Data Analysis & Probability

1. Data Analysis & Probability

2. Robert Stephanie Mark What is the typical number of letters in a person’s first or last name? Joseph Juanita Katie Meghan Valente Will Xiu Sean

3. Lengths of Names • On a sticky note, write the number of letters in your first name • Put the sticky note on the number line above the corresponding number to create a dot plot, or line plot • On a sticky note of a different color, write the number of letters in your last name • Put the sticky note on the number line above the corresponding number to create a dot plot, or line plot

4. Measures of Center • What is the mode for the number of letters in first/last names? How do you know? • What is the median for the number of letters in first/last names? How do you know? • What is the mean for the number of letters in first/last names? How do you know?

5. Conceptualizing the Mean We want to deepen students conceptual understanding of the mean • The mean as a balancing point • The mean as the equal share amount

6. Mean as a Balancing Point The X’s represent the sticky notes and the mean of this data set is 4. How can you show 4 as a “balancing point”?

7. Mean as a Balancing Point 6 – 2 = 4 6 – 2 = 4 Take away 2 twice -2 -2

8. Mean as a Balancing Point 3 + 1 = 4, 3 + 1 = 4, 2 + 2 = 4 Add 1 twice and add 2 once +1 +2 +1

9. Mean as a Balancing Point • Look at the data we collected about our names • With a partner, talk about how you could illustrate the mean as a balancing point with this data

10. Mean as Equal Groups Three people have first names containing 4, 8, and 6 letters - how can you show the mean as the equal share amount?

11. Mean as Equal Groups The mean is 6

12. Mean as Equal Groups • Look at the data we collected about our names • With a partner, talk about how you could illustrate the mean as an equal share amount using these data

13. Typical or Average Lengths • What is the typical number of letters in a person’s first/last name? • How do you know? Justify your response with the data

14. What Is an “Average”? • The term “average” is used frequently to mean a “measure of center” • An average could be a mean, median, or mode

15. Which “Average” is Needed? • Salaries at a large company • Salaries at a small company • Test scores for a student • Annual salaries for graduates from Bill Gates’ high school • Size of shoes sold in a shoe store • Cost of a ticket to the movies

16. Big Ideasin Data& Probability • Multiple counting strategies and sample space representations are used to determine theoretical probabilities; experimental and theoretical probabilities can be computed and compared

17. Big Ideasin Data& Probability • Collection, analysis, and interpretation of univariate data are used to make decisions and solve problems • Analysis of data includes understanding relationships among mean, median, mode, range, distribution, inter-quartile ranges, and outliers • Interpretation of data includes relating results of analysis back to the purpose of collecting data and making decisions about representations of data

18. Big Ideasin Data& Probability • Bivariate data may be displayed and then analyzed within the rectangular coordinate plane, where a linear equation may or may not be a good model for the relationship between the two attributes

19. Big Ideasin Data& Probability • PCAI is an important model for statistical investigations that highlights a process Posea question Collectdata Analyzedata Interpretresults

20. Where does it all fit? • Using your copy of the , decide how the objectives fit under the umbrella of the Big Ideas • Write the objective numbers on your chart • Be prepared to share observations with the whole group

21. Big Ideas in K-5 • Data are attributes collected about individual objects (people, cities, prices, opinions, etc.) and can be either categorical or numerical • Pose, Collect, Analyze, Interpret (PCAI cycle) is a model for the process of statistical investigations • Different representations and graphs communicate and classify data • Simple probability allows us to make more accurate predictions

22. Read and Enjoy! • The next item in your handouts is the article “Statistical Investigation” • With your AdultLearner Hat on, read the article - it is for your knowledge, not a student article • As you read, make notes on concepts that are new to you and any questions you may have

23. Pose a question • Collect data • Analyze data • Interpret results PCAI Model (Graham, 1987) • A framework for engaging in statistical investigations • Components may emerge in order or may include revisiting and making connections among components

24. Pose a question • Collect data • Analyze data • Interpret results PCAI Model • Pose a question - Why is it important for students to write and refine questions? • Collect data - What problems might students encounter as they collect data? • Analyze data - What are different ways to look at the “shape of the data”? • Interpret results - How might “bias” influence an interpretation?

