1.2 Displaying and Describing Categorical

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1.2 Displaying and Describing Categorical
Quantitative Data
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You should be able to

• Recognize when a variable is categorical or
  quantitative
• Choose an appropriate display for a categorical
  variable and a quantitative variable
• Summarize the distribution with a bar, pie chart,
  stem-leaf plot, histogram, dot plot, box plots
• Know how to make a contingency table
• Describe the distribution of categorical
  variables in terms of relative frequencies
• Be able to describe the distribution of
  quantitative variables in terms of its shape,
  center and spread
• Describe abnormalities or extraordinary features
  of distribution
• Discuss outliers and how they deviate from the
  overall pattern

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3 RULES of EXPLORATORY DATA ANALYSIS

1. MAKE A PICTURE find patterns in difficult to
   see from a chart
2. MAKE A PICTURE show important features in graph
3. MAKE A PICTURE- communicates your data to others

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Concepts to know!

• Bar graph
• Histogram
• Dot plot
• Stem leaf plot
• Scatterplots
• Boxplots

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Which graph to use?

• Depends on type of data
• Depends on what you want to illustrate

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Categorical Data

• The objects being studied are grouped into
  categories based on some qualitative trait.
• The resulting data are merely
• labels or categories.

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Categorical Data(Single Variable)
Eye Color BLUE BROWN GREEN
Frequency (COUNTS) 20 50 5
Relative Frequency 20/75 .27 50/75 .66 5/75 .07
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Pie Chart(Data is Counts or Percentages)
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Bar Graph(Shows distribution of data)
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Bar Graph

• Summarizes categorical data.
• Horizontal axis represents categories, while
  vertical axis represents either counts
  (frequencies) or percentages (relative
  frequencies).
• Used to illustrate the differences in percentages
  (or counts) between categories.

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Contingency Table(How data is distributed
across multiple variables)
Class Class Class Class Class Class
Survival First Second Third Crew Total
Survival ALIVE 203 118 178 212 711
Survival DEAD 122 167 528 673 1490
Survival Total 325 285 706 885 2201
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What can go wrong when working with categorical
data?

• Pay attention to the variables and what the
  percentages represent
• (9.4 of passengers who were in first class
  survived is different from 67 of survivors were
  first class passengers!!!)
• Make sure you have a reasonably large data set
  (67 of the rats tested died and 1 lived)
  

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Analogy
Bar chart is to categorical data as histogram is
to ...
quantitative data.
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Histogram
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Histogram

• Divide measurement up into equal-sized categories
  (BIN WIDTH)
• Determine number (or percentage) of measurements
  falling into each category.
• Draw a bar for each category so bars heights
  represent number (or percent) falling into the
  categories.
• Label and title appropriately.
• http//www.stat.sc.edu/west/javahtml/Histogram.ht
  ml

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Histogram
Use common sense in determining number of
categories to use. Between 6 15 intervals is
preferable
(Trial-and-error works fine, too.)
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Too few categories
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Too many categories
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Dot Plot
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Dot Plot

• Summarizes quantitative data.
• Horizontal axis represents measurement scale.
• Plot one dot for each data point.

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Stem-and-Leaf Plot
Stem-and-leaf of Shoes N 139 Leaf Unit
1.0 12 0 223334444444 63 0
55555555555556666666667777777888888888888899999999
9 (33) 1 000000000000011112222233333333444
43 1 555555556667777888 25 2
0000000000023 12 2 5557 8 3 0023
4 3 4 4 00 2 4 2 5 0
1 5 1 6 1 6 1 7
1 7 5
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Stem-and-Leaf Plot

• Summarizes quantitative data.
• Each data point is broken down into a stem and
  a leaf.
• First, stems are aligned in a column.
• Then, leaves are attached to the stems.

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Box Plot
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Box Plot

• Summarizes quantitative data.
• Vertical (or horizontal) axis represents
  measurement scale.
• Lines in box represent the 25th percentile
  (first quartile), the 50th percentile
  (median), and the 75th percentile (third
  quartile), respectively.

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5 Number Summary

• Minimum
• Q1 (25th percentile)
• Median (50th percentile)
• Q3 (75th percentile)
• Maximum

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An aside...

• Roughly speaking
• The 25th percentile is the number such that 25
  of the data points fall below the number.
• The median or 50th percentile is the number
  such that half of the data points fall below the
  number.
• The 75th percentile is the number such that 75
  of the data points fall below the number.

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Box Plot (contd)

• Outliers are drawn to the most extreme data
  points that are not more than 1.5 times the
  length of the box beyond either quartile(IQR).
• IQR Q3 - Q1
• Outliers(upper) gt Q31.5 IQR
• Outliers(lower)ltQ1-1.5IQR

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Using Box Plots to Compare
Outliers
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Strengths and Weaknesses of Graphs for
Quantitative Data

• Histograms
• Uses intervals
• Good to judge the shape of a data
• Not good for small data sets
• Stem-Leaf Plots
• Good for sorting data (find the median)
• Not good for large data sets

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Strengths and Weaknesses of Graphs for
Quantitative Data

• Dotplots
• Uses individual data points
• Good to show general descriptions of center and
  variation
• Not good for judging shape for large data sets
• Boxplots
• Good for showing exact look at center, spread and
  outliers
• Not good for judging shape

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Analogy
Contingency table is to categorical data with two
variables as scatterplot is to ..
quantitative data with two variables.
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Scatter Plots
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Scatter Plots

• Summarizes the relationship between two
  quantitative variables.
• Horizontal axis represents one variable and
  vertical axis represents second variable.
• Plot one point for each pair of measurements.

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No relationship
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Summary

• Many possible types of graphs.
• Use common sense in reading graphs.
• When creating graphs, dont summarize your data
  too much or too little.
• When creating graphs, label everything for
  others.
• Remember you are trying to communicate something
  to others!

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• GRAPHICAL ANALYSIS

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• CENTER
• SPREAD

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Graphical Analysis
Center (Location)
Spread (Variation)
Shape
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Interesting Features Identified by Graphs

• Center (Location)
• Spread (Variability)
• Shape
• Individual Values
• Compare Groups
• Identify Outliers
• LOOK FOR PATTERNS, CLUSTERS, GAPS!!!!!
• LOOK FOR DEVIATIONS FROM THE GENERAL PATTERN!!!!!
• http//bcs.whfreeman.com/bps3e/

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Shape of Graph
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Shape of Graph

• Symmetry-Skewness
• Modes(peaks)

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SYMMETRY SKEWNESS

• Skewness - a measure of symmetry, or more
  precisely, the lack of symmetry.
• A distribution, or data set, is symmetric if it
  looks the same to the left and right of the
  center point.

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Kurtosis

• Kurtosis - a measure of whether the data are
  peaked or flat relative to a normal distribution.
  
• High kurtosis tend to have a distinct peak near
  the mean, decline rather rapidly, and have heavy
  tails. L
• Low kurtosis tend to have a flat top near the
  mean rather than a sharp peak.

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Symmetric -- Mesokurtic
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Symmetric -- Platykurtic
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Symmetric -- Leptokurtic
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Nonsymmetric Skewed Positive -Skewed Right
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Nonsymmetric Skewed Negative -- Skewed Left
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Symmetric -- Bimodal

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Describing Distributions SOCS Rock!

• Shape
• Outliers
• Center
• Spread

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Graphical Analysis - Scale
SOL Scores Class 1
SOL Scores Class 2
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Graphical Analysis - Scale
SOL Scores Class 1
SOL Scores Class 2
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SOL Scores Class 1
SOL Scores Class 2
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• Comparing Histograms