Data Analysis. Jerry Bellefeuille Matt Nohner

Transcription

1 Data Analysis Jerry Bellefeuille Matt Nohner

2 Overview The following unit will allow students to explore distributions that are not symmetric. It begins with a lesson that reviews dot plots and histograms and leads into skewness and how the measures of central tendencies relate to each other in these skewed distributions. It is followed by lessons that allow students to develop and demonstrate an understanding of simulations in real-world situations via the Cereal Box Problem. Table of Contents Skewness Exercise Page 3 Skewness Exercise Worksheets Page 4 Cereal Box Problem Part 1... Page 7 Cereal Box Problem P1 Worksheets. Page 9 Cereal Box Problem Part 2... Page 12 Cereal Box Problem P2 Worksheets. Page 14 Pre/Post Assessment. Page 17 Scoring Rubric for Assessment. Page 20 Assessment of Instructional Changes... Page 22 2

3 Unit: Data Analysis Standards Addressed: Data Analysis and Probability Describe a data set using data displays, such as box-and-whisker plots; describe and compare data sets using summary statistics, including measures of center, location and spread. Measures of center and location include mean, median, quartile and percentile. Measures of spread include standard deviation, range and inter-quartile range. Know how to use calculators, spreadsheets or other technology to display data and calculate summary statistics. Objective: Students will use descriptive statistics, box plots, histograms and distribution to describe data. Activity: Skewness Exercise In groups of 3-4 and using technology (calculator, Excel ), find the descriptive statistics for each set of data. Symmetric: Mean, median and mode are the same Skewed Right: Mode highest bump Median middle number Mean center of mass (pulled to right by big values) Skewed Left: Mode highest bump Median middle number Mean center of mass (pulled to left by big values) 3

4 Name: Data Set ) Find the mean, median and mode. 3.) Using appropriate intervals, make a histogram. 4.) Label the mean, median and mode on the histogram. 5.) Looking at the histogram, describe how is the data distributed. 6.) Describe how that distribution has affected the mean, median and mode. 4

5 Name: Data Set ) Find the mean, median and mode. 3.) Using appropriate intervals, make a histogram. 4.) Label the mean, median and mode on the histogram. 5.) Looking at the histogram, describe how is the data distributed. 6.) Describe how that distribution has affected the mean, median and mode. 5

6 Name: Data Set ) Find the mean, median and mode. 3.) Using appropriate intervals, make a histogram. 4.) Label the mean, median and mode on the histogram. 5.) Looking at the histogram, describe how is the data distributed. 6.) Describe how that distribution has affected the mean, median and mode. 6

7 Unit: Data Analysis Standards Addressed: Data Analysis and Probability Describe a data set using data displays, such as box-and-whisker plots; describe and compare data sets using summary statistics, including measures of center, location and spread. Measures of center and location include mean, median, quartile and percentile. Measures of spread include standard deviation, range and inter-quartile range. Know how to use calculators, spreadsheets or other technology to display data and calculate summary statistics. Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Understand that the Law of Large Numbers expresses a relationship between the probabilities in a probability model and the experimental probabilities found by performing simulations or experiments involving the model. Use random numbers generated by a calculator or a spreadsheet, or taken from a table, to perform probability simulations and to introduce fairness into decision making. For example: If a group of students needs to fairly select one of its members to lead a discussion, they can use a random number to determine the selection. Objective: Students will use box plots, distributions and descriptive statistics to describe data. Students will use a simulation to represent a problem. Activity: Cereal Box Problem Suppose there are 6 different prizes inside your favorite box of cereal. How many boxes would you expect to have to eat before you collected all 6 prizes? Part 1: Have students write their guesses on the board. Find the measures of central tendency for the guesses. Make a box plot for the guesses. Find inner fence for possible outliers. Use box plot to describe distribution of data. Find standard deviation using technology and use standard deviation to create a confidence interval. Compare standard deviation to the IQR. Part 2: Break class into groups of 3-4. Each group create a simulation for this problem (dice, random number generator, spinner, Excel ). Have each group run their simulation 5 times. Record the group results on a dot plot on the board. What kind of distribution is this? How does this skewness affect the mean, median and mode. Find mean, median, mode and outliers. 7

8 Part 3: Use the simulations to describe different probabilities. P(X = 6) P(X = 20) P(X is at least 25) P(X is less than 15) Part 4: Use the simulations to find the expected value (mean) of the data. How does this relate to our problem? What is the theoretical expected value? Have students work in groups to come up with possible solutions. Expected Mean: 6/6 + 6/5 + 6/4 + 6/3 + 6/3 + 6/2 + 6/1 What is the theoretical probability of getting 6 prizes in exactly 6 boxes? 6! / 6^6 =.015 = 1.5% 8

