Why AFM?

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Why AFM?

• This class will provide you with a variety of
  Math topics that will prepare you for your
  college classes, as well as the fact that many of
  you need this class either in order to graduate
  from High School or get into college.
• This class will prepare you for Honors Pre
  Calculus or Discrete Mathematics (the next course
  in your math sequence).
• This course will demonstrate some practical
  applications of algebra, your use beyond this
  class will be somewhat determined by your college
  major and/or career choice.

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What will we study?

• Review of Functions including linear, quadratic,
  polynomial, and piecewise.
• Exponents and Logarithms.
• Trigonometry.
• Introduction to Statistics.
• Sequences and Series.
• Probability and Odds.

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Splash!
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Calculator Exercise!
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Statistics

• The Frequency Distribution

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Essential Question

• What are some different ways that numerical data
  can be represented?

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4 Main Ways
Bar Graph
Frequency Distribution
Histogram
Frequency Polygon
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Bar Graphs

• Bar graphs are good for a variety of data
  including categories and numerical data
• Bar graphs are also good if you are wanted to
  compare data with either a back to back bar graph
  or side to side bar graph

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Bar Graphs continued

• In 1973 the music industry sold the following
  quantities 280 million albums, 228 million 45s,
  15 million cassette tapes and 0 CDs
• How would you represent this data in a bar graph?

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Bar Graphs continued

• In 1990 the music industry sold the following
  quantities 12 million albums, 28 million 45s,
  442 million cassette tapes and 287 CDs
• How would you compare this data to the previous
  data?
• How might this graph look different today?

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Bar Graphs continued
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Frequency Distribution

• Used when lots of numerical data needs to be
  grouped together
• Divides the numbers into classes of equal value
• 5 10
• 10 15
• 15 - 20

Class Interval

• Range UL LL
• All range values are the same
• Class Mark (UL LL)/2
• The Class Mark is the middle of each class find
  each

Upper Limit
Lower Limit
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Example of a Frequency Distribution
Tree Diameter (in) Frequency
0 2 6
2 4 30
4 6 38
6 8 33
8 10 0
10 - 12 4
Frequency can be zero
Includes trees 10 inches wide up to but not
including 12 inches
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Making a Frequency Distribution

• Find the range of values for the data set
  (highest number lowest number)
• Divide the range by the number of classes you
  want (for small sets of data lets say 5 or 6)
  this is the class range
• Create your classes by adding the class range to
  the lowest number (or something smaller)
• Continue until you reach a number bigger than
  your largest numer
• Tally up the numbers in each class

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Now lets try (Scores on a Test)
95 52 72 85 89 78 65 77 77 85
70 82 75 96 35 88 77 92 81 72
30 70 55 76 83 61 85 68 73 99
72 65 74 54 70 91 75 67 71 80
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Histogram

• Bar graph in which the width of each bar
  represents a class interval and the height
  represents the frequency of the data (must be
  numerical in value.

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Example of a Histogram
Tree Diameter (in) Frequency
0 2 6
2 4 30
4 6 38
6 8 33
8 10 0
10 - 12 4
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Create your own Histogram
Salary (Thousands of dollars Frequency
20 26 6
26 32 30
32 38 38
38 44 33
44 50 0

• What are the class marks?
• What percent earned between 38,000 and 44,000?
• What percent earned at least 38,000?

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Frequency Polygon

• Broken line graph created by connecting the class
  marks of each bar on a histogram

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Example of a Frequency Polygon
Tree Diameter (in) Frequency
0 2 6
2 4 30
4 6 38
6 8 33
8 10 0
10 - 12 4
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Add a Frequency Polygon to your created histogram
Salary (Thousands of dollars) Frequency
20 26 6
26 32 30
32 38 38
38 44 33
44 50 13
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Revisiting the EQ

• What are some different ways that numerical data
  can be represented?

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Homework

• Worksheet 1

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Statistics

• Measures of Central Tendency

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Essential Question

• What are the different ways to measure the
  central tendency of a data set?

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What one number best describes this data set?
Salary (Thousands of dollars) Frequency
20 26 6
26 32 30
32 38 38
38 44 33
44 50 13

• What statistical terms do we use to describe data
  so that one number represents the data?
• Mean (Average)
• Median (Middle)
• Mode (Most Frequent)

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Mean Median - Mode

• Arithmetic Mean ( )
• Median Middle Value, odd set exact middle,
  even set average of the two middle values
• Mode most frequent
• Bimodal 2 that tie for most frequent
• A data set can have no mode if there are 3 or
  more that are most frequent

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Find the Mean, Median, and Mode
95 52 72 85 89 78 65 77 77 85
70 82 75 96 35 88 77 92 81 72
30 70 55 76 83 61 85 68 73 99
72 65 74 54 70 91 75 67 71 80
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Find the Mean, Median and Mode
Country Number of Immigrants
China 41,700
Cuba 26,500
Dominican Republic 39,600
India 44,900
Jamaica 19,100
Mexico 163,600
Philippines 55,900
Russia 19,700

• Which measure of central tendency seem most
  representative of the set of data?
• Explain
• Are we OK with mean, median and mode?

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Whats the mean?
Weekly Wages Frequency
130 140 11
140 150 24
150 160 30
160 170 10
170 180 13
180 190 8
190 - 200 4

• Why would have trouble finding the mean of this
  frequency distribution?
• Can you find the mean without knowing the exact
  numbers?

