Statistics and Probability from the

High School CoreBuilding Foundational Skills

Module 1.A

Getting Started…Create as many data sets as you can having 8 elements with the following characteristics:• Median = 8• Mode = 8• Mean is between 6 and 10

Iowa Core Standard(s)

6.SP.36.SP.5(d)

I can explain the differences between the three measures of

center.

Foundation for:

S-ID.2S-ID.3

Learning TargetsUnderstand that data can be displayed in a variety of ways depending on the purpose of the display.

Understand that the study of statistics must be grounded in relationships among shape, center, and spread.

Understand that data can be compared in a variety of ways using a variety of tools.

Success Criteria• I can explain the differences

between the three measures of center.

• I can list two displays used for categorical data.

• I can list two displays used for quantitative data.

• I can describe why categorical and quantitative displays differ.

• I can list three terms used to describe variability.

• I can define the Five Number Summary and interpret its meaning.

Categorical vs. QuantitativeWhat types of displays are used for categorical data?What types of displays are used for quantitative data?Why are they different?

– List specific characteristics and limitations

Do both types of data show variability?

Make a displayLook at the data given and display the data in an appropriate way.• What observations can be made with your

display?• Are there strengths and weaknesses with your

display?

Share your display with the group.• Which display represented the data the best?

Iowa Core Standard(s)

6.SP.46.SP.5(c)

Foundation for:

S-ID.1S-ID.2S-ID.3

I can list two displays used for categorical data.I can list two displays used for quantitative data.

I can describe why categorical and quantitative displays differ.I can define the Five Number Summary and interpret its meaning.

Break

Statistics: Creating a New Way to Think About Mathematics

Statistics is multi-faceted; it is not just making calculations using algorithms.

Statistics calls for a transformed way of thinking that requires us to consider what the data is saying, to determine the available evidence, and to communicate the findings.

Statistical ReasoningFour-Step Investigative Process

– Formulating a statistical question- a question that can be answered with data

– Designing a plan for collecting useful data, implementing the plan, and collecting relevant data

– Analyzing the data- creating and exploring various representations of the distribution to identify and describe patterns in the variability in the data and summarize various features of the distribution with appropriate methods

– Interpreting the results- providing a statistical answer to the question posed that takes the variability in the data into account

Progressions OverviewRead Overview of Progressions.

Discuss as a group. Focus discussion on the content as well as the emphasis on shape, center, and spread that will now exist in grades 6-8.

Means & MADsAdapted from “Means and MADs”, Mathematics Teaching in the Middle Grades, NCTM, 1999

Launch

In a survey, nine people were asked, “How many people are in your family?” One result from the poll was that the average family size for the nine people was five.

Launch• In your group, determine some possibilities

for the distribution of the nine different family sizes utilizing the fact that the mean family size is 5?

• Display your distributions as line plots using the chart paper and post-its supplied by your teacher.

• For this problem, consider family sizes no smaller than 2 and no larger than 11.

Explore

Think-Pair-Share• What are the limitations of only

knowing the mean family size?

Given these 8 distributions with a mean of 5

A major goal of statistics is to offer ways to summarize and measure variation in data.

Of all these distributions, which distribution shows data values with the least variation? Explain.

Of all these distributions, which distribution shows data values with the most variation? Explain.

How would you order, from least variation to most variation, the 8 distributions?

• Record Individually• Come to a group

consensus• Share out

Because it can be difficult to come to a consensus on this single ordering, we need a number (a metric) to quantify variation in a set of data. In your group determine methods to do this…

(consider as many methods as you can)

Share Out

• Each group will select a

reporter.• Each reporter will describe one of the methods their group created unless it is a duplicate.

• Rotation among groups will continue until all methods are presented.

Calculate the MAD(Mean Absolute Deviation)

Distribution Number MAD

1 0

2

3

4

5

6

7

8

• Does the MAD ordering give you the same ordering your group got?

• If yours was the same (or close), given a different set of distributions, do you think you would always be close?

• At what number of data points does is become difficult to order visually?

