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## Exploring Data with Graphs and Numerical Summaries

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### EXPLORING DATA WITH GRAPHS AND NUMERICAL SUMMARIES Chapter 2 Math 1530 - Chapter 2 Math 1530 - Chapter 2 Math 1530 - Chapter 2 Math 1530 - Chapter 2 Math 1530 ... – PowerPoint PPT presentation

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Title: Exploring Data with Graphs and Numerical Summaries

1
Exploring Data with Graphs and Numerical
Summaries
• Chapter 2

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2.1 What Are the Types of Data?
4
2.1 Objectives
• Know the definitions of
• Variable
• Categorical versus quantitative variable
• Discrete versus continuous
• Frequency, proportion (relative frequencies), and
percentages
• Be able to create Frequency Tables.

www.managementscientist.org
5
Variable
• A variable is any characteristic that is recorded
for the subjects in a study
• Examples Marital status, Height, Weight, IQ
• A variable can be classified as either
• Categorical or
• Quantitative
• Discrete or
• Continuous

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Categorical Variable
• A variable is categorical if each observation
belongs to one of a set of categories.
• Examples
• Gender (Male or Female)
• Religion (Catholic, Jewish, )
• Type of residence (Apt, Condo, )
• Belief in life after death (Yes or No)

www.post-gazette.com
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Quantitative Variable
• A variable is called quantitative if observations
take numerical values for different magnitudes of
the variable.
• Examples
• Age
• Number of siblings
• Annual Income

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Quantitative vs. Categorical
• For Quantitative variables, key features are the
center and spread (variability).
• For Categorical variables, a key feature is the
percentage of observations in each of the
categories .

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Discrete Quantitative Variable
• A quantitative variable is discrete if its
possible values form a set of separate numbers
0,1,2,3,.
• Examples
• Number of pets in a household
• Number of children in a family
• Number of foreign languages spoken by an
individual

upload.wikimedia.org
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Continuous Quantitative Variable
• A quantitative variable is continuous if its
possible values form an interval
• Measurements
• Examples
• Height/Weight
• Age
• Blood pressure

www.wtvq.com
11
Proportion Percentage (Rel. Freq.)
• Proportions and percentages are also called
relative frequencies.

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Frequency Table
• A frequency table is a listing of possible values
for a variable, together with the number of
observations or relative frequencies for each
value.

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2.2 Describe Data Using Graphical Summaries
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Learning Objectives
• Understand distributions
• Graph categorical data bar graphs and pie charts
• Graph quantitative data dot plot, stem-leaf, and
histogram
• Construct histograms
• Interpret histograms
• Display data over time time plots

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Distribution
• Tells possible values of data as well as the
occurrence of those values (frequency or relative
frequency)
• Represented in
• Tables
• Graphs or Charts

www.gravic.com
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Graphs for Categorical Variables
• Use pie charts and bar graphs to summarize
categorical variables
• Pie Chart A circle having a slice of pie for
each category
• Bar Graph A graph that displays a vertical bar
for each category

wpf.amcharts.com
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Pie Charts
• Summarize categorical variable
• Drawn as circle where each category is a slice
• The size of each slice is proportional to the
percentage in that category

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Bar Graphs
• Summarizes categorical variable
• Bar height represents counts or percentages
• Easier to compare categories with bar graph than
with pie chart
• Called Pareto Charts when ordered from tallest to
shortest

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Graphs for Quantitative Data
• Dot Plot shows a dot for each observation
placed above its value on a number line
• Stem-and-Leaf Plot portrays the individual
observations
• Histogram uses bars to portray the data

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Which Graph?
• Dot-plot and stem-and-leaf plot
• More useful for small data sets
• Data values are retained
• Histogram
• More useful for large data sets
• Most compact display
• More flexibility in defining intervals

content.answers.com
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Dot Plots
• To construct a dot plot
• Draw and label horizontal line
• Mark regular values
• Place a dot above each value on the number line

