Transcript the median - Math GR. 6-8

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
•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
mode
range
•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.
MODEL: FINDING THE MEAN
• 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?
6
6 15
n
n
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?
6
6 15
n
n
6  6 15  n  n 27  2n

12
5
5
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?
6
6 15
a
b
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?
6
6 15
a
b
6  6 15  a  b 27  a  b

12
5
5
27  a  b  60
a  b  33
Missing Observations: Mean
Here are Jane’s scores on her first 4 math tests:
80 82 75 79
What score will she need to earn on the fifth test
for her test average (mean) to be an 80%?
80  82  75  79  n
 80
5
316  n
 80
5
400  316  n
n  84
Missing Observations: Mean
Here are Jane’s scores on her first 4 math tests:
80 82 75 79
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:
80 82 75 79
There 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
 90
5
316  n
 90
5
450  316  n
n  134
Missing Observations: Median
Here are Jane’s scores on her first 4 math tests:
80 82 75 79
What 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: 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 82
70?
75?
79?
80?
81?
82?
83?
84?
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• 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
Adding a constant k
• 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 6
mean = 5
median = 5.5
mode = 6
range = 8
Adding a constant k
5
6
7
9
2
4
1
6
mean = 5
median = 5.5
mode = 6
range = 8
5+2=
6+2=
7+2=
9+2=
2+2=
4+2=
1+2=
6+2=
7
8
mean = 7
9
11 median = 7.5
mode = 8
4
range = 8
6
3
8
Multiplying by a constant k
• 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 6
mean = 5
median = 5.5
mode = 6
range = 8
Multiplying by a constant k
5
6
7
9
2
4
1
6
mean = 5
median = 5.5
mode = 6
range = 8
5×2
=
6×2
=
7×2
=
9×2
=
2×2
=
10
12
14 mean = 10
18 median = 11
mode = 12
4
range = 16
8
2
12
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!
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.ht
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
n 1
the formula 2 , where n is the number of
observations in the data set.
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
1
2
3
4
5
6
7
8
9
Frequency
6
1
2
3
5
5
4
3
0
Watch out!
Reviewing How Frequency Tables are Made
Choosing an appropriate measure
of center
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
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= 12 - 5 = 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
http://bcs.whfreeman.com/fapp7e/content/c
at_010/meanmedian.html
Mean, Median or Mode?
Number
1
2
3
4
5
6
7
8
9
Frequency
6
1
2
3
5
5
4
3
0
mean = 4.896
median = 5
mode = 1
Mean, Median or Mode?
Number
1
2
3
4
5
6
7
8
9
Frequency
9
8
7
6
5
6
7
8
9
mean = 5
median = 5
modes = 1 and 9
Mean, Median or Mode?
Number
1
2
3
4
5
6
7
8
9
Frequency
7
20
15
11
8
3
2
0
15
mean = 4.58
median = 3
mode = 2
Mean, Median or Mode?
Number
1
2
3
4
5
6
7
8
9
Frequency
3
2
3
2
1
3
2
2
54
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
With
Outlier
mean
median
mode
range
Without
Outlier
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
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 MOST
Average is the mean
Median, median, median,
median
The number in between
NLVM: Bar Chart
Another Representation of the
Mean