Dana Cartier Illinois Center for School Improvement Julia Brenson

The New Illinois Learning Standards for Grades 6 - 8 Statistics and Probability
Dana Cartier Illinois Center for School Improvement Julia Brenson Lyons Township High School Tina Dunn This PowerPoint is intended to be used as a walk-through of the New Illinois Learning Standards for Statistics and Probability allocated to grades 6. In addition to providing example activities, guidance of key points to emphasize is provided for many of the standards.

The New Illinois Learning Standards
Agenda Resources Available Through ISBE Sixth Grade – Shape, Center, Spread Seventh Grade – Random Sampling for Inference and Simulation for Probability Eighth Grade – Bivariate Data

ILStats All materials from this session are available at this website. This website is currently under construction, but please keep checking back for more information about the Statistics Standards.

Sixth Grade

Statistics Standards for 6th Grade
Focus PBA EOY Calculator 6.SP.A.1 Additional X No 6.SP.A.2 6.SP.A.3 X 6.SP.B.4 Yes 6.SP.B.5 The PARCC Evidence Tables indicate that all statistics standards for sixth grade will be tested on the End-Of-Year Assessment. Please be sure to read the PARCC Evidence Tables for both the Performance Based Assessment (PBA) and End-Of-Year Assessment (EOY) for additional information about the assessment of these standards. 6.SP.A.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 are the students in my school?” is a statistical question because one anticipates variability in students’ ages. 6.SP.A.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. 6.SP.A.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. 6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 6.SP.B.5 Summarize numerical data sets in relation to their context, such as by: 6.SP.B.5a Reporting the number of observations. 6.SP.B.5b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. 6.SP.B.5c Giving quantitative measures of center (median 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. 6.SP.B.5d Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered

6.SP.A.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. 6.SP.A.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. 6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

6.SP.B.5 Summarize numerical data sets in relation to their context, such as by: 6.SP.B.5a Reporting the number of observations. 6.SP.B.5b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. 6.SP.B.5c Giving quantitative measures of center (median 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. 6.SP.B.5d Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered

Statistics Standards for 6th Grade Types of Graphs
Gaps Categories It is easy to confuse bar graphs with histograms. While both of them have bars, a bar graph is used to show the frequency or relative frequency(proportion) for categorical data. A histogram, like a dot plot and box plot, is a number line graph. A count (or proportion) of the numerical data that falls within an interval are represented by the height of the bar in a histogram. In a bar graph, we typically leave gaps between the bars. In a histogram, we do not leave gaps between the bars. No Gaps

Statistics Standards for 6th Grade Shape, Center and Spread
BIG IDEAS: When describing distributions, we talk about Shape, Center, and Spread in the context of the data. Try to use real life data rather than made up data sets whenever possible. In sixth grade, students first learn to describe a distribution in terms of its shape, center and spread relating each to the context of the situation. They learn to graph distributions using the number line graphs: dot plots, box plots and histograms. The shape of the distribution is used to guide decisions about which measures of center (mean vs. median) and spread (range, interquartile range, and mean absolute deviation – MAD) are appropriate for a given situation. Students should learn to interpret mean, median, mean absolute deviation (MAD), range and interquartile range (iqr) in the context of the problem. Students will have a stronger connection to activities if the data is from their own class, or if the data comes from an interesting real-life situation. For class data, first formulate a question of interest and then walk through the four steps of a statistical investigation. Class questions might include: How many hours of sleep did you get last night? How many jelly beans can each student hold in his hand? If a student grabs a handful of jelly beans, what proportion of the jelly beans will be black? What is a typical vertical jump for students? Do the boys have a greater vertical jump than the girls? How fast do students complete the memory test on the Census at School website ( Do the reaction times improve from the first trial to the fifth trial on the reaction time test for each of the students in our class ( The reaction test gives times to the nearest thousandth of a second. Teachers may want to have students convert their times to milliseconds. (.524 seconds = 524 ms) 7) When asked “What is the age of our current president?” will the answers given by students in our 6th grade class be closer to the correct age than students from a 5th grade class? Questions can also be asked that require collection data from the internet or a news source. For example: What is the average amount of snowfall per season, and how does the snowfall this winter compare to the average? What is the average length of reign for a British monarch (king/queen)? What is the shortest and what is the longest amount of time that a British monarch ruled? What is the typical age at which a U.S. president is elected? How old are players on the current Chicago Blackhawks team? As teachers, we may need to continue to shape the questions above. For example, if students are interested in how many states their classmates have typically been to, can they count a state where their only visit was when they changed planes? Students will also need guidance on respectful data use. We never used data to embarrass another student! Teachers should consider whether a statistical question is likely to be too embarrassing to some of the students in the class. Often, the very questions that students are most interested in may also be the questions that are most embarrassing to answer. Asking the students to record their responses anonymously on slips of paper which can then be recorded may allow more sensitive questions to be asked without making students uncomfortable.

