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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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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?

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.

1, 2, 4, 6

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

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 ?

Pie Chart :

1) A chart which represents the data using percentages.

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

13, 14, 16

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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4

5

5

Steps 1 and 2 :

6

6

Step 3 :

7

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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

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8

Steps 1 and 2 :

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Step 3 :

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2

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0

1

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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

- 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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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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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

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 :

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

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

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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 ?

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 :

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20, 22, 23, 26

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.

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.

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

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4

Example :

Suppose the final breakdown in grades looks like this :

50

60

70

80

90

100

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

31, 32

Variable

Time

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

the other axis represents the variable being measured.

89

90

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Year

HR

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39

22

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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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Year

70

HR

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89

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35, 36

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