Learning Math: Data Analysis, Statistics, and Probability
Probability Part A: Probability in Statistics (20 minutes)
In This Part: What Is Probability?
Make a list of the topics and ideas that come to mind when you think of probability, including both everyday uses of probability and mathematical or school uses.
What does probability have to do with statistics? Think about ways that statistics might use probability, and vice versa.
In this video segment, participants discuss probability and how it relates to statistics. Watch this segment after recording your own thoughts on Problems A1 and A2.
When many people think of probability, they think of rolling dice, picking numbers at random, or playing the lottery. In fact, games of chance, which often involve dice or other random devices, rely on the principles of probability.
What is a random event? Give an example of something that happens randomly and something that does not.
In This Part: A Game of Chance?
Can you improve the odds of a game with practice, or is it truly just a question of randomness? Let’s explore this question by playing Push Penny. See Note 2 below.
Make the Push Penny board by adding horizontal lines to a 36″ x 24″ sheet of poster board. (Despite its name, this game uses quarters rather than pennies, since they tend to slide better than other coins when pushed.) Draw the lines exactly two coin diameters apart, as illustrated below — uniform spacing on the lined poster board is crucial for meaningful analysis and interpretation of the results. Put the board on a flat surface, with a second sheet of blank poster board in front of it.
To play, push a quarter from the edge of the blank board onto the lined board. Each round of the game consists of four pushes. You score a “hit” if the quarter touches one of the lines when it stops. You “miss” if the quarter stops between the lines. (Remove the coin from the board between successive pushes.)
Suppose you wanted to find out whether you could develop skill at playing rounds of Push Penny. How might you design an experiment to test this idea?
Play 20 rounds of Push Penny (four pushes per round), and record your results from each round using the following format (five rounds are provided as an example):
a. Do the results from your 20 rounds suggest that you have developed skill in playing Push Penny? Describe the process you used to answer this question.
b. If you don’t think you’ve developed much skill in playing the game, do you think it is still possible to develop this skill?
c. Give an example of a game where it is not possible to increase your game-playing skills.
We will investigate Push Penny in more detail later in this session.
The primary problem of Session 8 is based on the question, “After several practices of Push Penny, have you developed skill in playing the game?”
This is a statistics problem; appropriate data consist of the results from several rounds of the game. The ultimate goal of this session is to compare the experimental results with the expected results, using a probability model.
First, you play 20 games and record the results from each game. Next, you consider whether the results indicate that you are a skillful Push Penny player. You then analyze the results for a person who played 100 games of Push Penny. Finally, based on these data, you try to determine whether the player did, in fact, develop any Push Penny skills.
Make sure that you allow time to develop your own ideas for analysis. In considering whether skill has been demonstrated, you may want to look at the average score or at the proportion of hits from all 400 pushes.
Both the average score and the proportion of hits depend on basic probability concepts. In order to determine whether the proportion of hits demonstrates skill, you must first consider the probability that a random push will hit a line. The “average” score depends on a concept of average other than the arithmetic mean: In this case, it is a weighted mean where the weights are probabilities.
The method of analysis investigated in this session — “goodness of fit” — uses the binomial probability model. At this point, it is important to consider how probability can be used in the analysis of this problem. You will return to the problem of determining skill after considering some basic ideas about probability.
Answers will vary. Some everyday uses of probability are predicting the weather, deciding which road is likely to have the least amount of traffic, and choosing a restaurant on the basis of how long you think the wait will be. Some mathematical uses include the probability of rolling a six on a die, the probability of tossing a coin and getting “heads,” and the probability of 1-2-3 coming up as the daily lottery number.
Statistical uses of probability include the probability that the estimate of a mean is accurate (this is known as a confidence interval). Places where probability uses statistics include taking experimental data and trying to create exact probabilities to match your data set.
A “random” event is entirely up to chance; there is no skill involved. A random event might be what appears as the top card after a thorough shuffling of a deck of cards. Most events are not random; for example, answering a question correctly on a test may happen randomly (as a guess) but usually is a result of skill.
You might look at your average score and determine whether your average score is improving over time. For example, if you played Push Penny 20 times a day for several days, you could compare your average first day’s score to your average last day’s score and see if there was any improvement.
a. Answers will vary.
b. Answers will vary.
c. One example of such a game is a game where you shuffle a deck of cards thoroughly and then try to guess the suit of the top card. Since the top card is completely random, there is no way to develop your skill in correctly guessing the suit of the card (without cheating or using ESP!). The card game War is another example.
Session 1 Statistics As Problem Solving
Consider statistics as a problem-solving process and examine its four components: asking questions, collecting appropriate data, analyzing the data, and interpreting the results. This session investigates the nature of data and its potential sources of variation. Variables, bias, and random sampling are introduced.
Session 2 Data Organization and Representation
Explore different ways of representing, analyzing, and interpreting data, including line plots, frequency tables, cumulative and relative frequency tables, and bar graphs. Learn how to use intervals to describe variation in data. Learn how to determine and understand the median.
Session 3 Describing Distributions
Continue learning about organizing and grouping data in different graphs and tables. Learn how to analyze and interpret variation in data by using stem and leaf plots and histograms. Learn about relative and cumulative frequency.
Session 4 Min, Max and the Five-Number Summary
Investigate various approaches for summarizing variation in data, and learn how dividing data into groups can help provide other types of answers to statistical questions. Understand numerical and graphic representations of the minimum, the maximum, the median, and quartiles. Learn how to create a box plot.
Session 5 Variation About the Mean
Explore the concept of the mean and how variation in data can be described relative to the mean. Concepts include fair and unfair allocations, and how to measure variation about the mean.
Session 6 Designing Experiments
Examine how to collect and compare data from observational and experimental studies, and learn how to set up your own experimental studies.
Session 7 Bivariate Data and Analysis
Analyze bivariate data and understand the concepts of association and co-variation between two quantitative variables. Explore scatter plots, the least squares line, and modeling linear relationships.
Session 8 Probability
Investigate some basic concepts of probability and the relationship between statistics and probability. Learn about random events, games of chance, mathematical and experimental probability, tree diagrams, and the binomial probability model.
Session 9 Random Sampling and Estimation
Learn how to select a random sample and use it to estimate characteristics of an entire population. Learn how to describe variation in estimates, and the effect of sample size on an estimate's accuracy.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in this course can be applied through a case study of a K-2 teacher, Ellen Sabanosh, a former course participant who has adapted her new knowledge to her classroom.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in this course can be applied through case studies of a grade 3-5 teacher, Suzanne L'Esperance and grade 6-8 teacher, Paul Snowden, both former course participants who have adapted their new knowledge to their classrooms.