Statistics Standards for 6th Grade Shape of the Distribution
Approximately Symmetrical Skewed Students should be able to describe the shapes of distributions as being skewed (having a longer tail on the right or left) or approximately symmetric. When describing a distribution, we describe not only its overall shape, but also any interesting or unusual features such as unusually large or small values, clusters or peaks. Note that the term mode does not appear in the Common Core Standards. unusually large value

Statistics Standards for Algebra I/Math I Shape of the Distribution
What would the shape be for the distribution of salaries of the 2013 Chicago Cubs? The distribution of salaries for the 2013 Chicago Cubs is skewed. Most players made less than $2 million. There are two players that made an exceptionally large salaries. (Alfonso Soriano made $19 million and Edwin Jackson made $13 million.)

Statistics Standards for 6th Grade Measures of Center
Mean = 𝑠𝑢𝑚 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑖𝑛 𝑑𝑎𝑡𝑎 𝑠𝑒𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑑𝑎𝑡𝑎 𝑠𝑒𝑡 Median = the center most value when observations in the data set are ordered BIG IDEA: The median is a better measure of center when the data is skewed. The median is resistant to unusual values, and therefore provides a better estimate of a typical value (the center of the distribution) when the distribution is skewed.

Statistics Standards for Algebra I/Math I Measures of Central Tendency
What was a typical salary for a baseball player on the 2013 Chicago Cub Team? Median = $1,550,000.00 Mean = $3,485,024.20 What is the better measure of center for this data? Why?

6th Grade & Algebra I / Math I Measures of Center
Demonstration: Comparing the Mean and Median NCTM Illuminations Mean and Median Applet This demonstration illustrates the impact of unusually large (or small) values on the value of the mean, while the median’s value remains unchanged. Students are shown a plot of a set of five numbers using either the applet from ICTM listed in the slide above or Fathom. While the teacher slowly drags the maximum to higher and higher values, students observe the mean increasing while the median remains unchanged. NCTM Illuminations also offers graphing tools for creating box plots, histograms, etc. at See Interactives. Fathom is a statistical software tool which can be used to display data in dot plots, bar graphs, and histograms as well as calculate mean, median, standard deviation, and the linear regression line of the data. Fathom is currently distributed by McGraw-Hill.

6th Grade & Algebra I / Math I Measures of Center
Comparing the Mean and Median This demonstration illustrates the impact of unusually large (or small) values on the value of the mean, while the median’s value remains unchanged. Students are shown a plot of a set of five numbers using either the applet from ICTM listed in the slide above or Fathom. While the teacher slowly drags the maximum to higher and higher values, students observe the mean increasing while the median remains unchanged. NCTM Illuminations also offers graphing tools for creating box plots, histograms, etc. at See Interactives. Fathom is a statistical software tool which can be used to display data in dot plots, bar graphs, and histograms as well as calculate mean, median, standard deviation, and the linear regression line of the data. Fathom is currently distributed by McGraw-Hill.

The Mean as Fair Share Dave, Sandy, Javier, and Maria have 12 cookies. How many cookies will each student have if each student receives a fair share? 3 3 12 3 3

The Mean as Fair Share What would each student’s fair share be if there are: 14 cookies? 9 cookies? 7 cookies? ? ? This activity can be simulated by giving groups of four students sets of paper cookies and a pair of scissors. Students fairly distribute the cookies, and then take any remaining cookies and determine how they can be cut so they are shared fairly four ways. See Grade 6 – Mean as a Fair Share - Cookie Page. ? ?

