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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?

Elementary Statistics

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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?

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.

Homework

1, 2, 4, 6

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

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 ?

Bar Graphs

15

12

9

Grade

A

B

C

D

Other

6

Count

6

12

15

9

3

3

A

B

C

D

F

Pie Charts

Pie Chart :

1) A chart which represents the data using percentages.

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

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

Homework

13, 14, 16

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.

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

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

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.

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

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

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 :

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

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

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 ?

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

Homework

20, 22, 23, 26

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.

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.

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

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

Homework

31, 32

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.

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.

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

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

Homework

35, 36