Chapter 17 To accompany Helping Children Learn Math Cdn Ed, Reys et al. ©2010 John Wiley & Sons Canada Ltd.

Guiding Questions 1.How do data analysis, statistics, and probability help children develop critical thinking skills? 2.How can you help children develop skills in analyzing data? 3.What descriptive statistics are appropriate to introduce in the elementary grades? What are some examples of ways they can be introduced? 4.What are some common misconceptions young students have about probability?

Why Teach Statistics and Probability? Children encounter ideas of statistics and probability outside of school every day. Data analysis, statistics and probability provide connections to other mathematics topics or school subjects. Data analysis, statistics and probability provide opportunities for computational activity in a meaningful context. Data analysis, statistics, and probability provide opportunities for developing critical thinking skills.

Three Steps of Data Analysis 1.Pose a question and collect data. 2.Display collected data. 3.Analyze data and communicate results.

Formulating Questions and Collecting Data There is real benefit to having students identify their own questions or problems, for they take ownership of the investigation and their motivation will be high. Once students have identified a suitable question, they will need to plan how to collect the data to answer the question. Students must clearly communicate with others to negotiate the details of the investigation.

Surveys Survey data result from collecting information. These data may range from a national census poll or observing cars pass the window, to simply tallying the ages of students in a class. A wide array of data are available from the Statistics Canada website.

Surveys (cont.) The actual data used depends on student interest and maturity. – To sharpen data-collecting techniques, students may consider the following questions: What questions will this survey answer? Where should I conduct my survey? When should I conduct the survey?

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

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. Sampling is another method of data collection that students can simulate – The whole group you are studying is called the population. – A sample is a sub-set of the population.

Analyzing Data: Graphical Organization What do you notice about these graphs? After data has been collected, graphs are often used to display data and help others digest the results

Quick and Easy Graphing Methods

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.

Plots Stem-and-leaf plots divide the data into tens and ones and arrange it in numerical order.

Plots Box plots (also called a box-and-whisker plot) summarize data and provide a visual means of showing variability— the spread of the data.

Picture Graphs

Bar Graph Bar graphs are used mostly for discrete or separate and distinct data; the bars represent these data

Histograms Although a histogram looks like a bar graph, a histogram is used with continuous data, not discrete data. Therefore the data are represented with connected bars, each representing an interval.

Pie Graphs A pie graph is a circle representing the whole, with wedges reporting percentages of the whole.

Line Graphs Line graphs are effective for showing trends over time. The data are continuous rather than discrete. Data can occur between points with continuous data. Change is accurately represented with linear functions rather than some other curve.

Graphical Roundup Each type of graph deserves instructional attention as students examine ways to display their data. Children need experience constructing them and interpreting information that is represented. The availability of graphing calculators and graphing programs allows for easy construction of a variety of graphs. It is vital that children are taught how to interpret and understand graphs.

Analyzing Data: Descriptive Statistics Another way to analyze data is to use descriptive statistics.

Measure of Variation: Range Range is the variation of a set of data or how spread out the data are. Once the range has been introduced in the elementary grades, middle school students often learn to measure variability with variance or standard deviations.

Measures of Central Tendency or Averages Any number that is used to describe the centre or middle of a set of values is called an average of those values. Many different averages exist, but we will look at the three main ones: – Mode: the value that occurs most frequently in a collection of data – Median: the middle value in a set of ordered data – Mean: the arithmetic average

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?”

Misleading Graphs These different graphs of the same data demonstrate how graphs can distort and sometimes misrepresent information.

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.

Probability Probability is used to predict the chance of something happening. The terms chance and probability are often applied to those situations where the outcome cannot be completely determined in advance.

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. In the previous examples, tossing a head, rolling a four, and having a birthday on February 30 are events or outcomes.

Probability of an Event Probability assigns a number (from zero to one) to an event. Long before children are ready to calculate probability of specific events, it is important that you introduce and discuss terms such as “certain,” “uncertain,” “impossible,” “likely,” and “unlikely.”

Sample Space The sample space for a probability problem represents all possible outcomes. What is the sample space for this spinner?

Randomness When something is random, it means that it is not influenced by any factors other than chance. It is helpful to have children discuss randomness in a specific context such as drawing a name out of a hat. – Through this notions of fair and unfair will be developed in a meaningful way.

Independence of Events If two events are independent, one event in no way affects the outcome of the other. If a coin is tossed, lands on heads, and then is rolled again, it is still equally likely to land on heads or tails. If the same result occurs repeatedly, students begin to doubt what they have been taught about probability and assume that a different outcome is inevitable.

Misconceptions about Probability 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.

A Probability Activity The following slides contain a probability activity which demonstrates to children how the more times a sample is taken, the closer you will get to knowing the actual population. This activity is suitable for students at all grade levels.

Mystery Bags Open your mystery bag, but do not look inside. Shake the contents, reach in and pull out one colour tile. Record the colour 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 colour of tile are in your bag? (Remember there are a total of 12 tiles in the bag.)

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 colour does your bag contain? Is this different from your previous guess?

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?

Copyright Copyright © 2010 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (The Canadian Copyright Licensing Agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information contained herein.