StatisticStatisticss

Middle School Middle School Content ShiftsContent Shifts

Concerning Concerning statistics, what have statistics, what have

you usually taught you usually taught or done? Share with or done? Share with

an elbow partner.an elbow partner.

Read “Data in Grades K – 5” handout Read “Data in Grades K – 5” handout and pages 2 – 3 of “6 – 8 Statistics and and pages 2 – 3 of “6 – 8 Statistics and Probability Progression Document”. Probability Progression Document”. Compare what you read to what you Compare what you read to what you have been teaching.have been teaching.

Critical Area for Grade Critical Area for Grade 66

Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students recognize that a data

distribution may not have a definite center and that different ways to measure center yield different

values. The median measures center in the sense that it is roughly the middle value. The mean

measures center in the sense that it is the value that each data point would take on if the total of the data

values were redistributed equally, and also in the sense that it is a balance point.

Iowa Core, page 41

Finding the Balance Finding the Balance PointPoint

Adapted from “Teaching Student Centered Mathematics, grades 5 - 8, Van de Walle and Lovin

Count the number of letters in your first and Count the number of letters in your first and last name. Place that number on a sticky last name. Place that number on a sticky note and post it on the class number line note and post it on the class number line graph.graph.

With a partner, create a number line like the With a partner, create a number line like the class one, but do not put any data on it.class one, but do not put any data on it.

Discuss with your Discuss with your partner….partner….

““Using the sticky notes, how Using the sticky notes, how could we find the balance could we find the balance

point of the data of our point of the data of our graph?”graph?”

How could we move some sticky notes to find the balance?

What would be another movement we can do?

What does this final What does this final display represent?display represent?

Will your set of data Will your set of data have the same have the same balance point?balance point?

This balance point is This balance point is called the mean. How called the mean. How

can you use your data to can you use your data to calculate the mean?calculate the mean?

Will your mean be the Will your mean be the same as other same as other partnerships?partnerships?

Iowa Core StandardsIowa Core Standards5.MD.2 5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

6.SP.36.SP.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.

Softball Softball Team Team

Weekly Weekly Study Study HoursHours

Make a displayMake 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 displayShare your display

Which display represented the data the best?

What other displays could have been made?

Iowa Core StandardsIowa Core Standards

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

Critical Area for Grade Critical Area for Grade 66Students recognize that a measure of

variability (interquartile range or mean absolute deviation) can also be useful for

summarizing data because two very different sets of data can have the same mean and

median yet be distinguished by their variability. Students learn to describe and

summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry,

considering the context in which the data were collected.

Iowa Core, page 41

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

1999

LaunchLaunch

In a survey, nine individuals were asked, “How many people are in your family?” One result from the poll was that the average family size for those asked was “5”.

If the mean size is “5”, how many members could be in each of those nine families? In your group, determine some possible distributions of the nine families.

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

Display one of your distributions as a dot plot using chart paper and post-its. When your group is finished, post the chart on the wall.

ExploreExplore

What are the limitations of only knowing the mean family size?What additional information about the data could be given to identify which of the distributions matched the results of the survey?

Given these 8 distributions with a mean of 5

A major goal of statistics is to offer ways to A major goal of statistics is to offer ways to summarize and measure this “spread” summarize and measure this “spread”

(or variability)(or variability)

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

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

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

distributions?distributions?

Individually (1 minute)

Group (2 minutes)

Share

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

….and consider as

many ways as you can

Share Out 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 MADCalculate the MAD

Mean Absolute Deviation:

The sum of the distances of each piece of data from the mean, divided by the number of

pieces of data

Distribution Number MAD

A 0

B

C

D

E

F

G

H

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?

SummarizeSummarizeHow can the MAD help you distinguish between Distribution C and Distribution E?

What does a MAD of 1.78 indicate about the set of data?

The mean and median are often referred to as “measures of center”. The notion of center is that of some halfway point, the median is the value at the middle. How is mean a center?

Check for UnderstandingCheck for UnderstandingAt exactly the same instant, Mike and Chuck checked the time on the clocks and watches at their homes. The times on Chuck’s ten clocks were: 8:16 8:10 8:14 8:16 8:12 8:15 8:13 8:17 8:15 8:23

1.Find the MAD (mean absolute deviation) of Chuck’s clocks.2.Mike had the same mean on his ten clocks as Chuck did, but his MAD was 10 minutes. Find an example of the ten times that could be the times on Mike’s clocks.3.What do the mean and the MAD tell you about how useful it is to look at a clock at Mike’s house and at Chuck’s house?

Iowa Core StandardsIowa Core Standards6.SP.2 6.SP.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.36.SP.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.56.SP.5 Summarize numerical data sets in relation to their context, such as by:a.Reporting the number of observations.b.Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.c.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. d.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.

Text Text MessagiMessagi

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Interpreting Box PlotsInterpreting Box PlotsBox plots are…

– Used for organizing and displaying data

– Easy to create but not always easy to understand

– Used to foster higher order thinking in students

– Part of the 6th grade CCSS

With a PartnerWith a PartnerWork through the activity “Text Messaging”.Ask for assistance as needed.Be prepared to discuss the results of your work.

SummarizeSummarizeWhat are the components of a box plot?

Where is the IQR and how could you calculate it?

Why would you use a box plot to represent data?

Iowa Core StandardsIowa Core Standards

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

7.SP.47.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

MakMake a e a

MatcMatchh

In partnerships….In partnerships….

Look at the 4 histograms on the top of the sheet. Below are 4 box plots. Decide which histogram and which box plot are using the same data.

Be prepared to explain your reasoning using specific evidence.

