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BPS - 5th Ed. Chapter 1 1

Chapter 1

Picturing Distributions with Graphs

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Statistics

Statistics is a science that involves the extraction of

information from numerical data obtained during an

experiment or from a sample. It involves the designof the experiment or sampling procedure, the

collection and analysis of the data, and making

inferences (statements) about the population based

upon information in a sample.

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Individuals and Variables

Individuals

the objects described by a set of data

may be people, animals, or thingsVariable

any characteristic of an individual

can take different values for differentindividuals

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Variables

Categorical

Places an individual into one of severalgroups or categories

Quantitative (Numerical)

Takes numerical values for whicharithmetic operations such as adding andaveraging make sense

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Case Study

The Effect of Hypnosison the

Immune System

reported in Science News, Sept. 4, 1993, p. 153

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Case Study

The Effect of Hypnosison the

Immune SystemObjective:

To determine if hypnosis strengthens the

disease-fighting capacity of immune cells.

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Case Study

65 college students.

33 easily hypnotized

32 not easily hypnotizedwhite blood cell counts measured

all students viewed a brief video about

the immune system.

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Case Study

Students randomly assigned to one ofthree conditions

subjects hypnotized, given mental exercise subjects relaxed in sensory deprivation

tank

control group (no treatment)

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Case Study

white blood cell counts re-measured after oneweek

the two white blood cell counts are compared

for each group

results

hypnotized group showed larger jump in white

blood cells easily hypnotized group showed largest immune

enhancement

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Case Study

Variables measured

Easyor difficultto achieve

hypnotic tranceGroup assignment

Pre-study white blood cell

count Post-study white blood cell

count

categorical

quantitative

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Case Study

Weight Gain Spells

Heart Risk for WomenWeight, weight change, and coronary heart disease

in women. W.C. Willett, et. al., vol. 273(6), Journalof the American Medical Association, Feb. 8, 1995.

(Reported in Science News, Feb. 4, 1995, p. 108)

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Case Study

Weight Gain Spells

Heart Risk for WomenObjective:

To recommend a range of body mass index

(a function of weight and height) in terms ofcoronary heart disease (CHD) risk in women.

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Case Study

Study started in 1976 with 115,818women aged 30 to 55 years and withouta history of previous CHD.

Each womans weight (body mass) was

determined.

Each woman was asked her weight atage 18.

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Case Study

The cohort of women were followed for14 years.

The number of CHD (fatal and nonfatal)cases were counted (1292 cases).

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Case Study

Age (in 1976)

Weight in 1976

Weight at age 18

Incidence of coronary heartdisease

Smoker or nonsmoker Family history of heart disease

quantitative

categorical

Variables measured

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Distribution

Tells what values a variable takes andhow often it takes these values

Can be a table, graph, or function

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Displaying Distributions

Categorical variables

Pie charts

Bar graphsQuantitative variables

Histograms

Stemplots (stem-and-leaf plots)

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Year Count Percent

Freshman 18 41.9%

Sophomore 10 23.3%

Junior 6 14.0%

Senior 9 20.9%

Total 43 100.1%

Data Table

Class Make-up on First Day

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Freshman41.9%

Sophomore

23.3%

Junior

14.0%

Senior

20.9%

Pie Chart


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41.9%

23.3%

14.0%

20.9%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

35.0%

40.0%

45.0%

Freshman Sophomore Junior Senior

Year in School

Percent


Bar Graph

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Example: U.S. Solid Waste (2000)

Data TableMaterial Weight (million tons) Percent of total

Food scraps 25.9 11.2 %

Glass 12.8 5.5 %

Metals 18.0 7.8 %

Paper, paperboard 86.7 37.4 %

Plastics 24.7 10.7 %

Rubber, leather, textiles 15.8 6.8 %

Wood 12.7 5.5 %

Yard trimmings 27.7 11.9 %

Other 7.5 3.2 %

Total 231.9 100.0 %

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Pie Chart

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Bar Graph

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Examining the Distribution of

