CHAPTER 21 Developing Concepts of Data Analysis Elementary and Middle School Mathematics Teaching Developmentally Ninth Edition Van de Walle, Karp and Bay-Williams Developed by E. Todd Brown /Professor Emeritus University of Louisville

Big Ideas Statistics is its own field distinct from mathematics; one key difference is focus on variability of data in statistical reasoning. Doing statistics involves a four-step process: formulating questions, collecting data, analyzing data, and interpreting results. Data are gathered and organized in order to answer questions about the populations from which the data come. Different types of graphs and other data representations provide different information about the data and, hence, the population from which the data were taken. Measures that describe data with numbers are called statistics. Both graphs and statistics can provide a sense of the shape of the data.

What Does It Mean to Do Statistics? Statistical literacy is needed by all students to interpret the world Statistics and mathematics are two different fields The shape of the data How data is spread out or grouped Characteristics about the data set as whole can be described

Process of doing statistics

Formulate Questions Data collection should be for a purpose, to answer a question. Students should have opportunities to generate their own questions. Student-generated questions make the data collection more meaningful. Student- or teacher-initiated questions should be well defined. Questions that can be answered using statistics.

Data Collection Two types of data- Categorical- data grouped by labels Favorite ice cream, color of car etc. Numerical- data that counts things or measures on a continuous scale How many miles to school, temperature over time, weight of student backpacks.

Sampling Statistics DOES NOT involve gathering from the “whole” population. Uses a representative sample. Sampling takes into consideration- variability Variability means gender, time of day when surveyed, culture Students need to Consider how they will gather data that will include a representative sample Asking- What is the population for your question? Who or what is the subject of your question?

Using Existing Data Sources Print resources Newspaper Almanacs Sports record book Maps Children’s literature Web Resources USDA Economic Research Service Food Consumption Google Public Data Explorer Better World Flux U.S. Census Bureau

Data Analysis: Classifications Making decisions about how to categorize things Attribute materials

Try this one Activity 21. 5 Guess My Rule Materials- use students in the class Directions- Decide on an attribute i.e. wearing jeans, glasses, hat Tell the students “I have a rule.” Call a person to the front that meets your rule and one that does not meet the rule. Call up more students and ask the students to predict whether the person meets or does not meet the rule. Before announcing the rule, give all students a chance to consider the possibilities.

Data Analysis: Graphical Representations Students should be involved in deciding how they want to represent their data. Creating graphs requires skill and precision. Choosing appropriate scale and labels

Data Analysis: Graphical Representations Object graph- small step from sorting, actual articles are used as the graph i.e. types of shoes, favorite fruit Picture graph- moves up a level of abstraction and used drawing or pictures to represent data i.e. book drawing could mean 5 books Bar graph-something is used to represent the data i.e. sticky note, multi-cube

Data Analysis: Graphical Representations Pie charts/ Circle graphs- generally used to show percentages Early pie chart Ratio table with percent and degrees

Continuous Data Graphs Line and dot plots Stem and Leaf plots Histogram Box Plots

Bivariate Graphs Line Graph- coordinate axis for plotting bivariate data Scatter plot - best fit is determined by the line you select that defines the observed relationship

Data Analysis: Measures of Center and Variability

Measures of Center Try this one Activity 21.18 You Be the Judge

Variability Focusing only on outliers or extremes. Considering change over time. Examining variability as the full range of data. Considering variability as the likely range or expected value. Looking at how far points are from the center. Looking at how far off a set of data is from some fixed value.

Variability Range- related to the median- difference between highest and lowest data points Mean absolute deviation-related to mean- tells how the spread of data- high MAD (lot of deviation between data points and mean)

Analysis and Interpretation of Statistics Questioning and assessment should focus on how effectively the graphical representations communicate the findings. Difference between actual facts and inferences that go beyond the data. Questions should focus on the mathematical ideas as well as the statistical ideas. Context of the situation What can be learned or inferred from the data

Ideas for Meaningful Discussion about Interpreting Data What do the numbers (symbols) tell us about our class (or other population)? How do the numbers in this graph (population) compare to this graph (population)? Where are the data “clustering”? How much of the data are in the cluster? How much are not in the cluster? What does the graph not tell us? What might we infer? What new questions arise from these data? What is the maker of the graph trying to tell us?