1、Teaching Data Analysis as an Investigative Process with Census at School,Rebecca Nichols and Martha Aliaga American Statistical Association,US Census at School Program,Free international classroom project that engages students in grades 4-12 in statistical problem solvingStudents complete an online
2、survey, analyze their class census data, and compare their class results with random samples of participating students in the United States and other countries. The project began in the United Kingdom in 2000 and includes Australia, Canada, New Zealand, South Africa, Ireland, Japan, and now the Unit
3、ed States. Teach statistical concepts in the Common Core Standards, measurement, graphing, data analysis, and statistical problem solving in context of students own data and data from their peers in the participating countrieswww.amstat.org/censusatschool,Statistical Problem Solving,Guidelines for A
4、ssessment and Instruction in Statistics Education (GAISE) Report: A Pre-K12 Curriculum Framework 1. Formulate Questions Clarify the problem at hand Formulate one (or more) questions that can be answered with data2. Collect Data Design a plan to collect appropriate data Employ the plan to collect the
5、 data3. Analyze Data Select appropriate graphical and numerical methods Use the methods to analyze the data4. Interpret results Interpret the analysis (in context) Relate the interpretation to the original questionSource: www.amstat.org/education/gaise,Common Core State Standards for Mathematics,Sou
6、rce: corestandards.org,Measurement & Data Grades 4 & 5,Grade 4 (4.MD)Measurement and Data Strand Common Core State Standards Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. Represent and interpret data. Grade 5 (5.MD) Measurement and Data Con
7、vert like measurement units within a given measurement system. Represent and interpret data.The U.S. Census at School questionnaire includes measurement questions (measuring height, arms pan, and foot length in centimeters, finger length in millimeters, etc.) and opportunities to represent and inter
8、pret real student data.Source: corestandards.org,Statistics & Probability Grade 6 (6.SP),Develop understanding of statistical variability1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How o
9、ld am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students ages. 2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, s
10、pread, and overall shape. 3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. Source: corestandards.org,Statistics & Probability Grade 6 (6.SP),Develop under
11、standing of statistical variability1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical
12、 question because one anticipates variability in students ages. 2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. 3. Recognize that a measure of center for a numerical data set summarizes al
13、l of its values with a single number, while a measure of variation describes how its values vary with a single number. Source: corestandards.org,Statistics & Probability Grade 6 (6.SP),Summarize and describe distributions 4. Display numerical data in plots on a number line, including dot plots, hist
14、ograms, and box plots. 5. Summarize numerical data sets in relation to their context, such as by: Reporting the number of observations. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. Giving quantitative measures of center (medi
15、an and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. Relating the choice of measures of center and variabilit
16、y to the shape of the data distribution and the context in which the data were gathered. Source: corestandards.org,Statistics & Probability Grade 7 (7.SP),Use random sampling to draw inferences about a population 1. Understand that statistics can be used to gain information about a population by exa
17、mining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.2. Use data from a random sample to draw inferen
18、ces about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner o
19、f a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. Source: corestandards.org,Statistics & Probability Grade 7 (7.SP),Draw informal comparative inferences about two populations 3. Informally assess the degree of visual overlap of two nume
20、rical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twic
21、e the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. 4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations
22、. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. Note: Grade 7 also includes probability standards Source: corestandards.org,Statistics & Probability Grade 8 (8.SP),Investigate patter
23、ns of association in bivariate data 1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
24、2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 3. Use the equation
25、 of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.
