grade 6 supporting idea 6: data analysis

Grade 6 Supporting Idea 6:Data Analysis

Grade 6 Supporting Idea: Data Analysis

• MA.6.S.6.1 Determine the measures of central tendency (mean, median, and mode) and variability (range) for a given set of data.

• MA.6.S.6.2 Select and analyze the measures of central tendency or variability to represent, describe, analyze and/or summarize a data set for the purposes of answering questions appropriately.

FAIR GAME: Prerequisite Knowledge

• MA.3.S.7.1: Construct and analyze frequency tables, bar graphs, pictographs, and line plots from data, including data collected through observations, surveys, and experiments.

• MA.5.S.7.1: Construct and analyze line graphs and double bar graphs.

FAIR GAME: Prerequisite Knowledge

Skills Tracemean

median

mode

range

•Add whole numbers, fractions, and decimals•Divide whole numbers, fractions, and decimals•Compare and order whole numbers, fractions, and decimals

•Add whole numbers, fractions, and decimals

•Divide whole numbers, fractions, and decimals•Compare whole numbers, fractions, and decimals

•Subtract whole numbers, fractions, and decimals

Measures of Center

meanmedianmode

MODEL: FINDING THE MEDIANFind the median of 2, 3, 4, 2, 6.

Participants will use a strip of grid paper that has exactly as many boxes as data values. Have them place each ordered data value into a box. Fold the strip in half. The median is the fold.

• Arrange interlocking/Unifix cubes together in lengths of 3, 6, 6, and 9.– Describe how you can use the cubes to find the

mean, mode, and median.– Suppose you introduce another length of 10 cubes. Is

there any change in i) the mean, ii) the median, iii) the mode?

MODEL: FINDING THE MEAN

MODEL: FINDING THE MEAN

Thinking about measures of center

The median of five numbers is 15. The mode is 6. The mean is 12. What are the five numbers?

1566 n n

Thinking about measures of centerThe median of five numbers is 15. The mode is 6. The mean is 12. What are the five numbers?

1566 n n

€

6 + 6 +15 + n + n

5=27 + 2n

5=12

€

27 + 2n = 60 2n = 33 n =16.5

Thinking about measures of center

The median of five numbers is 15. The mode is 6. The mean is 12. What are the five numbers?

1566 a b

Thinking about measures of centerThe median of five numbers is 15. The mode is 6. The mean is 12. What are the five numbers?

1566 a b

€

6 + 6 +15 + a+ b

5=27 + a+ b

5=12

€

27 + a+ b = 60 a+ b = 33

Missing Observations: MeanHere are Jane’s scores on her first 4 math tests:

80 82 75 79What score will she need to earn on the fifth test

for her test average (mean) to be an 80%?

€

80 + 82 + 75 + 79 + n

5= 80

316 + n

5= 80

400 = 316 + n

n = 84

Missing Observations: MeanHere are Jane’s scores on her first 4 math tests:

80 82 75 79There is one more test. Is there any way Jane can

earn an A in this class? (Note: An “A” is 90% or above)

What measure of center are we

asking students to consider?

Missing Observations: MeanHere are Jane’s scores on her first 4 math tests:

80 82 75 79There is one more test. Is there any way Jane can earn an A in this

class? (An “A” is 90% or above)

€

80 + 82 + 75 + 79 + n

5= 90

316 + n

5= 90

450 = 316 + n

n =134

Missing Observations: MedianHere are Jane’s scores on her first 4 math tests:

80 82 75 79What score will she need to earn on the fifth

test for the median of her scores to be an 80%?

7579 80 82

Missing Observations: MedianWhat score will she need to earn on the fifth test for the median

of her scores to be an 80%?

75 79 80 8270?75?79?80?81?82?83?84?

• Construct a collection of numbers that has the following properties. If this is not possible, explain why not.

mean = 6 median = 4 mode = 4

What is the fewest number of observations needed to accomplish this?

• Construct a collection of numbers that has the following properties. If this is not possible, explain why not.

mean = 6 median = 6 mode = 4

What is the fewest number of observations needed to accomplish this?

• Construct a collection of 5 counting numbers that has the following properties. If this is not possible, explain why not.

mean = 5 median = 5 mode = 10

What is the fewest number of observations needed to accomplish this?

• Construct a collection of 5 real numbers that has the following properties. If this is not possible, explain why not.

mean = 5 median = 5 mode = 10

What is the fewest number of observations needed to accomplish this?

