Chapter 1Picturing Distributions with Graphs

BPS - 5th Ed.Chapter 1 1

Statistics

BPS - 5th Ed.Chapter 1 2

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

Individuals and Variables

Individuals the objects described by a set of data may be people, animals, or things

Variable any characteristic of an individual can take different values for different individuals

BPS - 5th Ed.Chapter 1 3

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

BPS - 5th Ed.Chapter 1 4

Distribution

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

Can be a table, graph, or function

BPS - 5th Ed.Chapter 1 5

Displaying Distributions

Categorical variables Pie charts Bar graphs

Quantitative variables Histograms Stemplots (stem-and-leaf plots)

BPS - 5th Ed.Chapter 1 6

Class Make-up on First Day

BPS - 5th Ed.Chapter 1 7

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

BPS - 5th Ed.Chapter 1 8

Pie Chart

Class Make-up on First Day

BPS - 5th Ed.Chapter 1 9

Bar Graph

Example: U.S. Solid Waste (2000)

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 %

BPS - 5th Ed. Chapter 1 10

Data Table

Example: U.S. Solid Waste (2000)

BPS - 5th Ed.Chapter 1 11

Pie Chart

Example: U.S. Solid Waste (2000)

BPS - 5th Ed.Chapter 1 12

Bar Graph

Examining the Distribution of Quantitative Data

Overall pattern of graph Deviations from overall pattern Shape of the data Center of the data Spread of the data (Variation) Outliers

BPS - 5th Ed.Chapter 1 13

Shape of the Data

Symmetric bell shaped other symmetric shapes

Asymmetric right skewed left skewed

Unimodal, bimodal

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SymmetricBell-Shaped

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SymmetricMound-Shaped

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SymmetricUniform

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AsymmetricSkewed to the Left

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AsymmetricSkewed to the Right

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Outliers

Extreme values that fall outside the overall pattern May occur naturally May occur due to error in recording May occur due to error in measuring Observational unit may be fundamentally different

BPS - 5th Ed.Chapter 1 20

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

BPS - 5th Ed.Chapter 1 21

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)

BPS - 5th Ed.Chapter 1 22

Case Study

BPS - 5th Ed.Chapter 1 23

Weight Data

Introductory Statistics classSpring, 1997

Virginia Commonwealth University

Weight Data

BPS - 5th Ed.Chapter 1 24

Weight Data: Frequency Table

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

Weight Data: Histogram

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100 120 140 160 180 200 220 240 260 280Weight

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

Nu

mb

er

of s

tude

nts

Stemplots (Stem-and-Leaf Plots)

For quantitative variablesSeparate 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

BPS - 5th Ed.Chapter 1 27

Weight Data

BPS - 5th Ed.Chapter 1 28

12

Weight Data:Stemplot(Stem & Leaf Plot)

BPS - 5th Ed. Chapter 1 29

1011121314151617181920212223242526

Key

20|3 means203 pounds

Stems = 10’sLeaves = 1’s

192

2

1522

5

135

Weight Data:Stemplot(Stem & Leaf Plot)

BPS - 5th Ed. Chapter 1 30

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.

BPS - 5th Ed.Chapter 1 31

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:

BPS - 5th Ed.Chapter 1 32

151516161717

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