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Elementary Statistics Q: What is data? Q: What does the data look like? Q: What conclusions can we draw from the data? Q: Where is the middle of the data? Q: Why is the spread of the data important? Q: Can we model the data? Q: How do we know if we have a good model?

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Elementary Statistics

Q: What is data?

Q: What does the data look like?

Q: What conclusions can we draw from the data?

Q: Where is the middle of the data?

Q: Why is the spread of the data important?

Q: Can we model the data?

Q: How do we know if we have a good model?

Q: Is our data affected by other variables?


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Definitions

Individuals :

Objects described by a set of data. Individuals may be

people, but they may also be animals or things.

Variable :

Any characteristic of an individual. A variable can take

on different values for different individuals.

Categorical and Quantitative Variables

Categorical variable :

Places an individual into one or several

categories.

Quantitative variable :

Takes numerical values for which arithmetic

operations make sense.

Distribution :

Tells what values the data takes and how often it

takes these values.


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Homework

1, 2, 4, 6


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Exploring Data

Two Basic Strategies :

1) Begin by examining each variable by itself. Then move

on to study the relationships among variables.

2) Begin with a graph or graphs. Then add numerical

summaries of specific aspects of data.

Different types of graphs :

Bar graph, Pie chart, Stemplot, back-to-back Stemplot,

Histogram, Time plot


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Bar Graphs

Grade

A

B

C

D

Other

Count

Bar graph -

A graph which displays the data using heights of bars to

represent the counts of the variables.

Example : Consider the following grade distribution :

6

12

15

9

3

How could we display the data using a bar graph ?


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Bar Graphs

15

12

9

Grade

A

B

C

D

Other

6

Count

6

12

15

9

3

3

A

B

C

D

F


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Pie Charts

Pie Chart :

1) A chart which represents the data using percentages.

2) Break up a circle (pie) into the respected percentages.


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Pie Charts

Percent

Grade

A

B

C

D

Other

Count

6

12

15

9

3

13

27

33

20

7

B

A

C

D

F


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Homework

13, 14, 16


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Stemplot

How to make a Stemplot :

1) Separate each observation into a stem consisting of all

but the final (rightmost) digit, and a leaf, the final digit.

Stems may have as many digits as needed, but each leaf

contains only a single digit.

2) Write the stems in a vertical column with the smallest at

the top, and draw a vertical line at the right of this column.

3) Write each leaf in a row to the right of the stem, in increasing

order out from the stem.


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Stemplot

4

4

5

5

Steps 1 and 2 :

6

6

Step 3 :

7

7

8

8

9

9

10

10

Example: Here are the grades Max achieved while in school his

first two years.

Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77, 82, 100,

58, 76, 83, 88, 72, 66

5

8

6

2

7

7

6

2

8

3

3

6

2

3

8

1

0

4

1

0


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Stemplot

4

4

5

5

5

8

Steps 1 and 2 :

6

6

6

Step 3 :

7

7

2

2

6

7

7

8

8

2

3

3

3

6

8

8

9

9

0

1

1

4

10

10

0

Example: Here are the grades Max achieved while in school his

first two years.

Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77, 82, 100,

58, 76, 83, 88, 72, 66


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Back-To-Back Stemplot

  • This is a stemplot which allows you to see and compare the

  • distribution of two related data sets

Example : Here are the grades Lulu received during her first two

years at college :

Grades: 66, 77, 78, 84, 92, 90, 86, 78, 71, 93, 82, 55, 73, 95,

87, 76, 93, 82, 66, 75

  • To make a Back-To-Back Stemplot, you make the stem, and the

  • stems going off to the right and the left. You want the smaller

  • values closer to the stem.


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Back-To-Back Stemplot

4

5

5

8

6

6

7

2

2

6

7

7

8

2

3

3

3

6

8

8

9

0

1

1

4

10

0

Lulu’s Grades: 66, 77, 78, 84, 92, 90, 86, 78, 71, 93, 82, 55,

73, 95, 87, 76, 93, 82, 66, 75

Max’s Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77,

82, 100, 58, 76, 83, 88, 72, 66

5

6

6

5

6

3

1

8

8

7

2

7

2

6

4

3

5

3

0

2


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Back-To-Back Stemplot

4

5

5

5

8

6

6

6

6

8

8

7

6

5

3

1

7

2

2

6

7

7

7

6

4

2

2

8

2

3

3

3

6

8

8

5

3

3

2

0

9

0

1

1

4

10

0

Lulu’s Grades: 66, 77, 78, 84, 92, 90, 86, 78, 71, 93, 82, 55,

73, 95, 87, 76, 93, 82, 66, 75

Max’s Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77,

82, 100, 58, 76, 83, 88, 72, 66


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Splitting Stems

7

8

9

  • If you have a large data set (leaves), then sometimes a stemplot

  • will not work very well. For instance, if you have a large amount

  • of leaves, and only a few stems, you might want to split the stems.