9 Name: Cereal Box Problem Suppose there are 6 different prizes inside your favorite box of cereal. 1.) How many boxes would you expect to have to eat before you collected all 6 prizes? Record the class guesses here: 2.) Find the measures of central tendency of the guesses. What advantages and disadvantages does each value have? 3.) Make a box plot for the data and find the inner fence to find if there are any outliers. 4.) Find the standard deviation. What does the standard deviation tell you? How does the standard deviation compare to the IQR? 9

10 5.) In your groups, create a simulation to represent this situation. You can use any tool that we have used in class or something new. Write you idea here and have it approved before you conduct you simulation. 6.) Conduct your simulation 5 times and record your data here: 7.) Make a dot plot for the class data. 8.) Name the distribution for the class simulations. Approximate the mean, median and mode. Why did you choose those values? 9.) Find the actual mean, median and mode. How good was your approximation? 10

11 10.) Look back at the class data. Find the following probabilities. P(X=6) P(X=20) P(X at least 25) P(X is less than 15) 11.) Expected value is defined as the sum of the probability of each possible outcome of the experiment multiplied by the outcome value. When rolling a die, each number 1 through 6 has a probability of 1/6. The expected value for rolling a die is: Find the sample expected value for the class simulation. How does this relate to the original problem? 12.) The theoretical expected value is the population mean. What is the theoretical expected value? 13.) What is the theoretical probability of getting 6 different prizes in exactly 6 boxes? 11

12 Unit: Data Analysis Standard Addressed: Data Analysis and Probability Describe a data set using data displays, such as box-and-whisker plots; describe and compare data sets using summary statistics, including measures of center, location and spread. Measures of center and location include mean, median, quartile and percentile. Measures of spread include standard deviation, range and inter-quartile range. Know how to use calculators, spreadsheets or other technology to display data and calculate summary statistics. Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Understand that the Law of Large Numbers expresses a relationship between the probabilities in a probability model and the experimental probabilities found by performing simulations or experiments involving the model. Use random numbers generated by a calculator or a spreadsheet, or taken from a table, to perform probability simulations and to introduce fairness into decision making. For example: If a group of students needs to fairly select one of its members to lead a discussion, they can use a random number to determine the selection. Objective: Students will use box plots, distributions and descriptive statistics to describe data. Students will use a simulation to represent a problem. Activity: Cereal Box Problem (Part 2) Another cereal company has only one kind of prize. The odds of getting a prize are 1 in 6 boxes. The prizes are randomly distributed throughout the boxes of cereal. How many boxes of cereal do you have to open before you get a prize? Part 1: In same groups, create a simulation that represents this problem and conduct the simulation 5 times. Record the results on a line plot on the board. What kind of distribution is it? Where approximately are the mean, median and mode? What are the actual measures of central tendency using the simulations? Make a box plot. How does the box plot relate to the histogram? Part 2: Experimental probability. P(x = 1) P(x greater than 6) P(x is less than 3) What is the expected value for our simulation? 12

13 Theoretical probability. P(x = 1) = (1/6) P(x = 2) = (5/6)(1/6) P(x = 3) = (5/6)(5/6)(1/6) P(x = n) = (5/6)^(n 1)*(1/6) What is the theoretical expected value? E[x] = 1 / p 13

14 Name: Cereal Box Problem Part 2 Another cereal company has only one kind of prize. The odds of getting a prize are 1 in 6 boxes. The prizes are randomly distributed throughout the boxes of cereal. How many boxes of cereal do you have to open before you get a prize? 1.) In your groups, create a simulation to represent this situation. You can use any tool that we have used in class or something new. Write you idea here and have it approved before you conduct you simulation. 2.) Conduct your simulation 5 times and record your data here: 14

15 3.) Make a dot plot for the class data. 4.) Name the distribution for the class simulations. Approximate the mean, median and mode. Why did you choose those values? 5.) Find the actual mean, median and mode. How good was your approximation? 6.) Look back at the class data. Find the following probabilities. P(X=1) P(X=2) P(X=3) P(X at least 5) 15

16 7.) Find the theoretical probability of the following P(X=1) P(X=2) P(X=3) P(X=4) P(X=5) P(X=n) 8.) In which box do you have the greatest probability of getting the prize? Is this what you expected? 9.) What is the theoretical expected value for this problem? 16

17 Data Analysis Unit Assessment Name Please circle the value that best describes your response the following questions with 1 meaning Strongly Disagree and 5 meaning Strongly Agree. Strongly Agree Strongly Disagree 1. I feel confident finding and using the measures of central tendencies I feel confident developing and using simulations to represent real world situations I feel confident describing skewness in different distributions of data I feel confident in describing how skewness affects the measures of central tendencies I feel confident finding and describing outliers in a data set Answer the following questions completely. Describe or show the method(s) you used to obtain your result. Also, justify your result. 6. Find the measures of central tendencies for the following set of data, also draw a box plot of the data. [ 46, 42, 49, 48, 38, 48, 47, 40, 43, 37, 45, 48, 51, 48, 55, 47, 54, 47, 43, 52] 17

