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Organizing and describing Data. Instructor:. W.H.Laverty. Office:. 235 McLean Hall. Phone:. 966-6096. Lectures:. M W F 11:30am - 12:20pm Arts 143 Lab: M 3:30 - 4:20 Thorv105. Evaluation:. Assignments, Labs, Term tests - 40% Every 2nd Week (approx) – Term Test Final Examination - 60%.

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Presentation Transcript
slide2

Instructor:

W.H.Laverty

Office:

235 McLean Hall

Phone:

966-6096

Lectures:

M W F

11:30am - 12:20pm Arts 143

Lab: M 3:30 - 4:20 Thorv105

Evaluation:

Assignments, Labs, Term tests - 40%

Every 2nd Week (approx) – Term TestFinal Examination - 60%

techniques for continuous variables

Techniques for continuous variables

Continuous variables are measurements that vary over a continuum (Weight, Blood Pressure, etc.) (as opposed to categorical variables Gender, religion, Marital Status etc.)

to construct
To Construct
  • A Grouped frequency table
  • A Histogram
slide6
Find the maximum and minimum of the observations.
  • Choose non-overlapping intervals of equal width (The Class Intervals) that cover the range between the maximum and the minimum.
  • The endpoints of the intervals are called the class boundaries.
  • Count the number of observations in each interval (The cell frequency - f).
  • Calculate relative frequency

relative frequency = f/N

slide7

Data Set #3

The following table gives data on Verbal IQ, Math IQ,

Initial Reading Acheivement Score, and Final Reading Acheivement Score

for 23 students who have recently completed a reading improvement program

Initial Final

Verbal Math Reading Reading

Student IQ IQ Acheivement Acheivement

1 86 94 1.1 1.7

2 104 103 1.5 1.7

3 86 92 1.5 1.9

4 105 100 2.0 2.0

5 118 115 1.9 3.5

6 96 102 1.4 2.4

7 90 87 1.5 1.8

8 95 100 1.4 2.0

9 105 96 1.7 1.7

10 84 80 1.6 1.7

11 94 87 1.6 1.7

12 119 116 1.7 3.1

13 82 91 1.2 1.8

14 80 93 1.0 1.7

15 109 124 1.8 2.5

16 111 119 1.4 3.0

17 89 94 1.6 1.8

18 99 117 1.6 2.6

19 94 93 1.4 1.4

20 99 110 1.4 2.0

21 95 97 1.5 1.3

22 102 104 1.7 3.1

23 102 93 1.6 1.9

slide8

In this example the upper endpoint is included in the interval. The lower endpoint is not.

example
Example
  • In this example we are comparing (for two drugs A and B) the time to metabolize the drug.
  • 120 cases were given drug A.
  • 120 cases were given drug B.
  • Data on time to metabolize each drug is given on the next two slides
to construct1
To Construct
  • A Grouped frequency table
  • A Histogram
to construct a grouped frequency table
To Construct - A Grouped frequency table
  • Find the maximum and minimum of the observations.
  • Choose non-overlapping intervals of equal width (The Class Intervals) that cover the range between the maximum and the minimum.
  • The endpoints of the intervals are called the class boundaries.
  • Count the number of observations in each interval (The cell frequency - f).
  • Calculate relative frequency

relative frequency = f/N

to draw a histogram
To draw - A Histogram

Draw above each class interval:

  • A vertical bar above each Class Interval whose height is either proportional to The cell frequency (f) or the relative frequency (f/N)

frequency (f) or relative frequency (f/N)

Class Interval

some comments about histograms
Some comments about histograms
  • The width of the class intervals should be chosen so that the number of intervals with a frequency less than 5 is small.
  • This means that the width of the class intervals can decrease as the sample size increases
slide22
If the width of the class intervals is too small. The frequency in each interval will be either 0 or 1
  • The histogram will look like this
slide23
If the width of the class intervals is too large. One class interval will contain all of the observations.
  • The histogram will look like this
slide24
Ideally one wants the histogram to appear as seen below.
  • This will be achieved by making the width of the class intervals as small as possible and only allowing a few intervals to have a frequency less than 5.
slide25
As the sample size increases the histogram will approach a smooth curve.
  • This is the histogram of the population
slide31

Comment: the proportion of area under a histogram between two points estimates the proportion of cases in the sample (and the population) between those two values.

the characteristics of a histogram
The Characteristics of a Histogram
  • Central Location (average)
  • Spread (Variability, Dispersion)
  • Shape
the stem leaf plot

