Chapter 1

Chapter 1: January 29, 2004 1

Chapter 1

Picturing Distributions with Graphs

2Chapter 1: January 29, 2004

StatisticsStatistics is a science that involves the extraction of information from numerical data obtained during an experiment or from a sample. It involves:

1. the design of the experiment or sampling procedure,

2. the collection and analysis of the data, and 3. making inferences (statements, assumptions,

guesses) about the population based upon information in a sample.


Individuals and Variables Individuals

the objects described by a set of datamay be people, animals, or things

Variableany characteristic of an individualcan take different values for different

individuals


Variables Categorical

Places an individual into one of several groups or categories

Quantitative (Numerical)Takes numerical values for which

arithmetic operations such as adding and averaging make sense


Example 1.1

Individuals?

Variables?

Categorical?

Quantitative?


Distribution

Tells what values a variable takes and how often it takes these values

Can be a table, graph, or function


Displaying Distributions

Categorical variablesPie chartsBar graphs

Quantitative variablesHistogramsStemplots (stem-and-leaf plots)Time Plots


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

Example: Class Make-up on First Day


Freshman41.9%

Sophomore23.3%

Junior14.0%

Senior20.9%

Pie Chart



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

Per

cen

t


Bar Graph


Example: U.S. Solid Waste (2000)

Data Table

Material 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 %



Pie Chart



Bar Graph


Histograms

For quantitative variables that take many values

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

Count how many observations fall in each interval (may change to percents)

Draw picture representing distribution


Case Study

Weight Data

Introductory Statistics classSpring, 1997

Virginia Commonwealth University


Weight Data


Weight Data: Frequency Table

Weight Group Count 100 - <120 7 120 - <140 12 140 - <160 7 160 - <180 8 180 - <200 12 200 - <220 4 220 - <240 1 240 - <260 0 260 - <280 1

sqrt(53) = 7.2, or 8 intervals; range (260100=160) / 8 = 20 = class width


Weight Data: Histogram

Weight

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

0

2

4

6

8

10

12

14

Frequency

100 120 140 160 180 200 220 240 260 280

Nu

mb

er o

f st

ud

ents


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 only fall in exactly one interval (no overlap)


Examining the Distribution of Quantitative Data

Overall pattern of graphShape of the dataCenter of the dataSpread of the data (Variation)

Deviations from overall pattern Outliers


Shape of the Data Symmetric

bell shapedother symmetric shapes

Asymmetricright skewed left skewed

Unimodal, bimodal


Symmetric Bell-Shaped

Symmetric Mound-Shaped

Symmetric Uniform


Asymmetric Skewed to the Left

Asymmetric Skewed to the Right


Outliers Extreme values that fall outside the

overall patternMay occur naturallyMay occur due to error in recordingMay occur due to error in measuringObservational unit may be fundamentally

different


Stemplots(Stem-and-Leaf Plots)

For quantitative variables Separate each observation into a stem (first part

of the number) and a leaf (the remaining part of the number)

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

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


Weight Data


Weight Data:Stemplot(Stem & Leaf Plot)

Key

20|3 means203 pounds

Stems = 10’sLeaves = 1’s

1011121314151617181920212223242526

192

2

1522

5

135


Weight Data:Stemplot(Stem & Leaf Plot)

10 016611 00912 003457813 0035914 0815 0025716 55517 00025518 00005556719 24520 321 02522 023242526 0

Key

20|3 means203 pounds

Stems = 10’sLeaves = 1’s


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.


Extended Stem-and-Leaf PlotsExample: if all of the data values were between 150 and 179, then we may choose to use the following stems:

151516161717

Leaves 0-4 would go on each upper stem (first “15”), and leaves 5-9 would go on each lower stem (second “15”).


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), and

deviations from this trend. Connecting the data points by lines may emphasize this trend.

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


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

0%

10%

20%

30%

40%

50%

60%

70%

Percent of ClassThat Are Freshman

1985 1986 1987 1988 1989 1990 1991 1992 1993

Year of Fall Semester

Class Make-up On First Day


Average Tuition (Public vs. Private)

Chapter 1

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