18 7. For each graph, label the graphs as symmetric, skewed right, or skewed left. Then, sketch in where you would find the mean, median and mode of the data. Justify your response. a. b. c. 18

19 8. Describe a method to determine whether a value is an outlier or not. Give an example. 9. A pack of trading cards is advertised to have one of 12 rare foil cards in each pack. If there is an equal chance of getting one of the 12 rare cards, how many packs must you purchase in order to collect all 12 rare foil cards. Describe a simulation method you could use to obtain your result and justify your result. 19

20 Scoring Rubric for Pre/Post Test for Data Analysis Test questions 1 through 5 just measure the students confidence levels in various data analysis topics. The scores vary in range from 1 to 5 with a score of 1 reflecting low confidence in a subject and a 5 reflecting high confidence in the topic. The following questions were graded with a rubric: 1. Question 6 a. 5-Student accurately found the measures of central tendency as well as correct values for the box plot. Box plot was drawn to scale and accurate. The student clearly addressed how the values were found. b. 4- Student accurately found the measures of central tendency as well as correct values for the box plot. The box plot was drawn to scale. The student vaguely addressed method on how values were obtained. c. 3-Student found measures of central tendency and other values. A box plot was drawn. Some values may be incorrect. d. 2-Student found some measures of central tendency. Box plot is inaccurate. e. 1-Student attempted the problem. f. 0-No attempt was made by the student. 2. Question 7 a. 5-Student correctly labeled Mean, median, and mode on the graphs. The student s response was clear and concise. b. 4- Student correctly labeled Mean, median, and mode on the graphs. The student s response was vague. c. 3- Student correctly labeled Mean, median, and mode on the graphs. The student s response was not found. d. 2- Student correctly labeled Mean, median, and mode on most of the graphs. e. 1-Student attempted the problem. f. 0-No attempt was made. 3. Question 8 a. 5-Student clearly understands, demonstrates, and communicates a proper method for calculating an outlier. The student gives an example that contains an outlier. b. 4-Student gives an example that has an outlier, and shows some calculation to obtain an outlier. Little or no explanation is given. c. 3-The student gives a method, but no example. d. 2-The student gives an example of an outlier, but no explanation is given. e. 1-The student attempts the problem f. 0-No attempt is made. 20

21 4. Question 9 a. 5-Student gives a simulation method that would lead to a reasonable result. Justification is sound and may be generalized. Possibly has an example. b. 4- Student gives a simulation method that would lead to a reasonable result. Justification is sketchy. No generalizations are made. c. 3-Student made a simulation with a somewhat reasonable result. Justification is limited to this specific incident. d. 2-Student made a simulation with a somewhat reasonable result. No justification was given for the method. e. 1-Student attempted the problem. f. 0-No attempt was made. 21

22 Assessment of Instructional Changes 1. Lesson Plans what are you attempting to change or improve? We are attempting to increase student understanding and confidence of various data analysis topics which include: finding and understanding the purpose of the measures of central tendency, understanding skewness in data and how the measures of central tendency relate to it, understanding how to set up simulations and use them to represent real-world situations. 2. What actual changes are you making? We have changed our instructional delivery from lecture to a constructivist approach using meaningful data and situations. 3. What effect should these changes have? Students will have a stronger and more intuitive knowledge of the measures of central tendency, box plots, histograms, skewness and simulations of real-world situations. The students will also gain confidence in their abilities to analyze data. 4. Formulate hypotheses null and alternative. H o - There will be no change in the mean scores regarding student confidence. H a There will be an increase in the mean scores regarding student confidence. H o - There will be no change in the raw mean test scores. H a - There will be an increase in the raw mean test scores. 5. Experimental design for collecting data. A Pre and Post test will be given to the students. The first 5 questions refer to a student s confidence in their abilities to analyze data. For each question we will use a t-test for paired data using the first pair of hypotheses from above. The remaining 4 questions will be used to determine a student s understanding of the material. These questions will be graded against the rubric given. The students will be given a raw test score total from 0 to 20. The test scores will be compared using a t-test for paired data using the second set of hypotheses. We may also compare a student s confidence level against their actual score for each topic. (i.e. compare the value given in question 1 to the student s score in question 6). We would use a scatterplot and would hope to see a positive correlation. We will also do a two-sample t-test comparing the post test scores of all Teacher A students to all the Teacher B students. 6. Data is collected, reviewed for problems and documented 7. Data Analysis statistical tools you will use to analyze your data: Graphical tools: bar graphs, histograms, box plot, time plot, Statistical tools: mean, median, paired data t-test, two sample t-test 8. Statistical results and statements of conclusions 9. Interpretation in the appropriate context 10. Action and dissemination 22

23 Local students, administrators, parents, community, colleagues, department State conferences (MCTM, Service Coops) National conferences (NCTM), professional chat rooms and journals (Mathematics Teacher) 23