useful insights into sets of numbers and can start early with box-and-whisker plots for upper elementary students and progress to measures like standard deviation for older students.

The reason statistics grew as a branch of mathematics, however, is to provide tools that are helpful in analysis and inference and that focus should permeate everything we do with students in the area. Whenever they look at data, there should be some question they are trying to answer, some position they are trying to support. K-12 students should have many opportunities to look for patterns, draw conclusions, and make predictions about the outcomes of future experiments, polls, surveys, and so on. They should examine data to see whether it is consistent with some hypotheses that a classmate may already have made, and learn to judge whether the data is reliable or whether the hypotheses might need revision.

In summary, probability and statistics hold the key for enabling our students to understand, process, and interpret the vast amounts of quantitative data that exist all around them. To be able to judge the truth of a data-supported argument presented to them, to discern the believability of a persuasive advertisement that talks about the results of a survey of all of the users of a particular product, or to be knowledgeable consumers of the data-intensive government and electoral statistics that are ever-present, students need the skills that they can learn in a well-conceived probability and statistics curriculum strand.