26、5 cm in mature plant height. Source: corestandards.org,Statistics & Probability Grade 8 (8.SP),Investigate patterns of association in bivariate data4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-wa
27、y table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class o
28、n whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? Source: corestandards.org,Statistics & Probability High School Interpreting Categorical & Quantitative Data (S-ID),Summa
29、rize, represent, and interpret data on a single count or measurement variable1. Represent data with plots on the real number line (dot plots, histograms, and box plots). 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile ran
30、ge, standard deviation) of two or more different data sets. 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 4. Use the mean and standard deviation of a data set to fit it to a normal distributio
31、n and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. Source: corestandards.org,Statistics & Probability High School Interpreting Categorical & Qu
32、antitative Data (S-ID),Summarize, represent, and interpret data on two categorical and quantitative variables 5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative f
33、requencies). Recognize possible associations and trends in the data. 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given func
34、tions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. Source: corestandards.org,Statisti
35、cs & Probability High School Interpreting Categorical & Quantitative Data (S-ID),Interpret linear models7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 8. Compute (using technology) and interpret the correlation coefficient of a
36、 linear fit. 9. Distinguish between correlation and causation. Source: corestandards.org,Statistics & Probability High School Making Inferences & Justifying Conclusions (S-IC),Understand and evaluate random processes underlying statistical experiments1. Understand statistics as a process for making
37、inferences about population parameters based on a random sample from that population. 2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result
38、of 5 tails in a row cause you to question the model? Source: corestandards.org,Statistics & Probability High School Making Inferences & Justifying Conclusions (S-IC),Make inferences and justify conclusions from sample surveys, experiments, and observational studies3. Recognize the purposes of and di
39、fferences among sample surveys, experiments, and observational studies; explain how randomization relates to each. 4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. 5. Use data from a rand
40、omized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. 6. Evaluate reports based on data. Source: corestandards.org,US Census at School Program,www.amstat.org/censusatschool,US Census at School Program,Free international classroom pr
41、oject that engages students in grades 4-12 in statistical problem solvingStudents complete an online survey, analyze their class census data, and compare their class results with random samples of participating students in the United States and other countries. The project began in the United Kingdo
42、m in 2000 and includes Australia, Canada, New Zealand, South Africa, Ireland, Japan, and now the United States. Teach statistical concepts in the Common Core Standards, measurement, graphing, data analysis, and statistical problem solving in context of students own data and data from their peers in
43、the participating countrieswww.amstat.org/censusatschool,US Census at School Program,Students complete a brief online survey (classroom census) 13 international questions plus additional U.S. questions 15-20 minute computer sessionAnalyze your class results Use teacher password to gain immediate acc
44、ess to class data Formulate questions of interest that can be answered with Census at School data, collect/select appropriate data, analyze the data with appropriate graphs and numerical summaries, internet the results, and make appropriate conclusions in context relating to the original questionsCo
45、mpare your class with samples from the U.S. and other countries Download a random sample of Census at School data from U.S. students Download a random sample from participating international studentsInternational lesson plans are available, along with instructional webinars and other free resourcesw
46、ww.amstat.org/censusatschool,US Census at School Program,www.amstat.org/censusatschool,US Census at School Student Section,www.amstat.org/censusatschool,US Census at School Student Section,www.amstat.org/censusatschool,US Census at School Teacher Section,www.amstat.org/censusatschool,US Census at Sc
47、hool Resources,www.amstat.org/censusatschool,US Census at School Random Sampler,www.amstat.org/censusatschool,International Random Sampler,www.amstat.org/censusatschool,Statistical Investigations - Census at School,Formulate statistical questions of interest that can be answered with the Census at S
48、chool data. Collect/select appropriate Census at School data and write down the variable names and type for this investigation. Analyze the data. Include appropriate graphs and numerical summaries for the corresponding variables. Interpret the results and make appropriate conclusions in context. Be
49、sure to justify your results using your graphs and numerical summaries and relate your interpretation to the original question.For a demonstration of this process and software resources (some free) to analyze the data, watch the Census at School webinars posted under Resources at www.amstat.org/cens
50、usatschool.,Formulating a Statistical Question,Common Core Standards Grade 6 (6.SP) 1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old
51、 are the students in my school?” is a statistical question because one anticipates variability in students ages. A well-written statistical question anticipates answers that will vary and includes: Population of interest Measurement of interestExample from Common Core: How old are the students in my school? Population of interest: Students in my school Measurement: Age (measured in years) Student ages will vary,
