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### Techniques for continuous variables

### The Stem-Leaf Plot

### The two part stem leaf diagram

### Stem-leaf diagram for Initial Reading Acheivement

### The five-part stem-leaf diagram

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.

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 Construct

- A Grouped frequency table
- A Histogram

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

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

- 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.

- 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

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.

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

The Characteristics of a Histogram

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

An alternative to the histogram

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

- A stem
- A Leaf

To Construct a Stem- Leaf diagram

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

Example

The data on N = 23 students

Variables

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

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

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 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

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

Stem leaf Diagrams

Verbal IQ, Math IQ, Initial RA, Final RA

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

- Numerical Measures - Location

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