Iowa Core StandardsIowa Core Standards6.SP.4 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.6.SP.56.SP.5 Summarize numerical data sets in relation to their context, such as by:a.Reporting the number of observations.b.Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.c.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. d.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.

VariabilityVariability

On your own…On your own…For each of the four pairs of histograms, choose the statement that best describes the situation:

•A has more variability than BA has more variability than B

•B has more variability than AB has more variability than A

•Both graphs are equally variableBoth graphs are equally variable

Record your choice and thinking on the answer sheet.

With your table group…With your table group…

Discuss your thoughts and reasoning about the variability of each pair of graphs.

Variability:Variability:•How "spread out" a set of data is•The extent to which data points diverge from the average or mean value. •The extent to which data points differ from each other.

Iowa Core StandardsIowa Core Standards6.SP.2 6.SP.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.36.SP.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.4 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.6.SP.56.SP.5 d d Summarize numerical data sets in relation to their context, such as by: 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.

The Blue M&MThe Blue M&M™™ ProblemProblem

LauncLaunchhFinley loves to go to the grocery store with her dad, Mike,

and help him do the grocery shopping because she gets to pick out a treat for the car ride home. Her treat of choice is usually plain M&M’S™ Milk Chocolate Candies. Finley often dumps the entire bag out on her lap and sorts the candies by their color. She especially likes the blue ones.

Like any other weekend, Finley and Mike made a trip to the grocery store where Finley picked out a treat for the car ride home. Like any other car ride home, Finley dumped her bag of M&M’S™ out on to her lap and sorted them by color.

However, unlike any other car ride home from the grocery store, Finley was sad. She said “I need a new bag of M&M’s...this bag is bad…there are only 8 blue ones!”

Was Finley’s bag of M&M’S™ “bad”? How could you decide?

ExploreExploreHow can we randomly test bags of M&M’S as a class?

Please do not Please do not open bag or eat open bag or eat any candies until any candies until instructed to do instructed to do so!so!

ProcesProcess s

Open your bag of M&Ms™

Dump out your bag and all the M&Ms™ onto a paper plate (do not eat any at this time)

How many in excess of 50 do you have? Remember that number.

Mix them all up again.

Close your eyes and randomly remove enough M&Ms™ so that you have 50 left. This is your random sample of 50 M&Ms™.Count the number of blues in your random sample 50 M&Ms™.Record your “number of blues” on a post-it note.

Based on your sample, is Finley’s bag “bad”? Explain.

Based on the samples at your table, is Finley’s bag “bad”? Explain.

Hang your post-it at the appropriate spot on the class number line.

Based on the samples of ALL your classmates, is Finley’s bag “bad”? Explain.

Do you have enough information to make this decision now?

What Do You Think?What Do You Think?

Is it a rare event (meaning an uncommon occurrence) for a “good” bag of M&M’S™(same size as Finley’s) to have 8 blue candies?

You will explore some aspects of binomial distributions that will help you answer this question.

The The statistical questionstatistical question we are trying to answer is:we are trying to answer is:

Each observation falls into one of just two categories, which for convenience we call “success” or “failure”.

There is a fixed number n of observations.

The n observations are all independent. That is, knowing the result of one observation tells you nothing about the other observations.

The probability of success, call it p, is the same for each observation.

We are concerned with observing whether the

M&M™ is blue (success) or it is not (failure)

We are looking at 500 bags of candies.

One bag’s results does not depend upon another bag’s

results.

M&Ms™ company says 24 percent of the bag should

be blue. So the p of success (blue) is .24

Use the Technology

Open up Core Tools.

Work with a partner to simulate opening bags of M & M’s. What do you notice?

SummarizeSummarizeHow was the idea of spread spread displayed in this lesson? Explain.How was the idea of shapeshape shown in this lesson? Explain.How was the idea of centercenter used in this lesson? Explain.Is Finley’s bag of M&M’S™ “badbad”? Explain.

Check for UnderstandingCheck for Understanding

What will happen if we continue to

simulate bags of M&M’S™ over a long

period of time?

Iowa Core Standard(s)Iowa Core Standard(s)7.SP.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.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.

7.SP.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.

a.Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.

b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

ScatteredScattered

Bivariate Data….Bivariate Data….

• is data that has two variables

• deals with causes or relationships

• analysis purpose is to explain

• may have one variable that influences or determines the second variable

• can be represented using a scatter plot

→

Explore some data….Explore some data….

With a partner, work through “Using Bivariate

Data”.

Be prepared to share your results from

numbers 4 and 5 with the group.

Wrap upWrap upWas your line a “best fit”? Explain.

How does knowing the equation of the line and the slope help your analysis?

What other bivariate relationships could you explore?

Iowa Core Standard(s)Iowa Core Standard(s)8.SP.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.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 line, and informally assess the model fit by judging the closeness of the data points to the line.

Comparing Categorical Comparing Categorical DataData

LaunchLaunch

Data can be organized in many ways. Use the information on your handout and organize it any way you would like. Be

prepared to share your organization with the class and describe why you chose to

organize it in this way.

ExploreExplore

Complete “Two-Way Table

Exploration” to solidify your

understanding.

Relative Relative FrequencyFrequency

Relative FrequencyRelative Frequency is the ratio of the number of

observations in a statistical category to the

total number of observations.

Using “Survey Sheet”, determine the relative

frequencies for the table and then complete the

questions.

SummarizSummarizee

How can relative frequencies be How can relative frequencies be used to determine any possible used to determine any possible associations in the data set you just associations in the data set you just finished? finished?

How is it different than just using How is it different than just using the raw numbers? Explain.the raw numbers? Explain.

When and why are two-way tables When and why are two-way tables used?used?

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

Iowa Core Standard(s)Iowa Core Standard(s)

8.SP.48.SP.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.