Learning Math: Data Analysis, Statistics, and Probability
Data Organization and Representation Part A: Patterns in Variation (10 minutes)
To answer a statistical question, we collect data, which consist of measurements of a variable. When the measurements we collect differ from each other, variation exists. We can examine this variation in several ways to find interesting answers to the statistical questions we pose.
For example, we can look at the distribution of a data set — the different values and how often each occurs — to draw out informative patterns in the variation. Tabular and graphical representations of the data can help to display these patterns.
In the video segment below, Professor Kader and the participants examine the distribution of a data set to help explain an intriguing aspect of the variation they find. Professor Kader also introduces the uses of graphical and tabular representations. See Note 2.
In Problem H2 of Session 1, you looked at the mint marks of 100 Jefferson nickels. Then you answered a series of questions about those coins. In this video segment, the onscreen participants examine the distribution of coins to come up with a hypothesis about where the coins without mint marks (N) were minted. After you watch the video segment, reflect on the questions below.
You can find this segment on the session video approximately 8 minutes and 30 seconds after the Annenberg Media logo. The second part of this segment begins approximately 10 minutes and 3 seconds after the Annenberg Media logo.
Did you arrive at the same conclusions about where the coins were minted as the onscreen participants did? Was your reasoning similar or different from theirs?
If you’re working with a group, you may want to organize a coin activity based on Problem H2 from Session 1 before watching the video segment. Here’s how the activity can be organized:
Label the corners of a large square poster board with the letters P (for Philadelphia), D (for Denver), S (for San Francisco), and N (for none). Separate 100 nickels according to mint mark (using magnifying glasses, if needed), and place them on the corner of the poster board corresponding to their mint location. Discuss the following question:
- Based on what you see, what can you say about the way the coins are distributed among the four different mint marks?
Statisticians find it useful to think in fractional terms, i.e., proportions or percentages. In order to start thinking about the coin data in fractional terms, cross two pieces of string to divide the poster board into four sections (one for each mint location). Adjusting the strings to keep the coins from each of the four mint marks separate, slide the coins to form a circle. This will result in a pie chart. Discuss the following questions:
- Based on what you see in the pie chart, what can you say about the way the coins are distributed among the four different mint marks? Specifically, what fraction of the total would you guess is represented by each group?
- Which location has the most coins? Why do you think this location is the most common?
- Which location has the least coins? Why do you think this location is the least common?
Watch the video segment, and compare your hypotheses with those of the online participants.
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.