Frequency Distributions

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Frequency Distributions Graphing
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Nomenclature

• Frequency number of cases or subjects or
  occurrences
• represented with f
• i.e. f 12 for a score of 25
• 12 occurrences of 25 in the sample

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Nomenclature

• Percentage number of cases or subjects or
  occurrences expressed per 100
• represented with P or
• So, if f 12 for a score of 25 when n 25,
  then...
• 12/25100 48

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Caveat (Warning)

• Should report the f when presenting percentages
• i.e. 80 of the elementary students came from a
  family with an income lt 25,000
• different interpretation if n 5 compared to n
  100
• report in literature as
• f 4 (80) OR
• 80 (f 4) OR 80 (n 4)

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Frequency Distribution of Test Scores
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• 40 items on exam
• Most students gt34
• skewed (more scores at one end of the scale)
• Cumulative Percentage how many subjects in and
  below a given score

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Eyeball check of data intro to graphing with
SPSS
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• Stem and Leaf Plot quick viewing of data
  distribution
• Boxplot visual representation of many of the
  descriptive statistics discussed last week
• Bar Chart frequency of all cases
• Histogram malleable bar chart
• Scatterplot displays all cases based on two
  values of interest (X Y)
• Note compare to our previous discussion of
  distributions (normal, positively skewed, etc)

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Stem and Leaf(SPSS Explore command)
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Frequency Stem Leaf 2.00 Extremes
(lt25.0) 2.00 28 . 00 2.00 29 .
00 1.00 30 . 0 1.00 31 . 0
3.00 32 . 000 1.00 33 . 0 6.00 34
. 000000 3.00 35 . 000 4.00 36 .
0000 8.00 37 . 00000000 Stem width 1
Each leaf 1 case

• Fast look at shape of distribution
• shows f numerically graphically
• stem is value, leaf is f

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Stem and Leaf Plots

• Another way of doing a stemplot
• Babe Ruths home runs in each of 14 seasons with
  the NY Yankees
• 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46,
  41, 34, 22

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2 25 3 45 4 1166679 5 449 6 0
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Stem and Leaf Plots

• Back-to-back stem plots allow you to visualize
  two data sets at the same time
• Babe Ruth vs. Roger Maris

Maris Ruth
0 1 2 25 3 45 4 1166679 5
449 6 0
8 643 863 93 1
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Boxplots
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Maximum Q3 Median Q1 Minimum
Note we can also do side-by-side boxplots for a
visual comparison of data sets
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Format of Bar Chart
Y axis (ordinate)
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f
X axis (abcissa)
Individual scores/categories
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Test score data as Bar Chart
Note only scores with non-zero frequencies are
included.
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Bar chart in PASW

• Using the height file on the web

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Bar chart in SPSS

• Gives

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Bar chart in PASW

• Note you can use the same command for pie charts
  and histograms (next)

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Format of Histogram
Now the X-axis is groups of scores, rather than
individual scores gives a better idea of the
distribution underlying the data.
Y axis (ordinate)
f
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X axis (abcissa)
Can be manipulated
Groups of scores/categories
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Test score data as Histogram
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Test score data as revised Histogram
With an altered number of groups, you might get a
better idea of the distribution
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Scatterplot
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• Quick way to visualize the data see trends,
  patterns, etc
• This plot visually shows the relationship between
  undergrad GPA and GRE scores for applicants to
  our program

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Scatterplot
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• Heres the relationship between undergrad GPA
  (admitgpa) and GPA in our program

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Scatterplot
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• Finally, heres the relationship between GRE
  scores and GPA in our program

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Scatterplot in PASW
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• Use graphs_scatter/Dot

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Scatterplot in PASW

• Choose simple scatter

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Scatterplot in PASW

• Choose the variables (here Ive used a 3rd
  variable too youll see why in a moment)

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Scatterplot in PASW
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As you can see, there are rather different values
for males and females
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Bottom line

• First step should always be to plot the data and
  eyeball it...following is an example of what can
  happen when you do.

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One use of Frequency Distribution Skewness
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Expected distribution of agent-paid claims (State
Farm)
high
low
amount
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One use of Frequency Distribution Skewness
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f
Observed distribution of an agent-paid claims
(hmmm)
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1
high
low
amount