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Organizing and describing DataPowerPoint Presentation

Organizing and describing Data

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### Techniques for continuous variables

### The Grouped frequency table:The Histogram

### The Grouped frequency table: interval. The lower endpoint is not.The Histogram

### The Stem-Leaf Plot than 0.34

### The two part stem leaf diagram than 0.34

### Stem-leaf diagram for Initial Reading Acheivement than 0.34

### The five-part stem-leaf diagram than 0.34

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%

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

- A Grouped frequency table
- A Histogram

- 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

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

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

Histogram – Verbal IQ interval. The lower endpoint is not.

Histogram – Math IQ interval. The lower endpoint is not.

Example interval. The lower endpoint is not.

- 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

Drug A interval. The lower endpoint is not.

Drug B interval. The lower endpoint is not.

Grouped frequency tables interval. The lower endpoint is not.

Histogram – drug A interval. The lower endpoint is not.(time to metabolize)

Histogram – drug B interval. The lower endpoint is not.(time to metabolize)

To Construct interval. The lower endpoint is not.

- A Grouped frequency table
- A Histogram

To Construct - A Grouped frequency table interval. The lower endpoint is not.

- 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 interval. The lower endpoint is not.

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 interval. The lower endpoint is not.

- 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

- 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

- 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

- Ideally one wants the histogram to appear as seen below. interval will contain all of the observations.
- 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.

- As the sample size increases the histogram will approach a smooth curve.
- This is the histogram of the population

N = 25 smooth curve.

N = 100 smooth curve.

N = 500 smooth curve.

N = 2000 smooth curve.

N = smooth curve.∞

Comment: smooth curve. 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.

Example: smooth curve. The following histogram displays the birth weight (in Kg’s) of n = 100 births

The Characteristics of a Histogram than 0.34

- Central Location (average)
- Spread (Variability, Dispersion)
- Shape

Central Location than 0.34

Spread, Dispersion, Variability than 0.34

Shape – Bell Shaped (Normal) than 0.34

Shape – Positively skewed than 0.34

Shape – Negatively skewed than 0.34

Shape – Platykurtic than 0.34

Shape – Leptokurtic than 0.34

Shape – Bimodal than 0.34

An alternative to the histogram

Each number in a data set can be broken into two parts than 0.34

- A stem
- A Leaf

To Construct a Stem- Leaf diagram than 0.34

- Make a vertical list of “all” stems
- Then behind each stem make a horizontal list of each leaf

Example than 0.34

The data on N = 23 students

Variables

- Verbal IQ
- Math IQ
- Initial Reading Achievement Score
- Final Reading Achievement Score

Data Set #3 than 0.34

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

8 than 0.34

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The leafs may be arranged in order than 0.34

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 than 0.34

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 than 0.34

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

Sometimes you want to break the stems into two parts

for leafs 0,1,2,3,4

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

01234444455556666677789

0

This diagram as it stands does not

give an accurate picture of the

distribution

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 than 0.34

five parts

1.* 01

1.t 23

1.f 444445555

1.s 66666777

1. 89

2.* 0

Stem leaf Diagrams than 0.34

Verbal IQ, Math IQ, Initial RA, Final RA

Some Conclusions than 0.34

- 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 than 0.34

- Numerical Measures - Location

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