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Picturing Distributions with Graphs

Stat 1510 Statistical Thinking & Concepts

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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 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.

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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

Example – Temperature, Pressure, Weight

Height, Sex, Major Course, etc.

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Variables Categorical

– Places an individual into one of several groups or categories

Examples – Sex, Grade (A, B, C..), Number of Defects, Type of Defects, Status of application

Quantitative (Numerical)– Takes numerical values for which arithmetic

operations such as adding and averaging make sense

Examples – Height, Weight, Pressure, etc.

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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

Weight Gain SpellsHeart Risk for Women

“Weight, weight change, and coronary heart disease in women.” W.C. Willett, et. al., vol. 273(6), Journal of the American Medical Association, Feb. 8, 1995.

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

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Case Study

Weight Gain SpellsHeart Risk for Women

Objective:To recommend a range of body mass index (a function of weight and height) in terms of

coronary heart disease (CHD) risk in women.

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Case Study

Study started in 1976 with 115,818 women aged 30 to 55 years and without a history of previous CHD.

Each woman’s weight (body mass) was determined.

Each woman was asked her weight at age 18.

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Case Study

The cohort of women were followed for 14 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 heart

disease Smoker or nonsmoker Family history of heart disease

quantitative

categorical

Variables measured

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Study on Laptop Objective is to identify the type of laptop

computers used by university students. A random sample of 1000 university

students selected for this study Each student is asked the question

whether s/he have a laptop and if yes, the type of laptop (brand name)

Variables ?

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Distribution

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

Can be a table, graph, or function

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Displaying Distributions

Categorical variables– Pie charts– Bar graphs

Quantitative 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%

Sophomore23.3%

Junior14.0%

Senior20.9%

Pie Chart

Class Make-up on First Day

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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

Per

cen

t

Class Make-up on First DayBar 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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Example: U.S. Solid Waste (2000)

Pie Chart

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Example: U.S. Solid Waste (2000)

Bar Graph

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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), 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).

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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

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Average Tuition (Public vs. Private)

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Examining the Distribution of Quantitative Data

Observe overall pattern 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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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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Color Density of SONY TV

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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

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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

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Histograms: Class Intervals

How many intervals?– One idea: Square root of the sample size ( round

the value)

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)

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Usefulness of Histograms

To know the central value of the group To know the extent of variation in the group To estimate the percentage non-

conformance, if some specified values are available

To see whether non-conformance is due to shift In mean or large variability

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Case Study

Weight Data

Introductory Statistics classSpring, 1997

Virginia Commonwealth University

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Weight Data

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Weight Data: Frequency TableWeight 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

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Weight Data: Histogram

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

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Histogram of Soft Drink Weight

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Histogram of Soft Drink Weight

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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

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Weight Data12

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Weight Data:Stemplot

(Stem & Leaf Plot)

1011121314151617181920212223242526

Key

20|3 means203 pounds

Stems = 10’sLeaves = 1’s

192

2

1522

5

135

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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

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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:

151516161717

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