25. Pose a question • Collect data • Analyze data • Interpret results Pose a Question Select a question that is… • motivated by describing summarizing, comparing, and generalizing data within a context • measurable—variables (numerical or categorical) should be able to be measured • based on data available within the time frame of the investigation

26. Pose a question • Collect data • Analyze data • Interpret results Collect Data Determine… • The population, methods of collecting data, and • If a sample will be collected • Consider the type of sample (random or convenience) • Consider representativeness and bias • Consider the sample size • Will more than one sample be collected

27. Pose a question • Collect data • Analyze data • Interpret results Analyze Data • Describe and summarize data • using relevant summary statistics, such as the mean, median, mode and • using tables, diagrams, graphs, or other representations • Describe variation

28. Pose a question • Collect data • Analyze data • Interpret results Interpret Results • Relate analysis to original question and context • Make decisions about the question posed within the context of the problem based on data collection and analysis

29. Variability in the MS How can students in grades 6-8 attend to variability? • Minimum and maximum values • Unusual data points, such as outliers • Interquartile-range • Range

30. Old Faithful Old Faithful During the tenth episode of the Amazing Race 8, teams had to observe an eruption of the Old Faithful geyser, in Yellowstone National Park in order to get their next clue. Some teams arrived at the geyser just missing the eruption. How long would you have to wait before the next eruption? Make a prediction.

31. Collect Data • How might students suggest we collect data in order to make a prediction? • What factors would make one method better than another?

32. Old Faithful Our Data Is Collected Using data that has already been collected (see your Old Faithful Handout), answer the following question with the Day 1 data values: How long should we tell teams to expect to wait between eruptions of Old Faithful?

33. Old Faithful Student Analysis • What types of student responses would you want or expect to see from students? • Students may base their initial prediction on a measure of central tendency, like the mean or median • Students may find other statistics, such as the minimum and maximum values, the lower and upper quartiles, the inter-quartile range, the range, etc.

34. Old Faithful Student Analysis What types of student responses would you want or expect to see from students? • Students may represent the data as a histogram, stem-and-leaf plot, dot plot, bar graph, box plot, or other graphical representation • Students may also respond with a combination of the previous

35. Old Faithful One Number Summaries Is any one number summary helpful in predicting the wait time between eruptions on its own?

36. Histogram • What does the histogram tell us about the wait time? • What might this representation hide?

37. Box Plot • What does the box plot tell us about the wait time? • What might this representation hide?

38. Dot Plot • What does the dot plot tell us about the wait time? • What might this representation hide?

39. Stem-and-Leaf Plot • What does the stem-and-leaf plot tell us about the wait time? • What might this representation hide?

40. Many Toolsfor Analysis • Which of the previous representations would be the best? • What reasons might we want to move student thinking beyond a one-number summary of the data set?

41. Highlighting and Masking • Each type of representation highlights or masks patterns in the data • The stem-and-leaf plot, dot plot, and histogram yield a more clear picture of the data’s distribution and shape than the box plot

42. Let’s Justify … • The stem and dot display the actual data, whereas box plots involve data reduction • The 3 others reveal the gap masked by box plots • All 3 highlight variation which is an essential signature of the data set • All illustrate a physical center or balance point, but show how it relates to actual data values

43. Interpret the Data • Now back to the question that started all this (or coming full circle in the process of statistical investigation)… • How long should we tell the Amazing Race teams to expect to have to wait between Old Faithful eruptions? (Justify your response with the data)

44. Grappling with Graphs • With a partner discuss the problems in the handout Grappling with Graphs • Be ready to justify your responses to the whole group

45. Box Plot Trot

46. Box Plot Trot 1

47. Box Plot Trot 2

48. Box Plot Trot 3

49. # 2 Box- Betterthan Dot Plot? • Explain your reasoning • Any other reasons to choose a Box plot over a Dot plot?

50. #3 Histogram vs Box Plot • What would the representations look like? • Explain how you created each representation