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1
Picturing Distributions with Graphs
Stat 1510 Statistical Thinking & Concepts
2
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
39
40
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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
45
Weight Data:Stemplot
(Stem & Leaf Plot)
1011121314151617181920212223242526
Key
20|3 means203 pounds
Stems = 10’sLeaves = 1’s
192
2
1522
5
135
46
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
47
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.
48
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”).