SummarizeA. How can the MAD help you distinguish

between Distribution 3 and Distribution 5?

B. What does a MAD of 1.78 indicate about the set of data?C. The mean and median are often referred to as “measures of center”. The median is the value in the middle (the idea of halfway). How is mean a center?

Extension(s)• Given the Standard Deviation formula.

Compare it to the MAD.

• Research the development of the Standard Deviation formula. Explain.

1 -

-∑n

)xx( 2

=n

xxMAD

∑ -=

Iowa Core Standard(s)

6.SP.26.SP.36.SP.5

Foundation for:

S-ID.1S-ID.2S-ID.3S-ID.4

I can list three terms used to describe variability.

VariabilityFor each of the four pairs of histograms,

choose the statement that best describes the situation.

A has more variability than BB has more variability than ABoth graphs are equally variable

Iowa Core Standard(s)6.SP.26.SP.36.SP.4

6.SP.5(d)

Foundation for:

S-ID.1S-ID.2S-ID.3S-ID.4

I can list three terms used to describe variability.

Success Criteria• I can list the differences between

the three measures of center.• I can list two displays used for

categorical data.• I can list two displays used for

quantitative data.• I can describe why categorical and

quantitative displays differ.• I can list three terms used to

describe variability.• I can define the Five Number

Summary and interpret its meaning.

Reflection3 things I can take away from this learning.

2 things I want to implement in my teaching.

1 question I still have.

Statistics and Probability from the

High School CoreBuilding Foundational Skills

Module 1.B

Sampling TechniquesThe school food service wants to increase the number of students who eat hot lunch in the cafeteria. The student council has been asked to conduct a survey of the student body to determine the students’ preferences for hot lunch. They have selected three ways to do the survey. Which survey option should the student council use and why?

1. Obtain a class list of the freshman, number the students starting at 1, then choose every student whose number ends in a 0 (For example: 10, 20, 30, etc.). Repeat this process for the 10th, 11th, and 12th grade classes.

2. Survey the first 30 students that enter the lunch room.

3. Write all of the students’ names (9-12) on cards and pull out 25 names to determine who will complete the survey.

Iowa Core Standard(s)

7.SP.1

Foundation for:

S-ID.1S-ID.2S-ID.3S-ID.6

I can utilize a random sample to answer questions about a data set.

Learning TargetsUnderstand that data can be displayed in a variety of ways depending on the purpose of the display.

Understand that the study of statistics must be grounded in relationships among shape, center, and spread.

Understand that data can be compared in a variety of ways using a variety of tools.

Success Criteria• I can develop a probability model and

use it to create relative frequencies.• I can find probabilities of compound

events using multiple strategies.• I can utilize a random sample to

answer questions about a data set.• I can form comparative inferences

using measures of center and spread.• I can determine an appropriate model

for bivariate data.• I can compare categorical data using a

two-way table.

Comparative InferencesCompare the random sampling of

heights of soccer players and basketball players using the line plots and statistical

information given. Make sure to compare the two data sets in terms of

shape, center, and spread.

By the standard…Iowa Core states we compare two data sets by measuring the difference between the centers and expressing it as a multiple of a measure of variability.

But why?

Our dataThe difference between the means is 7.68. Knowing the difference between centers is nice, but it is more meaningful when it is made relative to the variation in the data.In this case, it is logical to wonder if soccer players are shorter than basketball players. It looks to be likely by the naked eye, but what data is there to support that?Taking the difference and dividing by the Mean Absolute Deviation will give us numerical evidence that this is true. In this case, 7.68/2.53 gives us a value of 3.04. The larger this number, the less likely that the data sets will share values in common.

Check for UnderstandingWhich pair of data sets is most likely to have the greatest number of values in common?

A BData set 1: mean = 7 Data set 1: mean = 7Data set 2: mean = 15 Data set 1: mean = 15MAD for both data sets is 4 MAD for both data sets is 8

C DData set 1: mean = 10 Data set 1: mean = 10Data set 2: mean = 15 Data set 1: mean = 18MAD for both data sets is 1 MAD for both data sets is 2

Iowa Core Standard(s)

7.SP.37.SP.4

Foundation for:

S-ID.5S-ID.6

I can form comparative inferences using measures of center and spread.