From the PARCC Grade 6 EOY Evidence Table Evidence Statement Key 6.SP.3 Rate the following statement as True/False/Not Enough Information. “The average height of trees in Watson Park is 65 feet. Are there any trees in Watson Park taller than 65 feet?”

Statistics Standards for 6th Grade Measures of Spread
Range = maximum value – minimum value Interquartile Range = Quartile3 – Quartile1 Interquartile Range (iqr) is the spread of the middle 50% of the data. Mean Absolute Deviation (MAD) = sum of the distances of each data value from the mean divided by the total number of observations. Big Idea: The mean absolute deviation (MAD) is the average distance (deviation) of data values from the mean. If a distribution is highly skewed, the median is a better measure of center and the interquartile range (iqr) is a better measure of spread. The iqr is discussed with box plots.

The Mean as a Balance Point (An Introduction to MAD) From Engage NY Grade 6 Module 6 Lesson 7 Sabina wants to know how long it takes students to get to school. She asks two students how long it takes them to get to school. It takes one student 1 minute and the other student 11 minutes. She thinks the mean is the balance point. What do you think? Engage NY features modules by grade or course designed to engage students in the Common Core State Standards. Roxy Peck, one of the leading statistics educators in the country, and a team of other top educators have created an entire unit for the statistics standards in Algebra I that is featured on the Engage NY website. The unit includes activities with explanations that can be used for class, homework and assessments. For more information about Roxy Peck and her contributions to statistics education, see the article Interview With Roxy Peck by Allan Rossman that appeared in Journal of Statistics Education (Volume 20, November 2012). Retrieved from

Introducing Deviations A deviation is the distance of a piece of data from the mean. A value that is below the mean has a negative deviation. A value above the mean has a positive deviation. The deviation of 1 to the mean is 1 – 6 = - 5 The deviation of 11 to the mean is 11 – 6 = 5 The sum of the deviations always equals zero. Questions: 1) What is the deviation from the mean for each of the pennies? 2) What is the sum of these two deviations?

Introducing Deviations Sabrina wants to know what happens if there are more than two data points. Suppose there are three students. One student lives 2 minutes from school, and another student lives 9 minutes from school. If the mean time for all three students is 6 minutes, she wonders how long it takes the third student to get to school. She tapes pennies at 2 and 9. - 4 + 3 +1 Questions: 1) Where should the third penny be placed to balance the ruler? 2) How can we use deviations to check this answer?

Introducing Mean Absolute Deviation (MAD) Activity: School Night Sleep How many hours of sleep do sixth graders get on a school night? Let’s make some predictions: Typically, how many hours of sleep do you think a sixth grader gets? How much will the number of hours of sleep vary if we asked a group of ten sixth graders? What do you predict will be the fewest hours? What do you predict will be the most hours? Before looking at the data, have students make predictions about the shape, center and spread.

On Monday morning, Carlos asked ten of his sixth grade classmates how many hours of sleep they usually get on school nights. He then created a dot plot of their answers. Answers: Students may look at the dot plot and approximate that the center is somewhere between 8 and 9 hours of sleep on a school night or may quickly find the median (8.5 hours). The number of hours of sleep on a school night had a range of 4.5 hours. The distribution of number of hours of sleep on a school night is skewed right (positive). Questions: Looking at the dot plot above, typically how much sleep did the ten sixth graders get on a school night? How much did the amount of sleep vary? What is the shape of this distribution?

Let’s look at another method of measuring the spread of the data. Mean Absolute Deviation (MAD) The mean absolute deviation (MAD) is the average distance of the data from the mean. We find MAD by doing these steps: Calculate the mean. Find the deviation for each data value. Take the absolute value of each deviation. Find the average of these absolute deviations (distances). Coherence across a grade level improves focus by linking supporting topics with major topics. In this case, MAD reinforces the work done in standard 6.NW.C.7 - the introduction of absolute value. (See 6.NS.C.7 below.) 6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars. 6.NS.C.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than -30 dollars represents a debt greater than 30 dollars.

