Topics: Descriptive Statistics

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Topics Descriptive Statistics

• A road map
• Examining data through frequency distributions
• Measures of central tendency
• Measures of variability
• The normal curve
• Standard scores and the standard normal
  distribution

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Raw Data Overachievement Study
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Frequency Distributions

• A method of summarizing and highlighting aspects
  of the data, showing the frequency with which
  each value occurs.
• Numerical Representations a tabular arrangement
  of scores
• Graphical Representations a pictorial
  arrangement of scores

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Tabular Frequency Distributions Single-Variable
(Univariate)
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Frequency Distribution MajorUngrouped

• MAJOR
• Valid Cum
• Value Label Value Frequency Percent Percent
  Percent
• PHYSICS 1.00 5 12.5 12.5 12.5
• CHEMISTRY 2.00 4 10.0 10.0 22.5
• BIOLOGY 3.00 7 17.5 17.5 40.0
• ENGINEERING 4.00 5 12.5 12.5 52.5
• ANTHROPOLOGY 5.00 5 12.5 12.5 65.0
• SOCIOLOGY 6.00 4 10.0 10.0 75.0
• ENGLISH 7.00 7 17.5 17.5 92.5
• DESIGN 8.00 3 7.5 7.5 100.0
• ------- ------- -------
• Total 40 100.0 100.0
• Valid cases 40 Missing cases 0

Cum
Valid
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Frequency Distribution Major Grouped

• MAJORGRP
• Valid Cum
• Value Label Value Frequency Percent Percent
• SCIENCE ENGINEERIN 1.00 21 52.5 52.5 52.5
• SOCIAL SCIENCE 2.00 9 22.5 22.5 75.0
• HUMANITIES 3.00 10 25.0 25.0 100.0
• ------- ------- -------
• Total 40 100.0 100.0

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Frequency Distribution SAT Ungrouped

• SAT
• Valid Cum
• Value Frequency Percent Percent
• 1000.00 2 5.0 5.0 5.0
• 1025.00 1 2.5 2.5 7.5
• 1050.00 2 5.0 5.0 12.5
• 1060.00 1 2.5 2.5 15.0
• 1075.00 1 2.5 2.5 17.5
• 1080.00 1 2.5 2.5 20.0
• 1085.00 1 2.5 2.5 22.5
• 1090.00 2 5.0 5.0 27.5
• 1100.00 7 17.5 17.5 45.0
• 1120.00 2 5.0 5.0 50.0
• 1125.00 3 7.5 7.5 57.5
• 1130.00 1 2.5 2.5 60.0
• 1150.00 5 12.5 12.5 72.5
• 1160.00 2 5.0 5.0 77.5
• 1175.00 3 7.5 7.5 85.0
• 1185.00 1 2.5 2.5 87.5

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Frequency Distribution SAT Grouped
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Graphical Frequency DistributionsA Picture is
Worth 1000 Words (or Numbers)

• Bar Graphs
• Histograms
• Stem and Leaf
• Frequency Polygons
• Pie Chart

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Graphical Frequency DistributionsSingle-Variab
le (Univariate)
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Bar Chart Major
Ordinate
Abscissa
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Histogram SAT(From Grouped Data)
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Frequency Polygon Overlay SAT(From Grouped Data)
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Frequency Polygon SAT(From Grouped Data)
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Frequency Polygon SAT Scores(From Ungrouped
Data)
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Stem and Leaf SAT
1000, 1000
1020
1050, 1050
Stem width 100 Each Leaf 1 case
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Graphical Frequency Distributions
Two-Variable (Joint or Bivariate)
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Relative Frequency Polygon GPAComparison of
Majors
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Relative Frequency Polygon GPA Comparison of
Gender
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What Can Be Seen in Frequency Distributions

• Shape
• Central Tendency
• Variability

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Shapes of Frequency Polygons
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Descriptive Statistics

• Central Tendency
• Mode
• Median
• Mean
• Variability
• Range
• Standard Deviation
• Variance

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Definitions Measures of Central Tendency

• Mean
• Arithmetic mean
• Median
• The number that lies at the midpoint of the
  distribution of scores divides the distribution
  into two equal halves
• Mode
• Most frequently occurring score

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Relative Position of Mode, Median, and Mean
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Mean, Median, ModeSAT Scores by Gender
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Mean, Median, ModeSAT Scores by Area
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Choosing Appropriate Measure of Central Tendency
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DefinitionsMeasures of Variability(Spread)

• Range
• Difference between highest and lowest score
• Inter-quartile Range
• The spread of the middle 50 of the scores
• The difference between the top 25 (Upper
  Quartile-Q3) and the lower 25 (Lower
  Quartile-Q1)
• Variance
• The average variability of scores (measured in
  squared units of the original scores (square of
  the standard deviation)
• Standard Deviation
• The average dispersion or deviation of scores
  around the mean (measured in original score
  units) (squareroot of the variance)

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Variance

• The average of each scores squared difference
  from the mean (mean of squared deviations)
• To calculate
• Find the mean
• Subtract the mean from each score
• Square each of the deviation (difference) scores
• Add up the squared deviations (sum of squared
  deviations)
• Divide by n-1

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Variance Calculation Teacher Service in
Particular School
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Standard Deviation

• Compute the variance and then take the square
  root
• Roughly the average amount that scores differ
  from the mean

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Range, Interquartile Range, and Standard
Deviation SAT Scores by Area
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Range, Interquartile Range, and Standard
Deviation SAT Scores by Gender
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Properties of Normal Distribution

• Bell-shaped (unimodal)
• Symmetric about the mean
• Mode, median, and mean are equal (approximately
  the same)
• Asymptotic (curve never touches the abscissa)

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Normal Curve
Areas Under the Curve
.3413
.3413
.1359
.1359
68
.0214
.0214
95
.0013
.0013
99
-1s
-2s
1s
2s
-3s
3s
X
s.d.
mean
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SD/Mean and Normal Curve

• Use s.d. relative to mean to determine if
  distribution of scores on a given variable is
  normal
• Determine possible range of scores
• Add 3 s.d.s to either side of the mean
• If the result is within range of possible scores,
  then distribution is likely normal
• If result is outside of range of possible scoes,
  then distribution is likely skewed.

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Definitions Standard Scores

• Standard Scores scores expressed as SD away
  from the mean (z-scores)
• Obtained by finding how far a score is above or
  below the mean and dividing that difference by
  the SD
• Changes mean to 0 and SD to 1, but does not
  change the shape (called Standard Normal
  Distribution)

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Standard Normal Distribution
.3413
.3413
.1359
.1359
68
.0214
.0214
95
.0013
.0013
99
-1
-2
1
2
-3
3
0
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Choosing Appropriate Measure of Variability