This document provides an overview of descriptive statistics concepts including frequency distributions, measures of central tendency, measures of variability, and the normal distribution. It discusses numerical and graphical representations of univariate and bivariate frequency distributions using examples from an overachievement study dataset. Key measures like the mean, median, mode, range, standard deviation, and normal curve are defined. Standard scores and their uses are also introduced.

1. 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

2. The Role of Description
• Description as a purpose of research
• Choosing the right statistical procedures

3. Raw Data: Overachievement Study

4. 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

5. Numerical Frequency
Distributions
• Ungrouped Frequency Distributions
• Grouped Frequency Distributions
• Relative Frequency Distributions
• Cumulative Frequency Distributions

6. Tabular Frequency
Distributions
Single-Variable (“Univariate”)

7. Frequency Distribution: Major
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

8. 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

9. Frequency Distribution: SAT
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
1200.00 5 12.5 12.5 100.0
------- ------- -------
Total 40 100.0 100.0
Valid cases 40 Missing cases 0

10. Grouped Frequency Distribution: SAT

11. Graphical Frequency
Distributions
• Bar Graphs
• Histograms
• Stem and Leaf
• Frequency Polygons
• Pie Chart

12. Graphical Frequency
Distributions:
Single-Variable (“Univariate”)

13. Bar Chart: Major
Bar Chart
MAJOR
DESIGN
ENGLISH
SOCIOLOGY
ANTHROPOLOGY
ENGINEERING
BIOLOGY
CHEMISTRY
PHYSICS
F
r
e
q
u
e
n
c
y
8
7
6
5
4
3
2
1
0

14. Histogram: SAT
(From Grouped Data)

15. Frequency Polygon Overlay: SAT
(From Grouped Data)

16. Frequency Polygon: SAT
(From Grouped Data)

17. Frequency Polygon: SAT Scores
(From Ungrouped Data)
Frequency Polygon: SAT
SAT
1200.00
1185.00
1175.00
1160.00
1150.00
1130.00
1125.00
1120.00
1100.00
1090.00
1085.00
1080.00
1075.00
1060.00
1050.00
1025.00
1000.00
C
o
u
n
t
8
7
6
5
4
3
2
1
0

18. Cumulative Frequency Polygon: SAT
Scores
SAT
1200.00
1185.00
1175.00
1160.00
1150.00
1130.00
1125.00
1120.00
1100.00
1090.00
1085.00
1080.00
1075.00
1060.00
1050.00
1025.00
1000.00
C
u
m
u
l
a
t
i
v
e
F
r
e
q
u
e
n
c
y
50
40
30
20
10
0

19. Stem and Leaf: SAT

20. 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)

21. Graphical Frequency
Distributions
Two-Variable (“Joint” or “Bivariate”)

22. Relative Frequency Polygon: GPA
Comparison of Majors
GPA
3.60
3.50
3.40
3.30
3.20
3.10
3.00
2.90
2.80
2.70
2.50
2.30
2.00
P
e
r
c
e
n
t
40
30
20
10
0
MAJORGRP
SCIENCE & ENGINEERIN
SOCIAL SCIENCE
HUMANITIES

23. Relative Frequency Polygon: GPA
Comparison of Gender
SEX
MALE
FEMALE
GPA
3.60
3.50
3.40
3.30
3.20
3.10
3.00
2.90
2.80
2.70
2.50
2.30
2.00
P
e
r
c
e
n
t
30
20
10
0

24. What Can Be Seen in Frequency
Distributions
• Shape
• Central Tendency
• Variability

25. Shapes of Frequency Polygons

26. Shapes of Distributions
Bell-S h
ap e d
Pro t o
ty p e :
No r
ma lDist r
ib u t i o n
SYMMETRIC
Hu m p in
Distr i b u
tio n
at Hig h S
co r
e End
Tail a
t L o
w Sc o
re End
NEGATIVELY SKEW E
D
Hu m p in
Distr i b u
tio n
at Lo wSc o
re E n d
Tail a
t High S
co r
e End
POSI T I
VELY SKEWED
Very P
ea k
ed in t h
e Cen t
er
C omp a
re d t o
No r
ma lDist r
ib u t i o n
LEPTO K
URTIC
Peak J u s t
Like
t he
No r
ma lDist r
ib u t i o n
MESOKURTIC
Fla tin t h e
Ce n
ter
C omp a
re d t o
No r
ma lDist r
ib u t i o n
PLATYKURTIC

27. Descriptive Statistics
• Central Tendency
– Mode
– Median
– Mean
• Variability
– Range
– Standard Deviation
– Variance

28. 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

29. 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

30. 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

31. Relative Position of Mode, Median,
and Mean

32. 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)

33. Range, Interquartile Range, and Standard
Deviation: SAT Scores by Area
Gro
up Range IQ Range Standard
Deviat
ion
Human
ities 200 35.00 55.88
Soci
al Science
s 95 15.00 28.59
Science
s 200 27.50 57.00

34. Range, Interquartile Range, and Standard
Deviation: SAT Scores by Gender
Group Range IQ Range Standard
Deviation
Males 200 100 60.92
Females 175 75 46.02
Total 200 70 54.02

35. 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)

36. .3413
.1359 .1359
.0214
.0214
.3413
Normal Curve
Areas Under the Curve
X
-1s
-2s +1s +2s
-3s +3s
.0013 .0013
68%
95%
99%

37. 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)

38. 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