Organizing Data

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Organizing Data

• Proportions, Percentages, Rates, and rates of
  change.

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Raw Data

• Often hard to interpret just a bunch of raw
  scores
• Raw scores can be transformed to show patterns
  and trends in the data
• Most useful is the frequency distribution or
  table

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Frequency Tables will have

• Informative title
• Two columns for nominal data
• (1) response and
• (2) frequency (How often did certain responses
  occur?)

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Standardizing data

• Proportion compare the number of cases for each
  response (frequency, f) with the total number of
  cases (N).
• Proportion
• frequency / number f / N
• In the previous example, 20 out of 45 students
  earned a B, so the proportion earning a B is
  20/45 .44444444, which (rounding to 2 decimals
  more than the original data) .44

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• Percentage is the frequency per 100 cases. (It is
  a special case of a proportion.)
• Percentage 100 (f / N)
• People are used to thinking in percentages
  (such as in cents per dollar....).

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Example

• 20 our of 45 students earned a B in a course.
• Proportion f / N 20/45 0.44
• Percentage 100 (20/35) 44
• (Per cent means per 100, and we write it 0/0. Per
  thousand would be 0/00)

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Ratios

• A ratio of a to b is the frequency of a
  compared to the frequency of b, with the
  frequency of a coming first, or in the
  numerator, just as it does in the sentence.
• a/b or sometimes expressed as ab

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Comparisons using the Frequency Ratio f1 / f2

• In a certain class, there were 15 women and 30
  men, in a class of 45. So, in the class,
• Proportion of women 15/45 0.33
• Percentage of women (100).33 33
• (note this is not 0.33)

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Ratio depends on how the question is stated.

• Ratio of women to men 15/30 1/2, or there
  was 1 woman for every 2 men.
• However, the ratio of men to women would be 30/15
  2 men for every woman.
• Note ratio is used differently than is the
  proportion in the class.

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Rate

• A rate indicates the number of actual cases
  compared to the number of potential cases. Pretty
  subtle, eh?
• For population studies, these are usually
  expressed as the number of actual cases per 1000
  potential cases (usually per 1000 people in the
  population).

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Example

• A town has 5000 people, of whom 450 have
  graduated from college.
• The towns college graduation rate is
  450/5000 .09 9 or
• 90 per thousand.
• (Why might I express this a per thousand? I chose
  the per part so the number was something easily
  visualized.)

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What denominators to use?

• per 100 percentage
• per 1000 commonly used for birth and death
  rates, divorces, etc.
• per 100,000 for lots of things determined in the
  U.S. census
• per 1,000,000 for things determined worldwide

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Generalization

• Use the denominator that gives you the simplest
  whole number, easiest for you to grasp. Usually
  this is a number between 1 and 100.
• Its hard for people to visualize the meaning of
  very small or large numbers such as 0.00123, or
  132,431,000

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Mortality Rates for example

• Mortality Rates per 1000 among blacks whites in
  Baltimore in 1972 were
• for whites, 15.2 per 1000 (or 1.52)
• for blacks, 9.8 per 1000 (or 0.98)
• Easier to visualize than .0152 for whites and
  .0098 for blacks. Do you agree?

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Powers of 10 Review

• Suppose a disease rate of .000567 per person (per
  capita).
• To convert into something more comprehensible,
  move the decimal point to the right 4 places, to
  5.67.
• 4 places 10,000 (4 zeroes),
• so this becomes 5.67 per 10,000. or go one step
  further to 56.7 per 100,000.

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Rates of change

• (100) Rate 2 Rate 1 / Rate 1
• then convert into the proper units (per 100,
  1000, etc.)
• Ex a towns population increases from 20,000 to
  30,000 between 1990 and 2005 (note rate of
  change can be positive or negative)
• (100) time2f - time1f (100) 30,000-20,000
  50
• time 1f
  20,000
• Increase of 50

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• Organizing the Data
• Review of Frequency Distributions
  Histograms

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Frequency Distributions

• List or plot data
• Nominal Data -- in any order
• Ordinal Interval Data Usually highest number
  at top of table to lowest number at bottom of the
  table

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Statistics Class Height Data Plotted from
shortest to tallest
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Intervals Grouping Data

• range of values in the data set
• numbers of class intervals desired
• size of class interval
• upper limit of a class interval
• lower limit of a class interval

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Statistics Class Height Data Grouped in 2 inch
intervals
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4 intervals
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6 intervals
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Cumulative

• Cumulative Frequencies number of cases at or
  below a given score.
• Cumulative Percentages percent of cases at or
  below a given score.
• Also percentile rank

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Class Limits

• Upper class limit the highest possible score
  which would round down to be included in that
  class.
• Lower class limit the lowest possible score
  which would round up to be included in that
  class.

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Midpoints of Intervals

• Lowest possible score for that interval
• plus highest possible score value
• Divided by 2

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Midpoints

• The interval of 58-61 actually has limits from
  57.5 to 61.5, so 57.5 61.5 119
• 119/2 59.5 is the midpoint.
• Yes, wed usually get the same answer by saying
  (58 61) / 2 however, for irregular classes,
  it is better if we get used to the lowest value
  being 57.5 and the highest being 61.5.

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Cumulative Frequency

• To expand our frequency table, add columns for
  cumulative frequency, percent, and cumulative
  percent.
• Arrange your scores from low at the bottom to
  high at the top. Then, the Cumulative Frequency
  is simply the frequency of scores at or below the
  value in question.

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Percentile Rank

• the cumulative percentage
• The at or below that score
• So for a height of 54, or 64, what is the
  percentile rank in our height data?
• The following chart shows frequency, cum. freq.,
  percentage, cumulative .

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Percentile Rank

• 64-65 has a cumulative percent of 59.37, so
  59.37 of class is in this category or shorter
  than this category.
• 62-63 has a cumulative percent of 40.62, so
  40.62 of class is in this category or shorter
  than this category
• So, percentile rank cumulative percent when
  looking at the raw data -- but it is more
  complex for grouped data, so be wary.

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• Cross-tabulations

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Cross-Tabulation

• Cross-tabulation review
• a table which presents the distribution of one
  variable (frequency and/or ) across the
  categories of one or more additional variables.

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Common Cross-Tab Example
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Cross-Tab Table 2.15

• If asking questions about the differences between
  males females in seat belt use, use column
  percents.
• If asking questions about different uses of seat
  belts by the population as a whole, use the row
  percents.
• Hint If totals are not given -- put them in
  before you start to evaluate.

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Cross-Tab Table 2.15
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Data Format on SPSS

• Note that when you are working with raw data sets
  on the computer, you will put each case in a row,
  rather than making a cross-tabulation table. We
  will do this when we work with SPSS.