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Data & Statistics

Explore the objectives and key concepts in the data and statistics strand for K-5 students. Learn about categorical and numerical data, the PCAI model for statistical investigations, and different data representations. Gain a deeper understanding of probability and how to pose, collect, analyze, and interpret data.

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Data & Statistics

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  1. Data & Statistics

  2. Data Objectives • Look at the Objectives for the data strand K-5 • Be prepared to discuss… • What concepts are common across the grades? • How does the data strand build?

  3. Data & Statistics Objectives Use your copy of the Standard Course of Study to identify these objectives

  4. Big Ideas in Data • Data can be either categorical or numerical • Pose, Collect, Analyze, Interpret (PCAI) is a model for the process of statistical investigations • Different representations and graphs classify and communicate data • Understanding basic concepts of probability allows us to make more accurate predictions

  5. Categorical & Numerical Data • Students need to distinguish between questions that give categorical and numerical data before they pose a question and try to interpret the data

  6. Categorical Data • Values that are often words and that represent possible responses with respect to a given category • Represent individuals or objects by one or more characteristics or traits that they share Examples: • Months in which people have birthdays • Favorite color T-shirt • Favorite fruits • Kinds of pets

  7. Numerical Data • Values that are numbers such as counts, measurements, and ratings • Represent objects or individuals by numbers assigned to certain measurable properties - Examples: • Number of children in families • Pulse rates of top athletes • Time in minutes that students spend watching television each day • Number of pets

  8. Creating Examples • Make a list of 10 questions that generate categorical data and 10 questions that generate numerical data that your students could investigate For which of your questions could mode, median, range, and/or mean be found?

  9. Processof Statistical Investigation • Pose the question • Collect the data • Analyze the data • Interpret the data

  10. PCAI Model

  11. Pose the Question • Step 1: Identify a specific question to explore, and decide what data to collect to address the question

  12. Pose the Question • All data investigations begin with a problem or question • Questions should be focused & precise • Questions may need to be refined • Students need multiple experiences with posing questions and carrying-out investigations • Each question should have a purpose

  13. Pose the Question Four Purposes for Investigations: • To describe or summarize what was learned from a set of data • To determine preference and opinions from a set of data • To compare and contrast two or more sets of data • To generalize or make predictions from a set of data

  14. Collect the Data • Step 2: Decide how to collect the data, and collect the data

  15. Collect the Data • Students should collect their own data • Students need to have a plan for collecting data and understand where data come from • Data collection methods include: observations, surveys, experiments, measurements, interviews, polls, simulations, examinations of past records, and searches of the internet, library, or other sources

  16. Collect the Data

  17. Statistical Investigation • Read the section titled “Analyzing the Data” in your Statistical Investigation handout • Individually note any questions you may have • When all have finished reading, discuss the main points of this section

  18. Analyze the Data • Step 3: Organize, summarize, describe, and display the data; and look for patterns in the data

  19. Analysis • Analysis is the process of reflecting on assigned values in a collection of data to obtain new information and gain understanding of the population from which the data were collected

  20. Analyze the Data • Representing the data in order to identify patterns of variation in the data • Manner of representation depends on why the data have been collected and what type of data have been collected • Graphical displays provide visual descriptions of variability in data (clusters, outliers) and ways to analyze the data (range, median, mode)

  21. Measures of Center • Mode: The most frequently occurring value in a data set • Median: The middle value in an ordered data set • Mean: The average - the value if each data in the set were the same value

  22. Median Exploration 1985 Statistic: The median age in the US was 31.5 years • List everything you know about the age of US citizens in 1985 • List 5 things you can not tell about the population using this statistic

  23. Data Set Comparisons Data for Number of Roofs Lived Under • Set 1: 3,3,4,4,4,5,6,6,6,7,7,7,11,11,12,12,12,12, 13,13,13,13,18,18,18,19,19,19,20,20 • Set 2: 2,3,4,5,5,6,6,7,7,7,9,9,9,10,10,10,10,10, 11,11,11,12,12,12,12,13,13,13,14,14,14, 15,19,21,43

  24. Mean Exploration • Explain that the mean is a balance point; a kind of middle point that has to do with how the data are spread out • Make a different colored tower showing the number of people in each family: 2, 3, 3, 4, 4, 4, 6, 6 What is the mode and median of this set? Can you determine the mean by using an evening out approach?

  25. Exploring “Mean • Create three data sets that have a mean price of $1.46 How did you think about creating these data sets? Did anyone think about this in a different way?

  26. Possible Solution Sets • $1.36, $1.46, $1.56 • $1.44, $1.45, $1.46, $1.47, $1.48 • $1.38, $1.42, $1.50, $1.54 • $1.41, $1.41, $1.56 How many others did we find?

  27. Shape of the Data • How can we use what we know about the shape of data to help us compare the two sets of data about roofs? Where are the clumps, clusters, gaps, outliers? • Does finding the range and median provide information to help us compare the data sets? How are the data spread out? What is the variability?

  28. Shape of the Data • Do we have any theories or experience that might account for how the data are distributed? What do the data tell us about the way people live? What is the context for the numbers in the data set?

  29. Interpret the Results • Step 4: Use the results from the analyses to make decisions about the original question

  30. Interpret the Results • Requires making sense from the analysis in order to address the question being asked “How do results from the data analysis relate to the original question?” • Reviews each stage of the process • May lead to more questions and investigations

  31. Interpret the Results Examples of “interpreting” questions: • How would we describe people in this group based on the data we collected? • If someone walked into this room, what might we predict would be…? • What can we say about the ways people chose to represent the data? • What might we want to ask next to refine our understanding of…..?

  32. Interpret the Data

  33. ??? Practicing the Process • How can we determine the typical time adults go to sleep at night? • Create an investigation that could help you make an educated guess for this question • Explore each step of the process of statistical investigations to gather data and answer this question

  34. ??? Practicing the Process • Analyze the data and create a representation to support the group’s answer • Interpret the representation for the entire group What other questions might this investigation lead to?

  35. Types of Graphs • Graphs provide a means for: • Communicating and classifying data • Comparing data and displaying mathematical relationships that often cannot be easily seen in numeric form • Integrating geometric ideas with computation skills and classification tasks with numeric understandings

  36. Pictorial Graphs • Use pictures to depict quantities of objects or people • Used with discrete data • Symbols must be the same size and shape • May represent real objects or be more abstract representations (such as shapes) • Legends or keys are used when the object ratio is not one-to-one

  37. Pictorial Graphs

  38. Pictorial Graphs :

  39. Line Plots • Quick, simple way to organize data • Uses X’s (or other same sized symbols) on a single horizontal axis • Includes all numbers within the range of the data set on the axis to show holes and the shape of the data

  40. Line Plots

  41. Venn Diagrams • Considered a graphic organizer • Show all possible mathematical relationships between sets (groups of objects) • Used to describe and compare attributes and characteristics of items

  42. Venn Diagrams Can you identify the three mystery labels?

  43. Bar Graph • Horizontal or vertical bars of uniform width • Compares frequencies of GROUPED, discrete quantities • Axes are labeled to indicate value or frequency

  44. Bar Graph Bar Graph showing ungrouped data

  45. Bar Graph

  46. Bar Graph

  47. Circle Graph • Summarizes data in categories • Used to show size of each category related to the whole (parts to whole) • Visual comparison of fractional parts • Often represented as percentages

  48. Circle Graph How could students who have not yet studied fractions or percents create circle graphs?

  49. Circle and Bar Comparisons Compare the information on these two graphs

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