Calculating Mean Absolute Deviation (MAD) Student Sleep Hours on School Night (Hours) Deviation Hours - Mean Absolute Deviation |Hours - Mean| Rachel 10 10 – 8.75 = 1.25 Gerty 9 9 – 8.75 = 0.25 Steve 11 Juan Michael 8 Josie 6.5 Philip Sergio Catherine Grace Total 87.5 Mean = 𝑇𝑜𝑡𝑎𝑙 ℎ𝑜𝑢𝑟𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 = = 8.75 hours 1.25 0.25 2.25 2.25 0.25 0.25 -0.75 0.75 MAD = 𝑇𝑜𝑡𝑎𝑙 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 = = hours -2.25 2.25 1.25 1.25 -0.75 0.75 -0.75 0.75 -0.75 0.75 0.00 10.5

Interpreting Mean Absolute Deviation (MAD) Mean = 8.75 hours MAD = 1.05 hours mean - MAD mean + MAD mean Notice that 6 of the 10 students’ number of hours of sleep fall within one MAD either side of the mean. The number of hours of sleep on a school night for these ten sixth graders varies1.05 hours, on average, from the mean of 8.75 hours.

Mean Absolute Deviation (MAD) Ten sixth graders are asked to report the number of hours of sleep they typically get on a school night. Their hours of sleep are shown on the dot plot below. Questions: What is the mean number of hours of sleep on a school night for these ten sixth graders? What is the median? How much variability is there amongst the ten sixth graders? What is the value of MAD for this data? Answers: 10 hours These ten sixth graders did not vary at all in the number of hours of sleep on a school night. MAD = 0 hours.

Statistics Standards for 6th Grade Shape, Center and Spread
Activity What’s Your Age? See 6.SP What’s Your Age?

Activities: Mean, Median, Mode, and Range ( Candy Bar ( How Long is 30 Seconds Statistics Education Web (STEW) ( The Mean as a Balance Point Engage NY Grade 6 Module 6 ( teacher-materials.pdf) What’s Your Age?

Seventh Grade

Focus PBA* EOY Calculator 7.SP.A.1 Supporting X Yes 7.SP.A.2 7.SP.B.3 Additional 7.SP.B.4 7.SP.C.5 7.SP.C.6 7.SP.C.7 X 7.SP.C.8 * See Evidence Statement 7.D.3 Micro-models from the PARCC Evidence Table – Grade 7 PBA There could be a question on the PBA that connects a mathematical concept such as proportional reasoning with the statistics standards. Probability, for example, is a topic that would connect well with the concept of proportional reasoning. Evidence Statement from PARCC Evidence Tables - Grade 7 PBA 7.D.3 Micro-models: Autonomously apply a technique from pure mathematics to a real-world situation in which the technique yields valuable results even though it is obviously not applicable in a strict mathematical sense (e.g., profitably applying proportional relationships to a phenomenon that is obviously nonlinear or statistical in nature). Content Scope: Grade 7 knowledge and skills articulated in the Evidence Statements on the PBA (excludes the Reasoning Evidence Statements which are those that start with 7.C). 7.SP.A.1 Understand that statistics can be used to gain information about a population by examining 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. 7.SP.A.2 Use data from a random sample to draw inferences 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 of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. 7.SP.B.3 Informally assess the degree of visual overlap of two numerical 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 twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. 7.SP.B.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. 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. 7.SP.C.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. 7.SP.C.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. 7.SP.C.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. 7.SP.C.7a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. 7.SP.C.7b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? 7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 7.SP.C.8a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. 7.SP.C.8b Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. 7.SP.C.8c Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

7.SP.A.1 Understand that statistics can be used to gain information about a population by examining 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. 7.SP.A.2 Use data from a random sample to draw inferences 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 of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

Statistics Standards for 7th Grade Random Sampling to Draw Inferences About a Population
Activity: Gettysburg Address Part I Judgment Sample Part II Simple Random Sample  Sampling Distribution Teacher Note: A statistic is a numerical summary computed from a sample. A parameter is a numerical summary computed from a population. A statistic will vary depending on the sample from which it was calculated, but a population parameter is a constant value that does not change. Sixth grade students do not need to know the term parameter of distinguish between a statistic and a parameter. However, teachers should be aware of the difference between these two terms. The term parameter is introduced in high school.

