STANDARD 12 - PROBABILITY AND STATISTICS
Probability and statistics are the mathematics used to understand chance and to collect, organize, describe, and analyze numerical data. From weather reports to sophisticated studies of genetics, from election results to product preference surveys, probability and statistical language and concepts are increasingly present in the media and in everyday conversations. Students need this mathematics to help them judge the correctness of an argument supported by seemingly persuasive data.
Meaning and Importance
Probability is the study of random events. It is used in analyzing games of chance, genetics, weather prediction, and a myriad of other everyday events. Statistics is the mathematics we use to collect, organize, and interpret numerical data. It is used to describe and analyze sets of test scores, election results, and shoppers' preferences for particular products. Probability and statistics are closely linked because statistical data are frequently analyzed to see whether conclusions can be drawn legitimately about a particular phenomenon and also to make predictions about future events. For instance, early election results are analyzed to see if they conform to predictions from pre-election polls and also to predict the final outcome of the election.
Understanding probability and statistics is essential in the modern world, where the print and electronic media are full of statistical information and 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. Students should not think that those people who did not win the lottery yesterday have a greater chance of winning today! They should not believe an argument merely because various statistics are offered. Rather, they should be able to judge whether the statistics are meaningful and are being used appropriately.
K-12 Development and Emphases
Statistics and probability naturally lend themselves to plenty of fun, hands-on cooperative learning and group activities. Activities with spinners, dice, and coin tossing can be used to investigate chance events. Students should discuss the theoretical probabilities of different events such as the possible sums of a pair of dice, andcheck them experimentally. They can choose topics to investigate, such as how much milk and juice the cafeteria should order each day, gather statistics on current orders and student preferences, and make predictions on future use. Connections between these topics and everyday experiences provide motivation and a sense of relevance to students.
In the area of probability, young children start out simply learning to use probability terms correctly. Words like possibly, probably, and certainly have definite meanings, referring to the 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 probability language like chances are one out of six, and finally to standard fractional, decimal, and percent notation for the expression of a probability. To motivate and foster that maturation, students should be regularly engaged in predicting and determining probabilities.
Experiments leading to discussions about the difference between experimental and theoretical probability should be done by older elementary and middle school students. The theoretical probability is the probability based on a mathematical analysis of the physical properties and behavior of the objects involved in the event. For instance, when a fair die is rolled each face is equally likely to wind up on top, and so the probability of any particular face showing is one-sixth. Experimental probabilities are determined by data gathered through experiments. For example, students may be able to compare the experimental probabilities of rolling a sum of seven vs. a sum of four with two dice long before they can explain why the first is twice as likely from a theoretical point of view.
Older students should understand the difference between simple and compound events, like rolling one die vs. rolling two dice, and the difference between independent and dependent events, like picking five marbles out of a bag of blue and green marbles one at a time with replacement vs. without replacement. 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. The 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 them 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, where information is obtained from only some of the members of a group. Identifying and obtaining data from a well-defined sample of the population is one of the most challenging tasks 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 the average or mean, the median, and the mode and understand 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. These can be introduced early with the range for the early grades, box-and-whisker plots showing quartiles of a data distribution for upper elementary school students, and progress to measures like standard deviation for older students.
The reason statistics grew as a branch of mathematics, however, was to provide tools that are helpful inanalysis and inference in situations of uncertainty, and that focus should permeate everything students do in this area. Whenever they look at data, they should be trying to answer a question, support a position, or discover a pattern. Students at all grade levels 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 they are consistent with some hypotheses that a classmate may already have made, and learn to judge whether the data are reliable or whether the hypothesis might need revision.
In summary, probability and statistics hold the key for enabling our students to better understand, process, and interpret the vast amounts of quantitative data that exist all around them, and to have a probabilistic sense in situations of uncertainty. To be able to judge the validity 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.
Note: Although each content standard is discussed in a separate chapter, it is not the intention that each be treated separately in the classroom. Indeed, as noted in the Introduction to this Framework, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences.
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