topics: descriptive statistics

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