Gettysburg Address Judgment Sample First ask students to take a quick look at the population of 268 words and select 5 words that they think form a representative sample of the length of words found in the Gettysburg Address. This is a judgment sample. Students record the five words and the number of letters in each word in the table provided. After calculating the mean of the sample, each student records his mean on the class dot plot on the chalkboard. We call this a judgment sample, because we are asking students to select 10 words, using their own judgment, that are representative of the population of all words in the Gettysburg Address. Generally, students tend to select a sample of words with a mean number of letters that is greater than the actual population mean. Their own biases cause them to overestimate the typical word length.

How do we ensure that we select a sample that is representative of the population? We choose a method that eliminates the possibility that our own preferences, favoritism or biases impact who (or what) is selected. We want to give all individuals an equal chance to be chosen. We do not want the method of picking the sample to exclude certain individuals or favors others. One method that helps us to avoid biases is to select a simple random sample. If we want a sample to have n individuals, we use a method that will ensure that every possible sample from the population of size n has an equal chance of being selected. We do not want the method of picking the sample to produce samples with characteristics that are different than the population characteristics.

Which of the following would produce a simple random sample of size 6 from the population of all students in our classroom? A. Select the first 6 students that enter the classroom. B. Put every student’s name in a hat, mix and draw 6 names. C. The classroom has 6 tables with three students per table. Randomly select two tables. The students at these two tables are the sample. D. The classroom has 6 tables of students. Randomly select one student from each table.

Back to Gettysburg Address Simple Random Sample
Statistics Standards for 7th Grade Random Sampling to Draw Inferences About a Population Back to Gettysburg Address Simple Random Sample Use a random number generator or a random digits table to select a simple random sample of size 5 from the population of 268 words. Be sure to show students how to select random samples using different methods. Drawing numbers from a hat (or bag), a random number table, and a random number generator are three methods that can be used to select a random sample.

Random Digits Table Suppose, for example that we wanted a sample of size 5. There are 268 words. First select a row to use in the table. Select three digits at a time, letting 001 represent 1, 002 represents 2, and so on. Skip 000 and numbers that are greater than Skip repeats. Our Sample: 32, 148, 238, 128, 104

Random Number Generator Random sample of 5 numbers representing the 5 words to be selected. For this activity, agree with your students how many words will be in a sample. Five or ten both work well, though any number in between may also be used. The random number generator above is shared with permission from Beth Chance and Allan Rossman. This applet can be found at

Gettysburg Address Sampling Words – Permission to share this applet was given by Beth Chance and Allan Rossman. Number of Letters for all Words in the Population Population Mean Last random sample of size 5 that was selected. Sample Mean

Gettysburg Address 100 random samples of size 5 mean = 4.46 Additional Information for Teachers Only: The distribution above is called a sampling distribution. It is the distribution of all sample means. The shape of the sampling distribution is approximately symmetrical. (In high school, Algebra II and Math III students learn that the distribution is approximately normal.) While there is a great deal of variability in the sample means, the behavior of the sampling distribution is very predictable. As more and more random samples are selected from the population, the mean (center) of this sampling distribution gets closer and closer to the population mean. This allow us to make inferences about the population mean. Seventh grade students do not need to know that this is called a sampling distribution. The big concept is that we want students to learn that, by taking multiple random samples of a population, the mean of all the sample means provides a good estimate of the population mean. This allows us to make inferences (draw conclusions) about the mean of the population.

Sampling Distribution
Statistics Standards for 7th Grade Random Sampling to Draw Inferences About a Population Gettysburg Address 500 random samples of size 5 Sampling Distribution mean = 4.313 The Sampling Word applet allows us to demonstrate the long term behavior: as more and more samples of size 5 are selected from the population, the mean of the sampling distribution gets closer and closer to the population mean.