The Right Fit

LaunchFred and Ginger have been given the following coordinates and are asked to find a linear function to model the data. The coordinates include: (0,2), (1,3), (2,5), (3,6), (4,8), and (5,9). Fred believes f(x) = 2x + 1 will be a good model while Ginger believes g(x) = x + 3 will be a better equation to model the data. Which equation is a better model to predict data?

Putting it into Context

The JRM Company has decided to start a big ad campaign to increase their customer base. This company currently has 20,000 customers. After 3 months of the campaign, or 1 quarter, the company has grown to 30,000 customers. As their ad campaign wraps up after 5 quarters, they have grown to a total of 90,000 customers.

Putting it into ContextWhat would be the labels of the x- and y-axis?

What do the numerical values of slope and y-intercept represent in Fred and Ginger’s equations?

In context, which model seems to fit the data the best?

ExploreIn order to determine whether Fred or Ginger has the better

model we will need to develop a way to measure how well

their equations fit the data.

Work TogetherAs a pair, create a metric that will quantify which equation better fits the data. Record the method that you will use and then test it out to determine if Fred or Ginger had the better model.

Share OutExplain your methodology for developing a metric to find the equation with the better fit.

Technology Abounds!

Use technology to fit the best line you can to the data.Write down the equation of this line and interpret the slope and y-intercept.Use the metric you created earlier to test your line.

SummarizeDid the metric created for the “best” fit line verify your visual intuition?

The majority of “best” fit lines did not start at the y-intercept of 2, is this a problem?

Why would we need a line to represent our data?

Iowa Core Standard(s)

8.SP.18.SP.28.SP.3

Foundation for:

S-ID.6S-ID.7S-ID.8

I can determine an appropriate model for bivariate data.

Break

Comparing Categorical Data

LaunchData can be organized in many

ways. Take the following information and organize it however you would like. Be

prepared to share your organization with the class and

describe why you chose to organize it in this way.

Explore

Complete the Worksheet to solidify your

understanding of Two-Way Tables.

SummarizeWhen and why are two-way tables used?

How can two-way tables be manipulated to answer multiple questions?

Check for Understanding

Complete the worksheet.

Iowa Core Standard(s)

8.SP.4Foundation

for:S-ID.5S-CP.4

I can compare categorical data using a two-way table.

Calculate the Probability…Getting an even number when rolling a fair dice

Getting one head and one tail when tossing two coins

Landing in the circle

IT TAKES TIMEProbability uses theory to create calculations, but those values show what will happen over many, many trials. In the short-run, the chance process to produce sample outcomes is not as predictable.

The question then becomes:“How long does it take to feel confident in a probability prediction?”

“In the Bag”

http://nrich.maths.org/6016

Iowa Core Standard(s)

7.SP.67.SP.7

Foundation for:

S-MD.7

I can develop a probability model and use it to create relative frequencies.

One and One

Equals Win

The SituationThe basketball team is down

by one with one second on the clock but a foul may save

them. Can a 60% free throw shooter

win the game for her team? How often?

Simulate the Situation

Using technology, perform a simulation

that will determine how often a 60% shooter can win the game.

The SummaryWhat connection is there between the free throw shooter’s percentage and the percent of wins? How was this seen in the organized list, table, or tree diagram?

What percentage of times did the shooter lose the game? What does that have to do with the shooter’s percentage?

Iowa Core Standard(s)

7.SP.8Foundation

for:S-CP.1-3S-CP.6-9

I can find probabilities of compound events using multiple strategies.

Success Criteria• I can develop a probability model and

use it to create relative frequencies.• I can find probabilities of compound

events using multiple strategies.• I can utilize a random sample to

answer questions about a data set.• I can form comparative inferences

using measures of center and spread.• I can determine an appropriate model

for bivariate data.• I can compare categorical data using a

two-way table.

Thank you

Your hard work and great participation

was much appreciated!