Quantitative DataOverall pattern of graph

Deviations from overall pattern

Shape of the data

Center of the data

Spread of the data (Variation)Outliers

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Shape of the Data

Symmetric

bell shaped

other symmetric shapes

Asymmetric

right skewed

left skewed

Unimodal, bimodal

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Symmetric

Bell-Shaped

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Symmetric

Mound-Shaped

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Symmetric

Uniform

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Asymmetric

Skewed to the Left

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Asymmetric

Skewed to the Right

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Outliers

Extreme values that fall outside theoverall pattern

May occur naturally

May occur due to error in recording

May occur due to error in measuring

Observational unit may be fundamentally

different

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Histograms

For quantitative variables that takemany values

Divide the possible values into classintervals (we will only consider equal widths)

Count how many observations fall in

each interval(may change to percents)

Draw picture representing distribution

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Histograms: Class Intervals

How many intervals?

One rule is to calculate the square root of the

sample size, and round up.

Size of intervals?

Divide range of data (maxmin) by number of

intervals desired, and round to convenient number

Pick intervals so each observation can onlyfall in exactly one interval (no overlap)

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Case Study

Weight Data

Introductory Statistics classSpring, 1997

Virginia Commonwealth University

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Weight Data

192 110 195 180 170 215

152 120 170 130 130 125

135 185 120 155 101 194

110 165 185 220 180

128 212 175 140 187180 119 203 157 148

260 165 185 150 106

170 210 123 172 180

165 186 139 175 127

150 100 106 133 124

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Weight Data: Frequency Table

Weight Group Count100 -

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Weight Data: Histogram

0

2

4

6

8

10

12

14

Frequency

100 120 140 160 180 200 220 240 260 280Weight

* Left endpoint is included in the group, right endpoint is not.

Numberofstudents

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Stemplots(Stem-and-Leaf Plots)

For quantitative variables

Separate each observation into a stem (firstpart of the number) and a leaf (the remaining

part of the number)

Write the stems in a vertical column; draw avertical line to the right of the stems

Write each leaf in the row to the right of itsstem; order leaves if desired

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Weight Data

192 110 195 180 170 215

152 120 170 130 130 125

135 185 120 155 101 194

110 165 185 220 180

128 212 175 140 187180 119 203 157 148

260 165 185 150 106

170 210 123 172 180

165 186 139 175 127

150 100 106 133 124

1

2

10

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Weight Data:

Stemplot(Stem & Leaf Plot)

10

11

12

13

1415

16

17

18

19

20

21

22

23

24

25

26

Key

20|3 means

203 pounds

Stems = 10sLeaves = 1s

192

2

1522

5

135

10 0166

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Weight Data:

Stemplot(Stem & Leaf Plot)

10 0166

11 009

12 0034578

13 00359

14 0815 00257

16 555

17 000255

18 000055567

19 245

20 3

21 025

22 0

23

24

25

26 0

Key

20|3 means

203 pounds

Stems = 10sLeaves = 1s

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Extended Stem-and-Leaf Plots

If there are very few stems (when the

data cover only a very small range of

values), then we may want to create

more stems by splitting the original

stems.

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Extended Stem-and-Leaf Plots

Example: if all of the data values were

between 150 and 179, then we may

choose to use the following stems:151516

161717

Leaves 0-4 would go on eachupper stem (first 15), and leaves

5-9 would go on each lower stem(second 15).

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Time Plots

A time plot shows behavior over time. Time is always on the horizontal axis, and the

variable being measured is on the vertical axis.

Look for an overall pattern (trend), anddeviations from this trend. Connecting the data

points by lines may emphasize this trend.

Look for patterns that repeat at known regularintervals (seasonal variations).

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Class Make-up on First Day(Fall Semesters: 1985-1993)

0%

10%

20%

30%

40%

50%

60%

70%

Percent of Class

That Are Freshman

1985 1986 1987 1988 1989 1990 1991 1992 1993

Year of Fall Semester

Class Make-up On First Day

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Average Tuition (Public vs. Private)

ma116+chapter 01

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