Data Analysis, Statistics and Probability CHAPTER 17 Tina Rye Sloan To accompany Helping Children Learn Math10e, Reys et al. ©2012 John Wiley & Sons

Instructional programs from pre-k through grade 12 should enable all students to: Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. Select and use appropriate statistical methods to analyze data. Develop and evaluate inferences and predictions that are based on data. Understand and apply basic concepts of probability. NCTM (2000). Principles and Standards for School Mathematics. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Focus Questions What different graphs can be used to represent data? How does the type of graph used relates to the data? What descriptive statistics are appropriate to introduce in the elementary grades? What are some examples of ways they can be introduced? What are some common misconceptions young students have about probability?? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Why Teach Statistics and Probability? 1.Children encounter ideas of statistics and probability outside of school every day. 2.Data analysis, statistics and probability provide connections to other mathematics topics or school subjects. 3.Data analysis, statistics and probability provide opportunities for computational activity in a meaningful context. 4.Data analysis, statistics, and probability provide opportunities for developing critical thinking skills. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Three Steps of Data Analysis 1.Pose a question and collect data. 2.Display collected data. 3.Analyze data and communicate results. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Surveys Survey data result from collecting information. These data may range from a national public opinion poll or observing cars pass the window, to simply tallying the ages of students in a class. Many different sets of real world data are available the U.S. Census Bureau Web site These data can be used for formulating questions, exploring trends, and statistical analysis. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Experiments Experiments may be somewhat more advanced than surveys. When students conduct experiments, in addition to using observation and recording skills, they often incorporate the use of the scientific method Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Simulations Although a simulation is similar to an experiment, random number tables or devices such as coins, dice, spinners, or computer programs are also used to model real-world occurrences. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Analyzing Data: Graphical Organization What do you notice about the graph? How many children prefer apples? What is the favorite fruit? How many different fruits are shown? How many children contributed to the graph? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Quick and Easy Graphing Methods Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Plots A plot is another type of graph used to visually display data. Line plots A line plot may be used to quickly display numerical data with a small range. Stem-and-leaf plots A stem-and-leaf plot is another quick way to display data and provides a quite different representation than when the data are arranged in a line plot or bar graph. Box plots A box plot (also called a box-and- whisker plot) summarizes data and provides a visual means of showing variability—the spread of the data. T Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Plots Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Box Plots Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Picture Graphs Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Bar Graphs and Histograms Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Pie Graphs Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Line Graphs Line graphs are effective for showing trends over time. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Analyzing Data: Descriptive Statistics Measure of Variation Range - the variation or spread or a set of data Measures of Central Tendency or Averages Mode-the value that occurs most frequently Median-the middle value in a set of data Mean-the arithmetic average Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Data Sense Reading the data. The student is able to answer specific questions for which the answer is prominently displayed. For example, “Which player averaged the most points?” Reading between the data. The student is able to find relationships in the data such as comparison, and is able to operate on the data. For example, “How many players had a median less than their mean?” Reading beyond the data. The student is able to predict or make inferences. For example, “Which player had the greatest range? The smallest range? What do these numbers tell you about the player?” Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Misleading Graphs These different graphs of the same data demonstrate how graphs can distort and sometimes misrepresent information. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Communicating Results Once data have been collected, analyzed, and interpreted it is appropriate for students to communicate their findings. Just as in problem solving, students should be encouraged to look back at their results. Communication can help students clarify their ideas during this process. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Probability Probability of an Event Events or Outcomes Certain/Uncertain Impossible Likely/Unlikely Sample Space Randomness Fair/Unfair Independence of Events Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Probability of an Event The probability of tossing a head on a coin is 1/2. The probability of rolling a four on a standard six-sided die is 1/6. The probability of having a birthday on February 30 is 0. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Sample Space The sample space for a probability problem represents all possible outcomes. What is the sample space for this spinner? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Randomness When something is random, it means that it is not influenced by any factors other than chance. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Independence of Events If two events are independent, one event in no way affects the outcome of the other. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Young children often hold common misconceptions about various aspects of probability. The more opportunities you give them to explore a variety of probability notions through hands- on activities, the better they will be able to develop and evaluate inferences and predictions that are based on data and apply basic concepts of probability. Misconceptions about Probability Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Helpful Resources There are a number of excellent Web sites that provide opportunities for children to simulate many trials with dice and spinners, and to integrate them into lessons: Go to the Math Forum Web site at and then go to Math Topics. Under Probability and Statistics you will find lessons focusing on various models, including randomness and probability models. A number of applets (Bar Graph Sorter and Circle Graph) for different grade levels are available. National Library of Virtual Manipulatives at There you will find a number of tools that help to simulate different probability models (Spinners, Coin Tossing, Box Model, and Histogram). National Council of Teachers of Mathematics Illuminations web site to find an array of lessons and tools, such as an Adjustable Spinner and Random Drawing Tool that allow children to gather data via simulations. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Mystery Bags Open your mystery bag, but do not look inside. Shake the contents, reach in and pull out one color tile. Record the color of the tile chosen and return it to the bag. Repeat this process 9 more times, recording each result. Based on these 10 trials, how many of each color of tile are in your bag? (Remember there are a total of 12 tiles in the bag.) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Mystery Bags Conduct 10 more trials using the same process of random selection from the bag. Now, based on 20 trials, how many of each color does your bag contain? Is this different from your previous guess? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Mystery Bags Based on your experiment, what is the probability that the tile chosen will be red? yellow? blue? green? Open the bag and look at the contents. How well did your experimentation reveal the actual contents of the bag? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012