7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 7.SP.C.8a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. 7.SP.C.8b Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. 7.SP.C.8c Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

Statistics Standards for 7th Grade Chance Processes and Probability Models
Example: Tree Diagram Michael and Gita would like to have three children. What is the probability that all three children will be boys? Third Child Possible Outcomes BBB (0.5)(0.5)(0.5) = 0.125 BBG BGB BGG GBB GBG GGB GGG First Child Second Child 0.5 B B 0.5 0.5 G B 0.5 0.5 0.5 B G 0.5 7.SP.C.7a & 8 This is a good basic illustration of a tree diagram. The tree diagram can help students determine all possible outcomes. Other Questions What is the probability that that Mike and Gita have no boys? What is the probability that Mike and Gita have 2 boys and a girl in any order? If the first child is a boy, is the second child more likely to be a girl? G 0.5 B 0.5 0.5 B G 0.5 G 0.5 0.5 B G 0.5 G

Three Children Continued… Another way to look at this problem is to create a list of all possible outcomes (the sample space). (B, B, B) (G, B, B) (B, B, G) (G, B, G) (B, G, B) (G, G, B) (B, G, G) (G, G, G) This is a uniform distribution in which every outcome has an equal chance of occurring. There are 8 outcomes and each outcome has a 1/8 chance of occurring. We can now answer questions like: What is the probability of the couple having 3 boys? What is the probability of having one boy? (B, B, B) 1/8 Three outcomes have one boy: (B, G, G), (G, B, G) and (G, G, B). The probability of having one boy in three children is 3/8.

Activity: Blood Type A If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? This is the example mentioned in standard 7.SP.C.8c.

Activity: Blood Type A Using a random digits table, let 1, 2, 3, 4 represent having type A blood. 0,5,6,7,8,9 represent not having type A blood. Select a row. Count how many digits it takes to reach a 1,2,3, or 4. Record this count with a tally mark in a table. Repeat many times to determine the long-run behavior.

Row Continue on to simulate the long run behavior or combine results with classmates. Questions 1) How could you use a random digit table to simulate the outcomes of heads or tails in the toss of a fair coin? Let odd numbers be tails and even numbers be heads or let the digits 1 – 5 be tails and 6-9,0 be heads. 2) What is the probability of getting at least 4 heads in a row? How could the random digits table be used to help answer this question?

Blood Type A - Part II Tree Diagram Let A = the event that a donor has blood type A Let O = the event that a donor has some other blood type.

Standard 7.SP.8c covers using simulation only to estimate probabilities such as the probability that it will take at least 4 donors to find a donor with blood type A. This standard does not require that students use a tree to determine this probability. Creating trees with 4 sets of branches (4 stages) to answer the question about blood types as a guided activity allows students to see what the actual, long-run (theoretical) probability is and to make the connection between simulation and their work with probability trees in standard 7.SP.8b. When students work independently, it is better to keep to examples requiring only two or three stages (two or three sets of branches). OOOA - The 4th donor is the first with blood type A OOOO - None of the first four have blood type A. The first donor with blood type A is the 5th or 6th or 7th … Add the probabilities of these two outcomes (OOOA and OOOO) to get the probability that it will take at least 4 donors to get a donor with blood type A. Other questions to ask for Blood Type A using either the results from the simulation or the probability tree: What is the probability that it will take 1 donor to find the a donor with blood type A? What is the probability that it will take 2 donors to find the a donor with blood type A? What is the probability that it will take 3 donors to find the a donor with blood type A? How likely is it to find one of the first four donors has blood type A? At the probabilities for AOOO, OAOO, OOAO and OOOA. How likely is it to find at least one donor (one or more) with blood type A in the first four donors ? This is a more challenging question. Every branch pathway with the exception of OOOO has at least one donor with blood type A. Add up the probabilities from the 15 other branch pathways. Ask students if they can think of a more efficient way to answer this. Answer: Find the probability of OOOO and subtract this from – P(OOOO) = = OOOA (0.6)(0.6)(0.6)(0.4) =