• Construct a collection of 4 numbers that has the following properties. If this is not possible, explain why not.

mean = 6, mean > mode

• Construct a collection of 5 numbers that has the following properties. If this is not possible, explain why not.

mean = 6, mean > mode

• Suppose a constant k is added to each value in a data set. How will this affect the measures of center and spread?

5 6 7 9 2 4 1 6mean = 5

median = 5.5mode = 6range = 8

Adding a constant k

Adding a constant k

mean = 5median = 5.5

mode = 6range = 8

56792416

5+2=6+2=7+2=9+2=2+2=4+2=1+2=6+2=

789114638

mean = 7median = 7.5

mode = 8range = 8

• Suppose a constant k is multiplied by each value in a data set. How will this affect the measures of center and spread?

5 6 7 9 2 4 1 6mean = 5

median = 5.5mode = 6range = 8

Multiplying by a constant k

Multiplying by a constant k

mean = 5median = 5.5

mode = 6range = 8

56792416

5×2=6×2=7×2=9×2=2×2=4×2=1×2=6×2=

1012141848212

mean = 10median = 11mode = 12range = 16

Watch out! Graphical Displays of Data and Measures of Center

TableBar graphs

Double bar graphsLine graphsLine plots

PictographFrequency table

Watch out! Graphical Displays of Data and Measures of Center

Watch out! Graphical Displays of Data and Measures of Center

Watch out! Graphical Displays of Data and Measures of Center

Watch out! Graphical Displays of Data and Measures of Center

Watch out! Graphical Displays of Data and Measures of Center

Watch out! Graphical Displays of Data and Measures of Center

Watch out! Graphical Displays of Data and Measures of Center

Watch out! Graphical Displays of Data and Measures of Center

Watch out! Graphical Displays of Data and Measures of Center

Watch out! Graphical Displays of Data and Measures of Center

Watch out! Graphical Displays of Data and Measures of Center

Watch out! Line Graphs and Measures of Center

The Location A sixth-grade teacher uses a secret location game to teach the class about statistics, connections, and reasoning.

• http://www.learner.org/resources/series33.html?pop=yes&pid=918

Watch out! Line Graphs and Measures of Center

The Location1. What is the value of having students generate,

record, and graph their own data?2. Discuss Mr. Stevenson’s decision not to supply grid

paper.3. How does Mr. Stevenson stimulate discussion and

statistical reasoning?4. How can background experiences affect a student’s

ability to understand and generalize about data?

Watch out! Frequency Tables and Measures of Center

• The position of the median can be found by the formula , where n is the number of observations in the data set.

€

n +1

2

Watch out! Frequency Tables and Measures of Center

Watch out! Frequency Tables and Measures of Center

Watch out! Frequency Tables and Measures of Center

Watch out! Frequency Tables and Measures of Center

Number Frequency1 62 13 24 35 56 57 48 39 0

Watch out! Reviewing How Frequency Tables are Made

The student must:• Distinguish between data sets that are

symmetrical and those that are skewed• Understand the effect of skewness on the mean• Recognize outliers• Understand why the median is outlier-resistant• Remember that the mode is particularly helpful

for categorical (vs. quantitative) data

Choosing an appropriate measure of center

Mean vs. Median

What is an outlier?

• An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs.

• Extreme observations• In the real world, statisticians either discard

them or use a robust (outlier-resistant) measure of center or spread.

What is an outlier?

How do we determine outliers?1.5*IQR (interquartile range)

2, 5, 7, 9, 10, 12, 20lower quartile: Q1= 5

median: 9upper quartile: Q2=12

IQR = Q2-Q1= 12 - 5 = 71.5*IQR= 10.5

In order to be called a mild outlier, we say an observation has to be more than this distance below Q1 or above Q2.

If an observation is 3 or more IQRs above/below Q1/Q3, we say an observation is an extreme outlier.

Outliers: What to do?