Example : Consider the following test scores :

71, 71, 72, 74, 75, 75, 75, 76, 77, 79, 80, 81, 81, 82, 83, 83, 83, 83, 84,

85, 85, 88, 89, 90, 90, 90, 91, 93, 95, 96, 97

Normally we would set up the stems as follows :


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Splitting Stems

7

8

9

  • If you have a large data set (leaves), then sometimes a stemplot

  • will not work very well. For instance, if you have a large amount

  • of leaves, and only a few stems, you might want to split the stems.

Example : Consider the following test scores :

71, 71, 72, 74, 75, 75, 75, 76, 77, 79, 80, 81, 81, 82, 83, 83, 83, 83, 84,

85, 85, 88, 89, 90, 90, 90, 91, 93, 95, 96, 97

Normally we would set up the stems as follows :

1, 1, 2, 4, 5, 5, 6, 7, 9

0, 1, 1, 2, 3, 3, 3, 3, 4, 5, 5, 8, 9

0, 0, 0, 1, 3, 5, 6 , 7


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Splitting Stems

7

7

This stem gets scores 70 - 74

8

8

This stem gets scores 75 - 79

9

9

  • If you have a large data set (leaves), then sometimes a stemplot

  • will not work very well. For instance, if you have a large amount

  • of leaves, and only a few stems, you might want to split the stems.

Example : Consider the following test scores :

71, 71, 72, 74, 75, 75, 75, 76, 77, 79, 80, 81, 81, 82, 83, 83, 83, 83, 84,

85, 85, 88, 89, 90, 90, 90, 91, 93, 95, 96, 97

However, we could set up the stems as follows :

1

1

2

4

5

5

5

6

7

9

0

1

1

2

3

3

3

3

4

5

5

8

9

0

0

0

1

3

5

6

7


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Rounding Stems

29.1

29.0

5.7

5.6

5.5

Q: What if we have a lot of stems, but not a lot of leaves?

A: One might want to join the stems into larger stems by rounding.

Example: Consider the following charges for filling a car with gas :

9.73 10.12 8.72 6.53 12.89 15.67 5.50 16.97

11.38 10.77 7.77 9.00 10.50 8.00 17.12 13.00

21.00 18.11 9.99 25.12 22.57 15.00 23.00 29.11

What would this stem look like ?


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Rounding Stems

2

1

0

Q: What if we have a lot of stems, but not a lot of leaves?

A: One might want to join the stems into larger stems by rounding.

Example: Consider the following charges for filling a car with gas :

9.73 10.12 8.72 6.53 12.89 15.67 5.50 16.97

11.38 10.77 7.77 9.00 10.50 8.00 17.12 13.00

21.00 18.11 9.99 25.12 22.57 15.00 23.00 29.11

We could round the stems to be $10 stems :

1

5

2

3

9

0

2

5

6

1

0

0

7

3

8

5

9

8

6

5

7

9

8

9


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Homework

20, 22, 23, 26


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Histograms

A histogram breaks the range of variables up into (equal) intervals,

and displays only the count or percent of the observations which

fall into the particular intervals.

Notes:

  • You can choose the intervals (usually equal)

  • Slower to construct than stemplots

  • Histograms do not display the individual observations

  • In case a score falls on an interval point, you must decide in

  • advance which interval in which the point will go.


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Histograms

Steps to drawing a histogram :

1) Divide the range into classes of equal width.

2) Count the number of observations in each class.

These are called frequencies.

3) Draw the histogram.


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Histograms

Grade

Amount

Percent

90 - 100 8 20

80 - 90 10 25

70 - 80 10 25

60 - 70 8 20

50 - 60 4 10

Frequency

Table

10

10

8

8

4

Example :

Suppose the final breakdown in grades looks like this :

50

60

70

80

90

100


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Histograms

Grade

Amount

Percent

90 - 100 8 20

80 - 90 10 25

70 - 80 10 25

60 - 70 8 20

50 - 60 4 10

25%

25%

20%

20%

10%

Example :

Suppose the final breakdown in grades looks like this :

50

60

70

80

90

100


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Homework

31, 32


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Time Plot

Variable

Time

A Time Plot is a graph with two axis. One axis represents time ,and

the other axis represents the variable being measured.


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Time Plot

89

90

91

92

93

94

95

96

97

98

Year

HR

33

39

22

42

9

9

39

52

58

70

Example : The following are homerun totals for a certain baseball

player the last 10 years :

Construct a timeplot for this data set.


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Time Plot

89

90

91

92

93

94

95

96

97

98

Year

HR

33

39

22

42

9

9

39

52

58

70

Home Run

Year


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Time Plot

89

90

91

92

93

94

95

96

97

98

Year

70

HR

33

39

22

42

9

9

39

52

58

70

60

50

40

30

20

10

89

90

91

92

93

94

95

96

97

98


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Homework

35, 36


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