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 Trace mean 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 mean median mode

MODEL: FINDING THE MEDIAN Find 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

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? nn

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? nn

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? ab

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? ab

Missing Observations: Mean Here are Jane’s scores on her first 4 math tests: What score will she need to earn on the fifth test for her test average (mean) to be an 80%?

Missing Observations: Mean Here are Jane’s scores on her first 4 math tests: There 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: Mean Here are Jane’s scores on her first 4 math tests: There is one more test. Is there any way Jane can earn an A in this class? (An “A” is 90% or above)

Missing Observations: Median Here are Jane’s scores on her first 4 math tests: What score will she need to earn on the fifth test for the median of her scores to be an 80%?

Missing Observations: Median What score will she need to earn on the fifth test for the median of her scores to be an 80%? ? 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? mean = 5 median = 5.5 mode = 6 range = 8 Adding a constant k

mean = 5 median = 5.5 mode = 6 range = = 6+2= 7+2= 9+2= 2+2= 4+2= 1+2= 6+2= mean = 7 median = 7.5 mode = 8 range = 8

Suppose a constant k is multiplied by each value in a data set. How will this affect the measures of center and spread? mean = 5 median = 5.5 mode = 6 range = 8 Multiplying by a constant k

mean = 5 median = 5.5 mode = 6 range = ×2= 6×2= 7×2= 9×2= 2×2= 4×2= 1×2= 6×2= mean = 10 median = 11 mode = 12 range = 16

Watch out! Graphical Displays of Data and Measures of Center Table Bar graphs Double bar graphs Line graphs Line plots Pictograph Frequency table

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. ml?pop=yes&pid=918 ml?pop=yes&pid=918

Watch out! Line Graphs and Measures of Center The Location 1.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.

Watch out! Frequency Tables and Measures of Center

NumberFrequency

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, 20 lower quartile: Q1= 5 median: 9 upper quartile: Q2=12 IQR = Q2-Q1= = 7 1.5*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 distribution mean = median = mode skewed left distribution mean < median < mode skewed right distribution mean > median > mode

Visualizing how the outlier pulls the mean at_010/meanmedian.html

NumberFrequency Mean, Median or Mode? mean = median = 5 mode = 1

NumberFrequency Mean, Median or Mode? mean = 5 median = 5 modes = 1 and 9

NumberFrequency Mean, Median or Mode? mean = 4.58 median = 3 mode = 2

NumberFrequency Mean, Median or Mode? mean = 8 median = 2 mode = 54

symmetric distribution mean = median = mode skewed left distribution mean < median < mode skewed right distribution mean > 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 NSX1724 Audi TT Roadster2028 BMW Z4 Roadster2028 Cadillac XLR1725 Chevrolet Corvette1825 Dodge Viper1220 Ferrari 360 Modena1116 Ferrari Maranello1016 Ford Thunderbird1723 Honda Insight6066 Lamborghini Gallardo915 Lamborghini Murcielago913 Lotus Esprit1522 Maserati Spyder1217 Mazda Miata2228 Mercedes-Benz SL Mercedes-Benz SL Nissan 350Z2026 Porsche Boxster2029 Porsche Carrera Toyota MR22632 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 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 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 3M Olympics

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 MOST Average is the mean Median, median, median, median The number in between

NLVM: Bar Chart

Another Representation of the Mean