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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. The Role of Description. Description as a purpose of research - PowerPoint PPT Presentation
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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
The Role of Description
• Description as a purpose of research
• Choosing the right statistical procedures
Raw Data: Overachievement Study
Frequency Distributions
• A method of summarizing and highlighting aspects of the data in a data matrix, showing the frequency with which each value occurs.
• Numerical Representations: a tabular arrangement of scores
• Graphical Representations: a pictorial arrangement of scores
Numerical Frequency Distributions
• Ungrouped Frequency Distributions
• Grouped Frequency Distributions
• Relative Frequency Distributions
• Cumulative Frequency Distributions
Tabular Frequency Distributions
Single-Variable (“Univariate”)
Frequency Distribution: Major
MAJOR
Valid Cum
Value Label Value Frequency Percent Percent Percent
PHYSICS 1.00 5 12.5 12.5 12.5
CHEMISTRY2.00 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
SOCIOLOGY6.00 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
Frequency Distribution: Major Group
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
Frequency Distribution: SATSAT
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
1200.00 5 12.5 12.5 100.0
------- ------- -------
Total 40 100.0 100.0
Valid cases 40 Missing cases 0
Grouped Frequency Distribution: SAT
Graphical Frequency Distributions
• Bar Graphs
• Histograms
• Stem and Leaf
• Frequency Polygons
• Pie Chart
Graphical Frequency Distributions:
Single-Variable (“Univariate”)
Graphical Frequency Distributions:
Single-Variable (“Univariate”)
Bar Chart: Major
Bar Chart
MAJOR
DESIGNENGLISH
SOCIOLOGYANTHROPOLOGY
ENGINEERINGBIOLOGY
CHEMISTRYPHYSICS
Frequency
8
7
6
5
4
3
2
1
0
Histogram: SAT(From Grouped Data)
Frequency Polygon Overlay: SAT(From Grouped Data)
Frequency Polygon: SAT(From Grouped Data)
Frequency Polygon: SAT Scores(From Ungrouped Data)
Frequency Polygon: SAT
SAT
1200.001185.00
1175.001160.00
1150.001130.00
1125.001120.00
1100.001090.00
1085.001080.00
1075.001060.00
1050.001025.00
1000.00
Count
8
7
6
5
4
3
2
1
0
Cumulative Frequency Polygon: SAT Scores
SAT
1200.001185.00
1175.001160.00
1150.001130.00
1125.001120.00
1100.001090.00
1085.001080.00
1075.001060.00
1050.001025.00
1000.00
Cumulative Frequency
50
40
30
20
10
0
Stem and Leaf: SAT
Stem and Leaf: SAT
SAT Stem-and-Leaf Plot
Frequency Stem & Leaf
3.00 10 . 002 8.00 10 . 55678899 13.00 11 . 0000000222223 11.00 11 . 55555667778 5.00 12 . 00000
Stem width: 100.00 Each leaf: 1 case(s)
Graphical Frequency Distributions
Two-Variable (“Joint” or “Bivariate”)
Graphical Frequency Distributions
Two-Variable (“Joint” or “Bivariate”)
Relative Frequency Polygon: GPAComparison of Majors
GPA
3.603.50
3.403.30
3.203.10
3.002.90
2.802.70
2.502.30
2.00
Percent
40
30
20
10
0
MAJORGRP
SCIENCE & ENGINEERIN
SOCIAL SCIENCE
HUMANITIES
Relative Frequency Polygon: GPA Comparison of Gender
SEX
MALE
FEMALE
GPA
3.603.503.403.303.203.103.002.902.802.702.502.302.00
Percent
30
20
10
0
What Can Be Seen in Frequency Distributions
• Shape
• Central Tendency
• Variability
Shapes of Frequency Polygons
Shapes of Distributions
Bell-Shaped
Prototype:
Normal Distribution
SYMMETRIC
Hump in Distribution
at High Score End
Tail at Low Score End
NEGATIVELY SKEWED
Hump in Distribution
at Low Score End
Tail at High Score End
POSITIVELY SKEWED
Very Peaked in the Center
Compared to
Normal Distribution
LEPTOKURTIC
Peak Just Like
the
Normal Distribution
MESOKURTIC
Flat in the Center
Compared to
Normal Distribution
PLATYKURTIC
Descriptive Statistics
• Central Tendency– Mode– Median– Mean
• Variability– Range– Standard Deviation– Variance
Definitions: Measures of Central Tendency
• Mean:
– “Arithmetic mean”
– “Center of gravity” such that the “weight” of the scores above the mean exactly balances the “weight” of the scores below the 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
Mean, Median, Mode:SAT Scores by Gender
Group Mode Median Mean
Male 1200 1112.50 1112.00
Female 1100 1122.50 1129.50
Total 1100.00 1122.50 1122.75
Mean, Median, Mode:SAT Scores by Area
Group Mode Median Mean
Humanities 1100 1092.50 1095.00
Social Sciences 1100 1100.00 1108.89
Sciences 1150,1200 1150.00 1138.10
Total 1100 1122.50 1122.75
Relative Position of Mode, Median, and Mean
Definitions:Measures of Variability
• 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)
• Standard Deviation:– The average dispersion or deviation of scores around the mean (measured
in original score units)
• Variance:– The average variability of scores (measured in squared units of the
original scores (square of the standard deviation)
Range, Interquartile Range, and Standard Deviation: SAT Scores by Area
Group Range IQ Range Standard
Deviation
Humanities 200 35.00 55.88
Social Sciences 95 15.00 28.59
Sciences 200 27.50 57.00
Range, Interquartile Range, and Standard Deviation: SAT Scores by Gender
Group Range IQ Range StandardDeviation
Males 200 100 60.92
Females 175 75 46.02
Total 200 70 54.02
Properties of Normal Distribution
• Bell-shaped (unimodal)
• Symmetric about the mean
• Mode, median, and mean are equal (though rarely occurs)
• Asymptotic (curve never touches the abscissa)
.3413
.1359 .1359
.0214.0214
.3413
Normal CurveAreas Under the Curve
X-1s-2s +1s +2s-3s +3s
.0013 .0013
68%
95%
99%
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)
Uses of Standard Normal Distribution
• What proportion of scores falls between the mean and a given raw score
• What proportion of scores falls above or below a given raw score
• What proportion of scores falls between two raw scores
• What raw score fall above (or below) a certain percentage of scores