Sodium in Cereals
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Stem-and-leaf plots
• Summarizes quantitative variables
• Separate each observation into a stem (first part
of ) and a leaf (last digit)
• Write each leaf to the right of its stem order
leaves if desired

Sodium in Cereals
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Histograms
• Graph that uses bars to portray frequencies or
relative frequencies of possible outcomes for a
quantitative variable

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Constructing a Histogram
Sodium in Cereals
• Divide into intervals of equal width
• Count of observations in each interval

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Constructing a Histogram
• Label endpoints of intervals on horizontal axis
• Draw a bar over each value or interval with
height equal to its frequency (or percentage)
• Label and title

Sodium in Cereals
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Interpreting Histograms
• Assess where a distribution is centered by
finding the median
• Assess the spread of a distribution
• Shape of a distribution roughly symmetric,
skewed to the right, or skewed to the left

Left and right sides are mirror images
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Examples of Skewness
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Shape Type of Mound
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Shape and Skewness
• Consider a data set containing IQ scores for the
general public. What shape?
• Symmetric
• Skewed to the left
• Skewed to the right
• Bimodal

botit.botany.wisc.edu
33
Shape and Skewness
• Consider a data set of the scores of students on
an easy exam in which most score very well but a
few score poorly. What shape?
• Symmetric
• Skewed to the left
• Skewed to the right
• Bimodal

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Outlier
• An outlier falls far from the rest of the data

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Time Plots
• Display a time series, data collected over time
• Plots observation on the vertical against time on
the horizontal
• Points are usually connected
• Common patterns should be noted

Time Plot from 1995 2001 of the worldwide who
use the Internet
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2.3 Describe the Center of Quantitative Data
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Learning Objectives
• Calculating the mean
• Calculating the median
• Comparing the mean median
• Definition of resistant
• Know how to identify the mode of a distribution

flowjo.typepad.com
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Mean
• The mean is the sum of the observations divided
by the number of observations
• It is the center of mass

47
Median
• Midpoint of the observations when ordered from
least to greatest
• Order observations
• If the number of observations is
• Odd, the median is the middle observation
• Even, the median is the average of the two middle
observations

48
Comparing the Mean and Median
• Mean and median of a symmetric distribution are
close
• Mean is often preferred because it uses all
• In a skewed distribution, the mean is farther out
in the skewed tail than is the median
• Median is preferred because it is better
representative of a typical observation

49
Resistant Measures
• A measure is resistant if extreme observations
(outliers) have little, if any, influence on its
value
• Median is resistant to outliers
• Mean is not resistant to outliers

www.stat.psu.edu
50
Mode
• Value that occurs most often
• Highest bar in the histogram
• Mode is most often used with categorical data

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2.4 Describe the Spread of Quantitative Data
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Learning Objectives
• Calculate the range
• Calculate the standard deviation
• Know the properties of the standard deviation
• Know how to interpret the magnitude of s The
Empirical Rule

www.math.armstrong.edu
56
Range
• Range max ? min
• The range is strongly affected by outliers.

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Standard Deviation
• Each data value has an associated deviation from
the mean,
• A deviation is positive if it falls above the
mean and negative if it falls below the mean
• The sum of the deviations is always zero

58
Standard Deviation
• Standard deviation gives a measure of variation
by summarizing the deviations of each observation
from the mean and calculating an adjusted average
of these deviations
• Find mean
• Find each deviation
• Square deviations
• Sum squared deviations
• Divide sum by n-1
• Take square root

59
Standard Deviation
• Metabolic rates of 7 men (calories/24 hours)

60
Properties of Sample Standard Deviation
• Measures spread of data
• Only zero when all observations are same
otherwise, s gt 0
• As the spread increases, s gets larger
• Same units as observations
• Not resistant
• Strong skewness or outliers greatly increase s

