PSY 307

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PSY 307 Statistics for the Behavioral Sciences

• Chapter 2 Describing Data with Tables and Graphs

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Class Progress To-Date
Math Readiness
Descriptives
Midterm next Monday
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Frequency Distributions

• One of the simplest forms of measurement is
  counting
• How many people show a characteristic, have a
  given value or are members of a category.
• Frequency distributions count how many
  observations exist for each value for a
  particular variable.

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Frequency Table

• A frequency table is a collection of
  observations
• Sorted into classes
• Showing the frequency for each class.
• A class is a group of observations.
• When each class consists of a single observation,
  the data is considered to be ungrouped.

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Creating a Table

• List the possible values.
• Count how many observations exist for each
  possible value.
• One way to do this is using hash-marks and
  crossing off each value.
• Figure out the corresponding percent for each
  class by dividing each frequency by the total
  scores.

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Unorganized Data

• 1, 5, 3, 3, 6, 2, 1, 5, 2, 1, 2, 6, 3, 4, 1, 6,
  2, 4, 4, 2
• A set of observations like this is difficult to
  find patterns in or interpret.

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Example
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When to Create Groups

• Grouping is a convenience that makes it easier
  for people to understand the data.
• Ungrouped data should have lt20 possible values or
  classes (not lt20 scores, cases or observations).
• Identities of individual observations are lost
  when groups are created.

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Guidelines for Grouping

• See pgs 29-30 in text.
• Each observation should be included in one and
  only one class.
• List all classes, even those with 0 frequency (no
  observations).
• All classes with upper lower boundaries should
  be equal in width.

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Optional Guidelines

• All classes should have an upper and lower
  boundary.
• Open-ended classes do occur.
• Select an interval (width) that is natural to
  think about
• 5 or 10 are convenient, 13 is not
• The lower boundary should be a multiple of class
  width (245-249).
• Aim for a total of about 10 classes.

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Gaps Between Classes

• With continuous data, there is an implied gap
  between where one boundary ends and the other
  starts.
• The size of the gap equals one unit of
  measurement the smallest possible difference
  between scores.
• That way no observations can ever fall within
  that gap.
• Class sizes account for this.

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Relative Frequency

• Relative frequency frequency of each class as a
  fraction () of the total frequency for the
  distribution.
• Relative frequency lets you compare two
  distributions of different sizes.
• Obtain the fraction by dividing the frequency for
  each group by the total frequency
• Total 1.00 (100)

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Example
4/20 .20 or 20
5/20 .25 or 25
3/20 .15 or 15
3/20 .15 or 15
2/20 .10 or 10
3/20 .15 or 15
Total 20
Total 1.0 or 100
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Cumulative Frequency

• Cumulative frequency the total number of
  observations in a class plus all lower-ranked
  classes.
• Used to compare relative standing of individual
  scores within two distributions.
• Add the frequency of each class to the
  frequencies of those below it.

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Relative Frequency (Percent) and Cumulative
Frequency
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Cumulative Proportion (Percent)

• The cumulative proportion or percent is the
  relative cumulative frequency.
• Percent proportion x 100
• It allows comparison of cumulative frequencies
  across two distributions.
• To obtain cumulative proportions divide the
  cumulative frequency by the total frequency for
  each class.
• Highest class 1.00 (100)

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Percentile Ranks

• Percentile rank percent of observations with
  the same or lower values than a given
  observation.
• Find the score, then use the cumulative percent
  as the percentile rank
• Exact ranks can be found from ungrouped data.
• Only approximate ranks can be found from grouped
  data.

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Qualitative Data

• Some categories are ordered (can be placed in a
  meaningful order)
• Military ranks, levels of schooling (elementary,
  high school, college)
• Frequencies can be converted to relative
  frequencies.
• Cumulative frequencies only make sense for
  ordered categories.

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Interpreting Tables

• First read the title, column headings and any
  footnotes.
• Where do the data come from, source?
• Next, consider whether the table is
  well-constructed does it follow the grouping
  guidelines.
• Finally, look at the data and think about whether
  it makes sense.
• Focus on overall trends, not details.

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Parts of a Graph
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Constructing Graphs

• Select the type of graph.
• Place groups on the x-axis.
• Place frequency on the y-axis.
• Values for the groups and frequencies depend on
  the data.
• Label the axes and give a title to the graph.

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Histograms

• For quantitative data only.
• Equal units across x axis represent groups.
• Equal units across y axis represent frequency.
• Use wiggly line to show breaks in the scale.
• Bars are adjacent no gaps.

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Histogram Applets

• http//www.stat.sc.edu/west/javahtml/Histogram.ht
  ml
• Uses Old Faithful geyser data
• http//www.shodor.org/interactivate/activities/his
  togram/?version1.6.0_11browserMSIEvendorSun_M
  icrosystems_Inc.
• Uses math SAT data
• Notice that bin width refers to class or
  interval size.
• SPSS automatically creates classes or intervals.