Describing Distributions

symmetric distributionmean = median = mode

skewed left distributionmean < median < mode

skewed right distributionmean > median > mode

Visualizing how the outlier pulls the mean

http://bcs.whfreeman.com/fapp7e/content/cat_010/meanmedian.html

Number Frequency1 62 13 24 35 56 57 48 39 0

Mean, Median or Mode?

mean = 4.896median = 5mode = 1

Number Frequency1 92 83 74 65 56 67 78 89 9

Mean, Median or Mode?

mean = 5median = 5

modes = 1 and 9

Number Frequency1 72 203 154 115 86 37 28 09 15

Mean, Median or Mode?

mean = 4.58median = 3mode = 2

Number Frequency1 32 23 34 25 16 37 28 29 54

Mean, Median or Mode?

mean = 8median = 2 mode = 54

symmetric distributionmean = median = mode

skewed left distributionmean < median < mode

skewed right distributionmean > median > mode

• Skew

Which measure of center is best for each data set?

Using Boxplots to Show the Robustness of the Median

Removing the Outlier,

Recalculating the Mean

Fuel Economy (Miles per Gallon)for Two-Seater Cars

Model City Highway

Acura NSX 17 24

Audi TT Roadster 20 28

BMW Z4 Roadster 20 28

Cadillac XLR 17 25

Chevrolet Corvette 18 25

Dodge Viper 12 20

Ferrari 360 Modena 11 16

Ferrari Maranello 10 16

Ford Thunderbird 17 23

Honda Insight 60 66

Lamborghini Gallardo 9 15

Lamborghini Murcielago 9 13

Lotus Esprit 15 22

Maserati Spyder 12 17

Mazda Miata 22 28

Mercedes-Benz SL500 16 23

Mercedes-Benz SL600 13 19

Nissan 350Z 20 26

Porsche Boxster 20 29

Porsche Carrera 911 15 23

Toyota MR2 26 32

With Outlier

Without Outlier

mean

median

mode

range

Removing the Outlier, Recalculating the Mean

Encouraging Critical and Statistical Thinking

What would you say to these students?

Gregory: "The boys are taller than the girls."

What would you say to these students?

Marie: "Some of the boys are taller than the girls, but not all of them."

What would you say to these students?

Arketa: "I think we should make box plots so it would be easier to compare the number of boys and girls."

What would you say to these students?

Michael: "The median for the girls is 63 and for the boys it's 65, so the boys are taller than the girls, but only by two inches."

What would you say to these students?

Paul [reacting to Michael's

statement]: "I figured out that the boys are two inches taller than the girls, too, but I figured out that the median is 62 for the girls and 64 for the boys."

What would you say to these students?

Kassie: "The mode for the girls is 62, but for the boys, there are three modes -- 61, 62, and 65 -- so they are taller and shorter, but some are the same."

What would you say to these students?

DeJuan: "But if you look at the means, the girls are only 62.76 and the boys are 64.5, so the boys are taller."

What would you say to these students?

Carl: "Most of the girls are bunched together from 62 to 65 inches, but the boys are really spread out, all the way from 61 to 68."

What would you say to these students?

Arketa: "There is a lot of overlap in heights between the boys and girls."

What would you say to these students?

Michael: "We can see that the median for the boys is higher than for the girls."

What would you say to these students?

Monique: "It looks like just 12.5% of the boys are taller than all of the girls, and maybe about 10% of the girls are shorter than the shortest boy."

What would you say to these students?

Gregory: "The boys are taller than the girls, because 50% of the boys are taller than 75% of the girls."

What would you say to these students?

Morgan: "You can see that the middle 50% of the girls are more bunched together than the middle 50% of the boys, so the girls are more similar in height."

What would you say to these students?

Janet: "Why isn't the line in the box for the boys in the middle like it is for the girls? Isn't that supposed to be for the median, and the median is supposed to be in the middle?

Discovering Math: Summary (3:45)

Generating Meaningful Data • Make and fly paper airplanes—how far do they go?• How long is a second?• How many jumping jacks can you do in a minute?• Handspan, arm span• Food nutrition label analysis• 3M Olympics: Peanut Flick, Cookie Roll,

Marshmallow Toss

Instructional Resources

Read the article "What Do Children Understand About Average?" by Susan Jo Russell and Jan Mokros from Teaching Children Mathematics.

a.  What further insights did you gain about children's understanding of average?b.  What are some implications for your assessment of students' conceptions of average?c.  What would be an example of a "construction" task and an "unpacking" task?d.  Why might you want to include some "construction" and "unpacking" tasks into your instructional program?

To the tune of “Row, Row, Row Your Boat”

Mode, mode, mode– THE MOSTAverage is the meanMedian, median, median, medianThe number in between

NLVM: Bar Chart

Another Representation of the Mean