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Empirical Rule Magnitude of s
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2.5 How Measures of Position Describe Spread
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Learning Objectives
• Obtaining quartiles and 5 number summary
• Calculating interquartile range and detecting
potential outliers
• Drawing boxplots
• Comparing distributions
• Calculating a z-score

math.youngzones.org
67
Percentile
• The pth percentile is a value such that p percent
of the observations fall below or at that value

68
Finding Quartiles
• Splits the data into four parts
• Arrange data in order
• The median is the second quartile, Q2
• Q1 is the median of the lower half of the
observations
• Q3 is the median of the upper half of the
observations

69
Measure of Spread Quartiles
• Quartiles divide a ranked data set into four
equal parts
• 25 of the data at or below Q1 and 75 above
• 50 of the obs are above the median and 50 are
below
• 75 of the data at or below Q3 and 25 above

Q1 first quartile 2.2
M median 3.4
Q3 third quartile 4.35
70
Calculating Interquartile Range
• The interquartile range is the distance between
the thirdand first quartile, giving spread of
middle 50 of the data IQR Q3 ? Q1

71
Criteria for Identifying an Outlier
• An observation is a potential outlier if it falls
more than 1.5 x IQR below the first or more than
1.5 x IQR above the third quartile.

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5 Number Summary
• The five-number summary of a dataset consists of
• Minimum value
• First Quartile
• Median
• Third Quartile
• Maximum value

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Boxplot
• Box goes from the Q1 to Q3
• Line is drawn inside the box at the median
• Line goes from lower end of box to smallest
observation not a potential outlier and from
upper end of box to largest observation not a
potential outlier
• Potential outliers are shown separately, often
with or

74
Comparing Distributions
Boxplots do not display the shape of the
distribution as clearly as histograms, but are
useful for making graphical comparisons of two or
more distributions
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Z-Score
• An observation from a bell-shaped distribution is
a potential outlier if its z-score lt -3 or gt 3

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2.6 How Can Graphical Summaries Be Misused?
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Learning Objectives
• Identify misleading data displays
• Create displays of data
• Compare displays of data

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Misleading Data Displays
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Guidelines for Constructing Effective Graphs
• Label axes and give proper headings
• Vertical axis should start at zero
• Use bars, lines, or points
• Consider using separate graphs or ratios when
variable values differ

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Image Sources
• Statistics The Art and Science of Learning from
Data, 2nd Edition, Agresti and Franklinhttp//www
.managementscientist.org/wp-content/uploads/2008/0
6/dice.jpg
• http//www.thewallstickercompany.com.au/dynamic_im
ages/images-2349.jpg
• http//www.post-gazette.com/pg/images/200708/20070
828BizReligion_dm_500.jpg
• http//www.linnealenkus.com/image/siblingsLLS15.jp
g
• http//www.visualizingeconomics.com/wp-content/upl
oads/2005_income_distribution.gif
• http//upload.wikimedia.org/wikipedia/commons/d/dc
/Cats_Petunia_and_Mimosa_2004.jpg
• http//www.wtvq.com/images/news/HEALTH/bloodpressu
re.jpg
• http//blueroof.files.wordpress.com/2006/10/pie-ch
art.jpg
• http//www.gravic.com/remark/higher-ed/exams/sampl
es/Class20Frequency20Distribution20Report.png
• http//wpf.amcharts.com/lib/screenshots/frontpagem
ix.png
• http//content.answers.com/main/content/img/oxford
/Oxford_Statistics/0199541454.dot-plot.1.jpg
• http//botit.botany.wisc.edu/TOMS_FUNGI/images/ein
stein3.jpg
• http//4.bp.blogspot.com/_zsIIz0xhqxA/SY-qc6M_HuI/
AAAAAAAAABg/HkohLKVMb9s/s400/exam-stress.jpg
• http//www.stat.psu.edu/online/program/stat504/01_
overview/graphics/mean_median.gif
• http//www.math.armstrong.edu/statsonline/3/emprul
e.gif
• http//math.youngzones.org/boxplot.gif
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