Activities: Gettysburg Address (A sampling activity) This activity is adapted from the Sampling Words activity by Beth Chance and Allan Rossman. Beth Chance and Allan Rossman have given permission for their Sampling Words applet to be shared with Illinois math teachers. ( ple.html ) Blood Type A

Eighth Grade

Focus PBA* EOY Calculator 8.SP.A.1 Supporting X No 8.SP.A.2 8.SP.A.3 Yes 8.SP.A.4 * See Evidence Statement 8.D.2 (content from Grade 7 including 7.SP.B) and 8.D.3 (Micro-models) from the PARCC Evidence Table – Grade 8 PBA Standards 8.SP.A.1 – 3 enhance students’ study of scatterplots and linear functions. Be sure to look at the PARCC Evidence Tables – Grade 8 PBA. In particular, read the following two evidence statements: Evidence Statement from PARCC Evidence Tables - Grade 8 PBA 8.D.2 Solve multi-step contextual problems with degree of difficulty appropriate to Grade 8, requiring application of knowledge and skills articulated in 7.RP.A, 7.NS.3, 7.EE, 7.G, and 7.SP.B 8.D.3 Micro-models: Autonomously apply a technique from pure mathematics to a real-world situation in which the technique yields valuable results even though it is obviously not applicable in a strict mathematical sense (e.g., profitably applying proportional relationships to a phenomenon that is obviously nonlinear or statistical in nature). Content Scope: Knowledge and skills articulated in the Evidence Statements on the PBA (excludes Reasoning Evidence Statements). 8.SP.A.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. 8.SP.A.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. 8.SP.A.3 Use the equation 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 1.5 cm in mature plant height. 8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way 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 on 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?

8.SP.A.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. 8.SP.A.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. 8.SP.A.3 Use the equation 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 1.5 cm in mature plant height.

Statistics Standards for 8th Grade Investigate Patterns of Association in Bivariate Data
What is bivariate data? bi - means two variate – means variable Bivariate data is data about two variables. If the two variables are numeric, we examine the relationship between the two variables using a scatterplot. If the two variables are categorical, we organize the data in a two- way frequency table and look for an association.

A look at sample activities for 8.SP.1-3 Oil Changes and Engine Repair Adapted from an NCTM Illuminations’ activity Bike Weights and Jump Heights NCTM Illuminations Animal Brains Illustrative Mathematics US Airports, Assessment Variation Patterns in Scatter Plots – Lesson 7 Classwork EngageNY Determining the Equation of a Line Fit to Data – Lesson 9 Problem Set

Questions In what order would you use these activities in a unit? What questions did you particularly like? What questions are different than what we might typically ask students? Do you have suggestions for improvement of or additions to these activities?

8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way 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 on 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?

8.SP.4 Summarize categorical data in two categories Big Ideas: Review the difference between numeric data and categorical data. Explain that frequency refers to the count of the data. Relative frequency is a proportion. Analyze relative frequencies and assess possible associations and trends in the data. In sixth grade, students should learn that statistical questions can have answers that are either numeric or categorical. Consider the following: A survey is given to a group of students. What type of data, numeric or categorical, would be collected from each of these questions? 1) What is your age? 2) What is your favorite color? 3) What is your favorite professional football team 4) How many computers do you have in your household? For questions 1 and 4 above, we expect numeric answers. For questions 2 and 3 we expect categorical answers: the answers form categories (like red, blue, green) and the number of each category can be counted.

Association of two categorical variables There is an association between two categorical variables if the row (or column) conditional relative frequencies are different from row to row (or column to column) in the table. The greater the difference between the conditional relative frequencies, the stronger the association. The association that we see between two categorical variables can only be extended to the population if the sample was selected randomly from the population. In Algebra I / Math I, students begin to look at the difference between an association and causation. Cause-and-effect can only be concluded by conducting an experiment where the participants are selected at random, conditions are controlled, and the treatment is assigned randomly. Take time to introduce and help eighth graders to understand that in their work with scatterplots they are looking for an association between the two variables, but cannot conclude a cause-and-effect relationship.