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Frequency Polygons

• Also called a line graph.
• A histogram can be converted to a frequency
  polygon by connecting the midpoints of the bars.
• Anchor the line to the x axis at beginning and
  end of distribution.
• Two frequency polygons can be superimposed for
  comparison.

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Creating a Line Graph from a Histogram
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Stem-and-Leaf Displays

• Constructing a display
• Notice the highest and lowest 10s
• Arrange 10s in ascending order.
• Copy right-hand digits as leaves.
• The resulting display resembles a frequency
  histogram.
• Stems are whatever digits make sense to use.

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Sample
Stem and leaf display showing the number of passing touchdowns.
32337
2001112223889
12244456888899
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Purpose of Frequency Graphs

• In statistics, we are interested in the shapes of
  distributions because they tell us what
  statistics to use.
• They let us identify outliers that might distort
  the statistics we will be using.
• They present data so that readers can quickly and
  easily grasp its meaning.

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Shapes of Distributions

• Normal bell-shaped and symmetrical.
• Bimodal two peaks.
• Suggests presence of two different types of
  observations in the same data.
• Positively skewed lopsided due to extreme
  observations in right tail.
• Negatively skewed extreme observations in left
  tail.

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Shapes of Graphs
bimodal
normal
positive skew
negative skew
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Heavy vs Light-tailed Distributions

• Heavy-tailed a distribution with more
  observations in its tails.
• Light-tailed a distribution with fewer
  observations in its tails and more in the center.
• Kurtosis a statistic that measures the shape of
  the distribution and the size of the tails.

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Other Kinds of Graphs

• Frequency is not the only measure that can be
  displayed on the y-axis.
• We are using a graph to explore the shape of a
  distribution in this chapter.
• Usually the y-axis shows the dependent variable
  while the x-axis shows groups (independent
  variable).
• Graphs can be visually interesting!

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Graphs Allow Visual Comparisons
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The Best Graph Ever Drawn
Source http//strangemaps.wordpress.com/
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Details About the Graph

• The map was the work of Charles Joseph Minard
  (1781-1870), a French civil engineer who was an
  inspector-general of bridges and roads, but whose
  most remembered legacy is in the field of
  statistical graphics
• The chart, or statistical graphic, is also a map.
  And a strange one at that. It depicts the advance
  into (1812) and retreat from (1813) Russia by
  Napoleons Grande Armée, which was decimated by a
  combination of the Russian winter, the Russian
  army and its scorched-earth tactics. To my
  knowledge, this is the origin of the term
  scorched earth the retreating Russians burnt
  anything that might feed or shelter the French,
  thereby severely weakening Napoleons army. It
  unites temperature, time, geography and number of
  soldiers, all in one picture.

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A Modern Version
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Qualitative Data

• Bar graphs similar to histograms.
• Bars do not touch.
• Categorical groups are on x-axis.
• Pie charts

Where tax money goes.
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Misleading Graphs

• Bars should be equal widths
• Bars should be two-dimensional, not
  three-dimensional
• When the lower bound of the y-axis (frequency) is
  cut-off (not 0), the differences are exaggerated.
• Height and width of the graph should be
  approximately equal.

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Graphs are Used to Persuade
Reagan
Bush
Clinton
Bush
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Gallups Terry Schiavo Poll
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Who Increased the Debt?
This chart is misleading because it includes
social security as debt. If expressed as a of
public debt, Bush Obama would be tied around
60-70. Obama would look 4 times worse than Bush
and twice as bad as Reagan if this were expressed
as a of income (GDP).
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Misleading Tables
Average score, reading literacy, PISA,
2009Korea 539Finland 536Canada 524New
Zealand 521Japan 520Australia 515Netherlands
508Belgium 506Norway 503Estonia
501Switzerland 501Poland 500Iceland 500United
States 500Sweden 497Germany 497Ireland
496France 496Denmark 495United Kingdom
494Hungary 494OECD average 493Portugal
489Italy 486Slovenia 483Greece 483Spain
481Czech Republic 478Slovak Republic 477Israel
474Luxembourg 472Austria 470Turkey 464Chile
449Mexico 425
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How Big are Crime Rates?
Source http//www.npr.org/templates/story/story.p
hp?storyId5480227
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How Many Groups (Categories)?
This graph is misleading because income above
200k is broken into many sub-categories, making
the 100-200k group look larger than higher income
groups.
How it would look if redrawn.
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Comparing Scales (OK)
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Misleading Scales
The range of the scales for these two variables
are too different to be compared visually without
being misleading. The crossover point at 2004
disappears when the same range is used on both
scales of the graph.
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More Misleading Graphs

• http//www.coolschool.ca/lor/AMA11/unit1/U01L02.ht
  m