10/7/03 Quant

Web study: Ex Post Facto design (descriptive design)

Quasi-experimental – when they have two groups they are comparing, given and instruction or not, and they've been given a pre-test to see if the groups were equivalent to start with. In fully experimental, you would randomize selection of groups.

Web-based study no randomization (existing groups) and no pre-test. To make it a quasi-experiment they would have needed a way of measuring them at the beginning in order to assert that they were equal at the start.

Variables: number of postings, number of hits (ratio scales), number of hours studied (ratio), final grade (in this study it is the dependent variable, interval), participation in study groups – did they quantify this? Possible control variable – all psych majors. Other controls (same instructor, textbook, homework assignments, final exam) Demographic variables (Number of children in household, distance lived from campus, number of hours worked each week)

When you read a study, start making a little table of everything you're reading – write down all variables and what you think the scale of measurement is.

Ordinate: vertical axis

Abscissa: horizontal axis

We want our data to look mesokurtic because the most powerful statistical techniques can be applied to data that are normally distributed.

How do you determine how far out is an outlier? There are statistical methods and there are judg\ment calls.

From here it is mainly the class slides – I'll be taking more notes than this in future courses, but I feel pretty confident with this material so far.

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

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

Tabular Frequency Distributions

Single-Variable (“Univariate”)

Frequency Distribution: Major
Ungrouped

• 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

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

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

• 1200.00 5 12.5 12.5 100.0

• ------- ------- -------

• Total 40 100.0 100.0

• Valid cases 40 Missing cases 0

Frequency Distribution: SAT Grouped

Graphical Frequency Distributions
A Picture is Worth 1000 Words (or Numbers)

• Bar Graphs

• Histograms

• Stem and Leaf

• Frequency Polygons

• Pie Chart

Graphical Frequency Distribtions:

Single-Variable (“Univariate”)

Bar Chart: Major

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)

Stem and Leaf: SAT

Graphical Frequency Distributions

Two-Variable (“Joint” or “Bivariate”)

Relative Frequency Polygon: GPA
Comparison of Majors

Relative Frequency Polygon: GPA
Comparison of Gender

What Can Be Seen in Frequency Distributions

• Shape

• Central Tendency

• Variability

Shapes of Frequency Polygons

Descriptive Statistics

• Central Tendency

– Mode

– Median

– Mean

• Variability

– Range

– Standard Deviation

– Variance

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

Relative Position of Mode, Median, and Mean

Mean, Median, Mode:
SAT Scores by Gender

Mean, Median, Mode:
SAT Scores by Area

Choosing Appropriate Measure of Central Tendency

Definitions:
Measures 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)

• Standard Deviation:

– The average dispersion or deviation of scores around the mean (measured in original score units) (squareroot of the variance)

• Variance:

– The average variability of scores (measured in squared units of the original scores (square of the standard deviation)

Variance

• The average of each score’s 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

Variance Calculation: Teacher Service in Particular School

Standard Deviation

• Compute the variance and then take the squareroot

• Roughly the average amount that scores differ from the mean

Range, Interquartile Range, and Standard Deviation: SAT Scores by Area

Range, Interquartile Range, and Standard Deviation: SAT Scores by Gender

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)

Normal Curve

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.

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)

Standard Normal Distribution

Choosing Appropriate Measure of Variability