Activity: Music and Sports
Statistics Standards for 8th Grade Investigate Patterns of Association in Bivariate Data Activity: Music and Sports Adapted from Illustrative Mathematics ( Is there an association between whether a student plays a sport and whether he or she plays a musical instrument? To investigate this question, each student in your class should answer the following two questions: 1. Do you play a sport? (yes or no) 2. Do you play a musical instrument? (yes or no) In this task, students collect data from their classmates and then summarize the data in a two-way frequency table.

Music and Sports Summarize the class data in a two way frequency table. Sport No Sport Total Musical Instrument 6 7 13 No Musical Instruments 8 3 11 14 10 24 Questions: 1. Of those students who play a sport, what proportion play a musical instrument? 2. Of those students who do not play a sport, what proportion play a musical instrument? 3. Based on the class data, do you think there is an association between playing a sport and playing an instrument? Questions 1 and 2 ask students to find column relative frequencies. Question 3 asks students if there is an association between playing a sport and playing a musical instrument. Students should be able to provide an explanation for their answer.

Activities 8-SP Oil Changes and Engine Repair Bike Weights and Jump Heights NCTM Illuminations 8.SP Animal Brains ( 8.SP US Airports, Assessment Variation ( Patterns in Scatter Plots – Lesson 7 Classwork ( Monopoly - Determining the Equation of a Line Fit to Data – Lesson 9 Problem Set

More Activities 8-SP-4 Music and Sports Activity adapted from: (

Statistics Standards for Grades 6 - 8
Census at School ( Statistics Education Web ( Census at School ( allows students to complete an online survey, take random samples of students in the United States, and compare their class’ responses with the responses from their random samples. Census at School also is a great tool to use for getting samples of data to use for class examples and activities. Statistics Education Web has peer-reviewed lessons that can be used in teaching the statistics standards. This is a great resource for sample activities that can be used at different grades. Some of the lessons go beyond the scope of the Common Core statistics standards and may need to be adapted before use in the classroom.

Acknowledgements and Resources
Chance, B. & Rossman, A. (Preliminary Edition). Investigating Statistical Concepts, Application and Methods. Duxbury Press. Chance, B., et al. Rossman/Chance Applet Collection. Retrieved from Chicago Tribune. (2014, April). Chicago Bears. Retrieved from teams.aspx?page=/data/nfl/teams/rosters/roster16.html Franklin, C., Kader, G., Mewborn, J. M., Peck, R., Perry, M. & Schaeffer, R. (2007) Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report: A Pre-K- 12 Curriculum Framework. Alexandria, VA: American Statistical Association. McCallum, B., et al. (2011, December 26). Progressions for the Common Core State Standards in Mathematics (draft) 6-8 Statistics and Probability. Retrieved from is.pdf. McCallum, B., et al. (2012, April 21). Progressions for the Common Core State Standards in Mathematics (draft) High School Statistics and Probability. Retrieved from content/uploads/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf.

Acknowledgements and Resources
Moore, D. & McCabe, P. (1989). Introduction to the Practice of Statistics. New York, NY: W. H. Freeman. Rossman, A. (2012). Interview With Roxy Peck. Journal of Statistics Education, 20(2). pp. 1 – 14. Retrieved from Rossman, A., Chance, B., & Von Oehsen, J. (2002). Workshop Statistics Discovery With Data and the Graphing Calculator. New York: Key College Publishing. Scheaffer, R., Gnanadesikan, M., Watkins, A., & Witmer, J. (1996). Activity-Based Statistics. New York: Springer-Verlag.

Online Resources Census at School. Consortium for the Advancement of Undergraduate Statistics Education. Engage NY. Illustrative Mathematics. Inside Mathematics. Mathematics Assessment Project. Math Vision Project. NCSSM Statistics Institutes. NCTM Core Math Tools – Data Sets

Online Resources PARCC Model Content Frameworks. er2012V3_FINAL.pdf PARCC Mathematics Evidence Tables. blueprints-test-specs Smarter Balanced Assessment Consortium. Statistics Education Web (STEW). The Data and Story Library (DASL). The High School Flip Book Common Core State Standards for Mathematics. book-usd pdf

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The New Illinois Learning Standards for Grades Statistics and Probability Thank you for joining us! Dana Cartier Julia Brenson Tina Dunn

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