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Mean of a Frequency Distribution

• f represents the frequency
• xi represents the class mark
• k represents the number of classes

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Now lets try finding the mean
Weekly Wages Frequency xi fixi
130 140 11
140 150 24
150 160 30
160 170 10
170 180 13
180 190 8
190 - 200 4

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Now you try finding the mean
Salary (Thousands of dollars) Frequency xi fixi
20 26 6 23 138
26 32 30 29 870
32 38 38 35 1330
38 44 33 41 1353
44 50 13 47 611
120 4302
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Season New Plays Season New Plays Season New Plays
1960 48 1973 43 1986 41
1961 53 1974 54 1987 32
1962 54 1975 55 1988 30
1963 63 1976 54 1989 35
1964 67 1977 42 1990 28
1965 68 1978 50 1991 37
1966 69 1979 61 1992 33
1967 74 1980 60 1993 37
1968 67 1981 48 1994 29
1969 62 1982 50 1995 38
1970 49 1983 36 1996 37
1971 55 1984 33 1997 33
1972 55 1985 33 1998 20
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From the Previous Data do the following

• Make a stem-and-leaf plot of the number of new
  productions for the seasons listed
• Find the mean of the data
• Find the median of the data
• Find the mode of the data
• What is a good representative number for the
  average of the new Broadway productions for the
  seasons 1960 1998?

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Revisiting the EQ

• What are the different ways to measure the
  central tendency of a data set?

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Homework
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Statistics

• Measures of Variability

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Essential Question

• What does the measure of variability indicate
  about a set of data?

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Introduction to Links
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Now its your turn
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Lets Digest

• How well did the mean, median, or mode represent
  the data sets we just looked at?
• What about these two
• A 35, 40, 45
• B 10, 40, 70
• Both have the same mean but in one the numbers
  are farther from the mean than in the other

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Main ways to represent variability
Range
Box and Whisker Plot
Standard Deviation
Practice!
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Range

• Difference between the greatest number and the
  least number in a set.
• What is the range of set A?
• 10
• What is the range of set B?
• 60
• In which set are the numbers more spread out?
• What if I told you a set of data has a range of
  127, is that good or bad?
• You dont know unless you know more about the
  data set.

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Box and Whisker Plots

• Graphical representation of variation
• Similar to how a Histogram lets us see what the
  average value is without actually knowing what it
  is.

10 20 30 40 50
60 70 80 90 100
110
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Box and Whisker Plots
Median (Q2)
Q1 Median of the lower half
Q3 Median of the upper half
Outlier
Extreme Value (Lower)
EV (Upper)
10 20 30 40 50
60 70 80 90 100
110
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Calories from 20 popular cereals
50 90 100 110
110 120 125 130
140 155 165 200
210 210 220 250
260 265 270 300
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Now you try

• First organize the data in increasing order
• Find the median
• Split the data into two sets and find the median
  of each
• See if there are any outliers
• Plot the extreme values

506 583
612 102
789 881
412 457
814 826
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Standard Deviation

• Numerical Method to show Variability we see
  more about Standard Deviation later

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Calories from 20 popular cereals
50 90 100 110
110 120 125 130
140 155 165 200
210 210 220 250
260 265 270 300
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Now you try
506 583
612 102
789 881
412 457
814 826
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Revisiting the EQ

• What does the measure of variability indicate
  about a set of data?

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Homework Link Sheets
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Statistics

• Review of Previous Material and a Bonus

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Essential Question

• How well can I interpret data from a data set or
  frequency distribution?

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Pairs of two, Pairs of four

• In your groups, you and your partner will work
  through the practice problems, comparing your
  answers (each person will work each problem.
• When you and your partner come to agreement on
  the correct answers you will compare your answers
  with the other group in your four.

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Review Sheet 1
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What can the calculator do?
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Revisiting the EQ

• How well can I interpret data from a data set or
  frequency distribution?
• What do you still need to work on?

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Statistics

• Modeling Day!

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Essential Question

• How well can I use what I have learned to
  interpret data?

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Univariate Data Analysis
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Revisiting the EQ

• How well can I use what I have learned to
  interpret data?

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Homework
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Statistics

• Normal Distribution

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Essential Question

• What is true about normally distributed data?

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Compare and Contrast
1 0 2 5 6 7 8 9
2 2 5 8 9 9
3 4
4 3 7
5 2 5 6
6 1 2 5 7 8
1 0 2
2 2 5 8
3 4 5 6 7 8 9
4 3 7 5 6
5 2 5
6 1
1 0
2 2 5
3 4 5
4 3 7 8 8 9
5 2 5 6 6 7 8 9
6 1 2 5 7 8 9 9
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Normal Distribution
What percent of the data would you expect in each
region of the normal distribution curve?
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Question

• The lifetime of 10,000 watch batteries are
  normally distributed. If the mean lifetime is
  500 days and the standard deviation is 60 days,
  sketch a normally distributed curve.
• What percentage of the batteries would last more
  than 500 days?
• How many batteries would last more than 500 days?
• What percentage of batteries would last between
  380 and 560 days?
• How many batteries would last between 380 and 560
  days?

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Now you try

• The scores of 500 freshman taking Psychology 101
  had an arithmetic mean of 60 with a standard
  deviation of 10.
• Sketch the curve (assume that the data is
  normally distributed)

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From the previous example

• Determine what percent and how many fit the
  following criteria
• Scores of 50 to 70
• Scores of 40 to 80
• Scores of 30 to 90
• Scores of more than 60
• Scores of less than 30
• Scores of less than 60

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Teacher Curve
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• Create a teacher curve for the 500 freshman
  taking Psychology 101.
• How many As, Bs, Cs, Ds and Fs are there?

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Revisiting the EQ

• What is true about normally distributed data?

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Homework
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Review
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Statistics

• Review for Test

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• Any Questions