Displaying and Organizing Data

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Displaying and Organizing Data

• Chapter 3

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Objectives

• Understand the basics of frequency distributions
• Be able to present frequency distributions using
  tables or figures
• Describe and identify a distribution by
  characteristics of its shape

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Definitions

• Frequency is the number of times an event occurs
  in the population/sample
• Ex Number of students with a GPA of 4.0
• Frequency Distribution
• A distribution in which the values of the
  dependant variable are tabled or plotted against
  their frequency of occurrence
• Frequency distributions are displayed via a table
  or a graphic plot

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Example Frequency Table

• Frequency distribution of GPA values
• Value Number of Students
  4.0
  250
• 3.5 350
  
• 3.0 600
  
  2.5 450

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How to make a frequency table

• Make a list of all of the possible values in the
  data (from highest to lowest)
• Go through the data and record how many times
  each value occurs
• Make a table listing the values and the frequency
  of occurrence

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Example

• Using the following data
• 40 40 41 41 41 42 43 43 45 48 50 52 52
  52 55 55 55 55 55 56
• Make a frequency table

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Grouped Frequency Tables

• Often we have too many values to make a frequency
  table that includes each value of the DV
• So, we group the values into intervals (or
  classes)
• Ex Instead of using 40, 41, 42, 43, 44, . We
  use 40-43, 44-47, .

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How to make a grouped frequency table

• Make a simple ordered list of all of the values
  and their frequency
• Subtract the lowest value from the highest to get
  the range of the values
• Divide the range into reasonable intervals
• Try to use intervals of 2, 3, 5, 10
• You should have more than 5 intervals but less
  than 15

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How to make a grouped frequency table (cont.)

• Intervals must be of equal size
• Intervals cannot overlap
• Make a list of the intervals from lowest to
  highest
• Using your list from step 1, record the frequency
  of each value in the interval

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Example

• Using the following data
• 40 40 41 41 41 42 43 43 45 48 50 52 52
  52 55 55 55 55 55 56
• Make a group frequency table

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A note about the limits of grouped frequency
tables

• Real upper and lower limit of an interval
• The points half way between the top of one
  interval and the bottom of the next
• Ex The real upper and lower limits for the
  interval from 35 to 40 are 34.5 40.5
• If your values contain decimals use the real
  limits to decide which interval to place a value
• The midpoint of an interval is the upper and
  lower limit divided by 2

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Example

• What are the real upper and lower limits of our
  group frequency table?
• What is the midpoint of each interval?

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Relative Percent (frequency)

• Relative percent is the percent of the data that
  fall within a value or interval
• Compute the relative percent by dividing the
  frequency by the sample size and then multiply by
  100
• Relative Percent (f/n)100
• Relative Portionf/n

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Example

• Using the group frequency table we created,
  compute the relative percent for each interval
• Answer the following question
• What percent of the scores fall between the 54
  55?

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Cumulative Frequencies

• The sum of the frequency of a value or interval
  plus the frequencies of the previous values or
  intervals
• The cumulative frequency of the last value or
  interval should equal the sample size
• Cumulative percent (cf/n)100
• An Ogive is a line graph that plots the upper
  limits by the cumulative frequency

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Example

• Using the group frequency table we created,
  compute the cumulative frequency and cumulative
  percent for each interval
• Answer the following question
• How many scores fall below 50-51 interval?

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Percentiles

• The percentile gives the score that exceeds a
  given percentage
• use the cumulative percent

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Example

• Using our group frequency table
• What is the percentile of the score 45?
  Interpret.
• What scores are associated with the 50th 95th
  percentiles?
• To solve this, use the upper limit of the
  interval
• Interpret each of the above percentiles

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Plotting Frequency Distributions

• A histogram is a graph in which rectangles are
  used to represent frequencies of observation
  within each interval

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How to make a histogram

• Make a frequency or group frequency table
• Place the intervals along the X axis (bottom of
  the graph)
• Start on the left side with the lowest value
• Along the Y axis (left side of graph) put the
  frequencies
• Make a bar for each interval that goes up to the
  frequency of that interval

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Example

• Using our grouped frequency table, create a
  histogram

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Rules of thumb for plotting data

• Remember the goal of graphics is to visually
  communicate your point
• If a graphic doesnt clearly make the point,
  dont use it
• Keep it simple
• Often the simplest graphs are the most effective

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Describing Distributions

• A distribution is the pattern of your scores
• We often are interested in the shape of the
  distribution, so we plot the scores
• The ends of a distribution are called the tails
• Symmetry
• Skewness
• Peakedness

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Symmetry

• In a symmetric distribution, the left and right
  side of the distribution are identical
• We use a perfectly symmetrical distribution as
  the standard of comparison

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Skewness

• A measure of the degree to which a distribution
  is symmetrical
• Two Types
• Positive Skew
• Negative Skew
• Positive and Negative refer to the direction the
  tail points to
• Outliers are extreme data points

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Negative Skew

• The tail points off to the left

Outliers
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How to determine negative skew from a frequency
distribution

• Score
• 0 0
• 1 1 2
  1 3
  2
• 4 7
• 5 10
  6 15
  7 11
• 8 9
• 9 5

• All of the larger frequencies are concentrated at
  the higher numbers the smaller frequencies at
  the lower numbers
• This tells us the distribution is negatively
  skewed

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Positive Skew

• The tail points off to the right

Outliers
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How to determine positive skew from a frequency
distribution

• All of the larger frequencies are concentrated at
  the lower numbers the smaller frequencies at
  the higher numbers
• This tells us the distribution is positively
  skewed

• Score
• 0 9
• 1 10 2
  15 3
  10
• 4 7
• 5 4
  6 2
  7 1
• 8 1
• 9 1

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Modality

• Modality is used to describe the number of peak
  of a distribution
• Unimodal Bimodal

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How to determine modality from a frequency
distribution

• Score 0
  0 1 17
  2 5 3
  7 4 10
  5 7
  6 4
  7 17
  8 2
  

• Look for the frequency with the highest
  occurrence
• Look if there is another frequency that equals it
• This tells us the distributions modality

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Kurtosis

• This is how peaked or flat a distribution is

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Tomorrow

• Read Chapter 4
• We will be discussing measures of central
  tendency