The Stem-Leaf Plot

An alternative to the histogram

slide46
Example

Verbal IQ = 84

84

  • Stem = 10 digit = 8
  • Leaf = Unit digit = 4

Leaf

Stem

slide47
Example

Verbal IQ = 104

104

  • Stem = 10 digit = 10
  • Leaf = Unit digit = 4

Leaf

Stem

to construct a stem leaf diagram
To Construct a Stem- Leaf diagram
  • Make a vertical list of “all” stems
  • Then behind each stem make a horizontal list of each leaf
example1
Example

The data on N = 23 students

Variables

  • Verbal IQ
  • Math IQ
  • Initial Reading Achievement Score
  • Final Reading Achievement Score
slide50

Data Set #3

The following table gives data on Verbal IQ, Math IQ,

Initial Reading Acheivement Score, and Final Reading Acheivement Score

for 23 students who have recently completed a reading improvement program

Initial Final

Verbal Math Reading Reading

Student IQ IQ Acheivement Acheivement

1 86 94 1.1 1.7

2 104 103 1.5 1.7

3 86 92 1.5 1.9

4 105 100 2.0 2.0

5 118 115 1.9 3.5

6 96 102 1.4 2.4

7 90 87 1.5 1.8

8 95 100 1.4 2.0

9 105 96 1.7 1.7

10 84 80 1.6 1.7

11 94 87 1.6 1.7

12 119 116 1.7 3.1

13 82 91 1.2 1.8

14 80 93 1.0 1.7

15 109 124 1.8 2.5

16 111 119 1.4 3.0

17 89 94 1.6 1.8

18 99 117 1.6 2.6

19 94 93 1.4 1.4

20 99 110 1.4 2.0

21 95 97 1.5 1.3

22 102 104 1.7 3.1

23 102 93 1.6 1.9

slide51
We now construct:

a stem-Leaf diagram

of Verbal IQ

slide52
A vertical list of the stems

8

9

10

11

12

We now list the leafs behind stem

slide53

8

6

10

4

8

6

10

5

11

8

9

6

9

0

9

5

10

5

8

4

9

4

11

9

8

2

8

0

10

9

11

1

8

9

9

9

9

4

9

9

8

9

10

11

12

9

5

10

2

10

2

slide54

8

6

10

4

8

6

10

5

11

8

9

6

9

0

9

5

10

5

8

4

9

4

11

9

8

2

8

0

10

9

11

1

8

9

9

9

9

4

9

9

8

9

10

11

12

9

5

10

2

10

2

slide55
8 6 6 4 2 0 9

9 6 0 5 4 9 4 9 5

10 4 5 5 9 2 2

11 8 9 1

12

the leafs may be arranged in order
The leafs may be arranged in order

8 0 2 4 6 6 9

9 0 4 4 5 5 6 9 9

10 2 2 4 5 5 9

11 1 8 9

12

the stem leaf diagram is equivalent to a histogram
The stem-leaf diagram is equivalent to a histogram

8 0 2 4 6 6 9

9 0 4 4 5 5 6 9 9

10 2 2 4 5 5 9

11 1 8 9

12

the stem leaf diagram is equivalent to a histogram1
The stem-leaf diagram is equivalent to a histogram

8 0 2 4 6 6 9

9 0 4 4 5 5 6 9 9

10 2 2 4 5 5 9

11 1 8 9

12

the two part stem leaf diagram

The two part stem leaf diagram

Sometimes you want to break the stems into two parts

for leafs 0,1,2,3,4

* for leafs 5,6,7,8,9

stem leaf diagram for initial reading acheivement

Stem-leaf diagram for Initial Reading Acheivement

01234444455556666677789

0

This diagram as it stands does not

give an accurate picture of the

distribution

we try breaking the stems into two parts 1 012344444 1 55556666677789 2 0 2
We try breaking the stems into

two parts

1.* 012344444

1. 55556666677789

2.* 0

2.

the five part stem leaf diagram

The five-part stem-leaf diagram

If the two part stem-leaf diagram is not adequate you can break the stems into five parts

for leafs 0,1

t for leafs 2,3

f for leafs 4, 5

s for leafs 6,7

* for leafs 8,9

we try breaking the stems into five parts 1 01 1 t 23 1 f 444445555 1 s 66666777 1 89 2 0
We try breaking the stems into

five parts

1.* 01

1.t 23

1.f 444445555

1.s 66666777

1. 89

2.* 0

stem leaf diagrams verbal iq math iq initial ra final ra
Stem leaf Diagrams

Verbal IQ, Math IQ, Initial RA, Final RA

some conclusions
Some Conclusions
  • Math IQ, Verbal IQ seem to have approximately the same distribution
  • “bell shaped” centered about 100
  • Final RA seems to be larger than initial RA and more spread out
  • Improvement in RA
  • Amount of improvement quite variable
next topic
Next Topic
  • Numerical Measures - Location
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